A unified principled framework for resampling based on pseudo-populations: asymptotic theory

TL;DR

This paper introduces a unified resampling framework for finite populations under complex sampling, combining pseudo-population construction with resampling, supported by asymptotic theory and simulation results.

Contribution

It proposes a novel two-step resampling method based on pseudo-populations, with new calibration techniques and theoretical validation for complex sampling designs.

Findings

The framework is theoretically justified using large sample theory.

New calibration methods improve pseudo-population accuracy.

Simulation studies demonstrate the effectiveness of the approach.

Abstract

In this paper, a class of resampling techniques for finite populations under complex sampling design is introduced. The basic idea on which it rests is a two-step procedure consisting in : (i) constructing a pseudo-population on the basis of sample data; (ii) drawing a sample from the predicted population according to an appropriate resampling design. From a logical point of view, this approach is essentially based on the plug-in principle by Efron, at the "sampling design level". Theoretical justifications based on large sample theory are provided. New approaches to construct pseudo-populations based on various forms of calibrations are proposed. Finally, a simulation study is performed.

Tables

Table 1. Table 1: Simulated scenarios, population parameters and estimators

Scenarios
$N=200,400$		$n=\left(0.2N\right)=40,80$
correlation between $𝒴$ and $𝒳$	$\simeq 0.8$
Parameters		Hájek Estimators
${\overline{Y}}_N={\sum }_{i=1}^N{y}_i/N$		${\hat{\overline{Y}}}_H={\sum }_{i=1}^N{D}_i{\pi }_i^{-1}{y}_i/{\sum }_{i=1}^N{D}_i{\pi }_i^{-1}$
$Q_N\left(p\right)=inf\{y:F_N\left(y\right)\ge p\}$		${\hat{Q}}_H\left(p\right)=inf\{y:{\hat{F}}_H\left(y\right)\ge p\}y\in ℝ$
with $p=0.5,0.75$

Table 2. Table 2: 𝒰 N ∗ ∗ subscript superscript 𝒰 superscript 𝑁 {\mathcal{U}}^{*}_{N^{*}} as a predictor of 𝒰 N ​ ( N = 200 ​ 𝟒𝟎𝟎 ) subscript 𝒰 𝑁 𝑁 200 400 {\mathcal{U}}_{N}\ \ (N=200\quad{\bf{400}})

	$RB[{\overline{X}}^{\ast };{\overline{X}}_N]$	$RB[N^{\ast };N]$	$Su{p}_{MC}\left|F_{N^{\ast }}^{\ast }\left(y\right)-F_N\left(y\right)\right|$
PARETO sampling design
$HT$	0,03  0,04	-0,44  0,38	0,87  0,51
$MUL$	5,46  3,39	0  0	0,93  0,54
$CPP$	5,46  3,38	0  0	0,88  0,52
$DCal$	0,02  -0,02	0,003  -0,01	0,55  0,46
$HD$	0  0	0  0	0,47  0,37
CPS sampling design
$HT$	-0,02  0,02	-1,05  -1,40	0,50  0,52
$MUL$	5,06  3,39	0  0	0,53  0,55
$CPP$	5,04  3,88	0  0	0,51  0,52
$DCal$	0,06  -0,04	0,04  -0,04	0,46  0,47
$HD$	0  0	0  0	0,48  0,33

Table 3. Table 3: 95 % percent 95 95\% Resampling CI - p ​ e ​ r ​ c ​ e ​ n ​ t ​ i ​ l ​ e 𝑝 𝑒 𝑟 𝑐 𝑒 𝑛 𝑡 𝑖 𝑙 𝑒 percentile method    ( N = 200 ​ 𝟒𝟎𝟎 ) 𝑁 200 400 (N=200\quad{\bf{400}})

PARETO	${\hat{\overline{Y}}}_H$		${\hat{Q}}_{N}0.5$		${\hat{Q}}_H\left(0.75\right)$
	$Cov$	$AL$	$Cov$	$AL$	$Cov$	$AL$
$HT$	0,89   0,90	0,23   0,17	0,88   0,91	0,33   0,22	0,91   0,93	0,37   0,28
$MUL$	0,87   0,89	0,23   0,02	0,73   0,79	0,33   0,03	0,82   0,79	0,38   0,03
$CPP$	0,89   0,90	0,23   0,17	0,89   0,92	0,33   0,22	0,92   0,94	0,38   0,29
$DCal$	0,95   0,95	0,24   0,18	0,95   0,97	0,33   0,23	0,95   0,95	0,39   0,30
$HD$	0,97   0,98	0,27   0,20	0,95   0,95	0,36   0,26	0,99   0,99	0,43   0,33
CPS
$HT$	0,90   0,91	0,24   0,17	0,91   0,92	0,33   0,22	0,90   0,93	0,38   0,29
$MUL$	0,89   0,92	0,24   0,17	0,73   0,81	0,34   0,22	0,82   0,79	0,39   0,29
$CPP$	0,90   0,90	0,24   0,17	0,91   0,92	0,34   0,22	0,91   0,94	0,38   0,29
$DCal$	0,96   0,96	0,25   0,18	0,98   0,97	0,34   0,23	0,95   0,94	0,40   0,30
$HD$	0,98   0,98	0,27   0,20	0,96   0,95	0,37   0,26	0,99   0,99	0,44   0,33

Table 4. Table 4: 95 % percent 95 95\% Standard Normal CI with resampling variance estimate ( N = 200 ​ 𝟒𝟎𝟎 ) 𝑁 200 400 \ \ (N=200\quad{\bf{400}})

PARETO	${\hat{\overline{Y}}}_H$		${\hat{Q}}_{N}0.5$		${\hat{Q}}_H\left(0.75\right)$
	$Cov$	$AL$	$Cov$	$AL$	$Cov$	$AL$
$HT$	0,90   0,91	0,24   0,17	0,90   0,91	0,36   0,24	0,93   0,91	0,40   0,30
$MUL$	0,90   0,91	0,24   0,17	0,89   0,92	0,36   0,24	0,92   0,91	0,41   0,30
$CPP$	0,91   0,92	0,24   0,17	0,89   0,92	0,36   0,24	0,93   0,92	0,40   0,30
$DCal$	0,84   0,86	0,25   0,18	0,85   0,89	0,38   0,25	0,88   0,90	0,43   0,33
$HD$	0,91   0,93	0,27   0,20	0,92   0,94	0,40   0,27	0,95   0,96	0,44   0,34
$Dir$	0,89   0,90	0,22   0,16	0,86   0,87	0,32   0,21	0,92   0,90	0,38   0,28
CPS
$HT$	0,91   0,91	0,24   0,17	0,89   0,90	0,37   0,24	0,92   0,90	0,40   0,30
$MUL$	0,90   0,92	0,24   0,17	0,89   0,92	0,38   0,24	0,90   0,90	0,41   0,31
$CPP$	0,91   0,92	0,24   0,17	0,90   0,91	0,37   0,24	0,92   0,91	0,40   0,31
$DCal$	0,85   0,87	0,25   0,19	0,87   0,87	0,39   0,25	0,89   0,89	0,44   0,33
$HD$	0,94   0,95	0,27   0,20	0,90   0,93	0,40   0,27	0,97   0,95	0,45   0,34
$Dir$	0,90   0,90	0,23   0,16	0,85   0,88	0,33   0,21	0,92   0,88	0,38   0,28

Table 5. Table 5: Bootstrap-unbiasedness ( N = 200 ​ 𝟒𝟎𝟎 ) 𝑁 200 400 \ \ (N=200\quad{\bf{400}})

	PARETO			CPS
	$RB\left[{\hat{\overline{Y}}}_{HT}^{\ast }-{\hat{\overline{Y}}}_{HT}\right]$	$RB\left[{\hat{\overline{Y}}}_H^{\ast }-{\hat{\overline{Y}}}_H\right]$		$RB\left[{\hat{\overline{Y}}}_{HT}^{\ast }-{\hat{\overline{Y}}}_{HT}\right]$	$RB\left[{\hat{\overline{Y}}}_H^{\ast }-{\hat{\overline{Y}}}_H\right]$
$HT$	0,06   -0,16	0,87   0,72		-0,16   -0,13	0,97   0,63
$MUL$	5,57   3,11	0,92   0,65		4,96   3,74	1,07   0,65
$CPP$	5,46   3,15	0,84   0,70		4,84   3,73	1,01   0,64
$DCal$	1,66   1,12	-0,33   0,20		1,38   1,36	-0,34   -0,11
$HD$	3,17   2,07	0,35   0,59		2,55   2,27	0,34   0,28
$Dir$	0,01   -0,01	0,70   0,41		-0,02   0,01	0,68   0,41

Equations

$$\displaystyle Pr_{Po}(\mbox{\boldmath$D$}_{N})=\prod_{i=1}^{N}p_{i}^{D_{i}}(1-p_{i})^{1-D_{i}}.$$

$$\displaystyle H(P)=E_{P}\left[\log Pr_{P}(\mbox{\boldmath$D$}_{N})\right]=\sum_{D_{1},\,\dots,\,D_{N}}Pr_{P}(\mbox{\boldmath$D$}_{N})\,\log\left(Pr_{P}(\mbox{\boldmath$D$}_{N})\right).$$

$$\displaystyle d_{H}(P,\,P_{R})=\sum_{D_{1},\,\dots,\,D_{N}}\left(\sqrt{Pr_{P}(\mbox{\boldmath$D$}_{N})}-\sqrt{Pr_{P_{R}}(\mbox{\boldmath$D$}_{N})}\right)^{2}.$$

$$\underset{N,n\to \infty }{\lim }\mathbb{E}[{\pi }_i(1-{\pi }_i)]=d>0.$$

$$\underset{N\to \infty }{\lim }\frac{n}{N}=f,0<f<1.$$

$$d_H(P,P_R)\to 0\mathrm{a}sN\to \infty ,a.s.-\mathbb{P}.$$

$$d=f\left(1-\frac{\mathbb{E}[X_1^2]}{\mathbb{E}[X_1{]}^2}\right)+f(1-f)\frac{\mathbb{E}[X_1^2]}{\mathbb{E}[X_1{]}^2}>0.$$

$$x_i=\frac{p_l}{w_l}\mathrm{if}\mathrm{unit}i\mathrm{is}\mathrm{within}\mathrm{stratum}l.$$

$${\pi }_i=\frac{n{p}_l}{n}=\frac{n_l}{n}\mathrm{if}\mathrm{unit}i\mathrm{is}\mathrm{within}\mathrm{stratum}l.$$

$${\Delta }_{KL}(P\Vert P_R)=H(P_R)-H(P)$$

$$F_N(y)=\frac{1}{N}\sum _{i=1}^N{I}_{(y_i\le y)},y\in \mathbb{R}$$

$${\theta }_N=\theta (F_N).$$

$${\hat{F}}_H(y)$$

$${\hat{\theta }}_H=\theta \left({\hat{F}}_H\right).$$

$$H(y,x)=\mathbb{P}(Y_i\le y,X_i\le x)$$

$$F(y)=\mathbb{P}(Y_i\le y)=H(y,+\infty ),G(x)=\mathbb{P}(X_i\le x)=H(+\infty ,x),$$

$$K_{\alpha }(y)=\mathbb{E}\left[X_1^{\alpha }|Y_1\le y\right],y\in \mathbb{R},\alpha =0,\pm 1,\pm 2$$

$$W_N^H(y)=\sqrt{n}({\hat{F}}_H(y)-F_N(y));y\in \mathbb{R}.$$

$$C^H(y,t)$$

$$-f\left\{\frac{\mathbb{E}[X_1]}{f}\left(K_{-1}(y)+K_{-1}(t)-\mathbb{E}\left[X_1^{-1}\right]-1\right)\right\}F(y)F(t),$$

$$f(A-1)(F(y\wedge t)-F(y)F(t))$$

$$A=\frac{\mathbb{E}[X_1]}{f}\mathbb{E}[X_1^{-1}]$$

$$\frac{1}{N}\sum _{i=1}^N\frac{1}{{\pi }_i}$$

$$(f(A-1)B(F(y));y\in \mathbb{R})$$

$$|{\hat{F}}_H(y)-F_N(y)|\stackrel{p}{\to }0\mathrm{as}N\to \infty$$

$${\Vert \frac{\theta (F+t{h}_t)-\theta (F)}{t}-{\theta }_F^{\prime }(h)\Vert }_E\to 0ast\downarrow 0,forevery{h}_t\to h.$$

$$T_N^H=\sqrt{n}\left(\theta ({\hat{F}}_H)-\theta (F_N)\right),N\ge 1.$$

$${\sigma }_{\theta }^2=\mathbb{E}\left[{\theta }_F^{\prime }(W^H{)}^2\right].$$

$$\left\{(N_i^{\ast }{D}_i,y_i,x_i);i=1,\dots ,N\right\}$$

$$F_{N^{\ast }}^{\ast }(y)=\frac{1}{N^{\ast }}\sum _{k=1}^{N^{\ast }}{I}_{(y_k^{\ast }\le y)}=\sum _{i=1}^N\frac{N_i^{\ast }}{N^{\ast }}{D}_i{I}_{(y_i\le y)},y\in \mathbb{R}$$

$$N^{\ast }=\sum _{i=1}^N{N}_i^{\ast }{D}_i.$$

Full text

A unified principled framework for resampling

based on pseudo-populations: asymptotic theory

Pier Luigi Conti111Pier Luigi Conti. Dipartimento di Scienze Statistiche; Sapienza Università di Roma; P.le A. Moro, 5; 00185 Roma; Italy. E-mail pierluigi.conti@uniroma1.it

Daniela Marella222Daniela Marella. Dipartimento di Scienze della Formazione; Università Roma Tre; Via D. Manin, 53; 00185 Roma; Italy. E-mail daniela.marella@uniroma3.it

Fulvia Mecatti333Fulvia Mecatti. Dipartimento di Sociologia e Ricerca Sociale; Università di Milano-Bicocca; Via Bicocca degli Arcimboldi, 8; 20126 Milano; Italy. E-mail fulvia.mecatti@unimib.it

Federico Andreis444Federico Andreis. Department of Policy Analysis and Public Management; Università Bocconi; Via Roentgen,1; 20136 Milano; Italy. E-mail federico.andreis@unibocconi.it

Abstract

In this paper, a class of resampling techniques for finite populations under complex sampling design is introduced. The basic idea on which they rest is a two-step procedure consisting in: $(i)$ constructing a “pseudo-population” on the basis of sample data; $(ii)$ drawing a sample from the predicted population according to an appropriate resampling design. From a logical point of view, this approach is essentially based on the plug-in principle by Efron, at the “sampling design level”. Theoretical justifications based on large sample theory are provided. New approaches to construct pseudo populations based on various forms of calibrations are proposed. Finally, a simulation study is performed.

Keywords. Bootstrap, calibration, complex sampling designs, confidence intervals, finite populations, variance estimation.

1 Introduction

The use of resampling methods in survey sampling has a long history, and several different techniques have been proposed in the literature. The common starting point consists in observing that the classical bootstrap method, as proposed by [17], does not work in survey sampling, because of the dependence among units due to the sampling design itself.

Adaptations taking into account the non i.i.d. nature of the data are required when the sample is collected through a general sampling design, possibly assigning different probability to every population unit to be included in the sample. The literature on resampling from finite populations is mainly devoted to estimate variances of estimators; crf. [25]. The main approaches are essentially two: ad hoc approaches and plug in approaches (cfr. [31], [11] and references therein).

The basic idea of ad hoc approaches consists in resampling from the original sample through a special design, that accounts for the dependence among units. This approach is pursued in [27], [32], where the re-sampled data produced by the “usual” $i.i.d.$ bootstrap are properly rescaled, as well as in [34], [2], [10], [14], where a “rescaled bootstrap process” based on asymptotic results is proposed. Among the ad hoc approaches we also quote the recent paper by [1], where an ingenious mixed resampling design is proposed to account for the dependence among observations.

Plug-in approaches are based on the idea of “expanding” the sample to a “pseudo-population” that plays the role of a “surrogate” (actually an estimate) of the original one. Then, bootstrap samples are drawn from such a pseudo-population according to some appropriate resampling design: cfr. [20], [9], [7], [23], [11], as well as [25].

Virtually all resampling techniques proposed for finite populations rest on the same justification: in case of linear statistics, the variance of the resampled statistic should match (or should be very close to) the “usual” variance estimator, possibly with approximated forms of the second order inclusion probabilities; cfr. [1]. This is far from the arguments commonly used to justify the classical bootstrap and its variants, that are based on asymptotic considerations involving the whole sampling distribution of a statistic (cfr., for instance, [5] and [24]): the asymptotic distribution of a bootstrapped statistic should coincide with that of the “original” statistic. This argument is actually used in [14].

In the present paper a class of resampling techniques for finite populations is proposed. It is based on a two-phase procedure. In the first phase, a pseudo-population, that can be viewed as a prediction of the population, is constructed. In the second phase, a (re)sample is drawn from the pseudo-population. In a sense, this approach parallels the plug-in principle by [18]. The pseudo-population is plugged in the sampling process, and acts as a surrogate of the actual finite population. In other terms, the predicted population mimics the real population, and the (re)sampling process from the predicted population mimics the (original) sampling process from the real population. From a formal point of view, the main justification of the whole procedure is based on large sample arguments. In this sense, the approach pursued in the present paper offers a principled framework for resampling from finite populations that parallels the arguments used for classical Efron’s bootstrap of i.i.d. data. For this reason, some preliminary developments of large sample theory for finite populations are needed. In particular, we consider here high entropy sampling designs, similar to those studied in [13], [14], but with an important addition: the possible relationships between the variable of interest and the design variables are explicitly taken into account. This dramatically changes the asymptotic results in [13]. As a matter of fact, the resampling method defined in [14], based on rescaling Efron’s bootstrap, does not work when there is dependence between the variable of interest and the design variables.

The paper is organized as follows. Section 2 contains the basic assumptions on which the paper rests. Sections 3, 4 are devoted to asymptotic results for a wide class of estimators of appropriate population parameters. Section 5 describes the proposed predictive resampling and the basic theoretical results. In Section 6 different strategies to construct pseudo-populations are introduced. In Section 7, such methods are compared via a Monte Carlo simulation study. Conclusions are provided in Section 8. Technical lemmas and proofs are gathered in Appendix.

2 Assumptions and preliminaries

Let ${\mathcal{U}}_{N}$ be a finite population of size $N$. A sample $s$ is a subset of ${\mathcal{U}}_{N}$. For each unit $i\in{\mathcal{U}}_{N}$, let $D_{i}$ be a Bernoulli random variable (r.v.), such that $i$ is (is not) in the sample $s$ whenever $D_{i}=1$ ($D_{i}=0$), so that \mbox{\boldmaths}=\{i\in{\mathcal{U}}_{N}:\;D_{i}=1\}. Denote further by \mbox{\boldmathD}_{N} the $N$-dimensional r.v. of components $(D_{1},\dots,D_{N})$. A (unordered, without replacement) sampling design $P$ is the probability distribution of the random vector \mbox{\boldmathD}_{N}. The expectations $\pi_{i}=E_{P}[D_{i}]$ and $\pi_{ij}=E_{P}[D_{i}\,D_{j}]$ are the first and second order inclusion probabilities, respectively. The suffix $P$ denotes the sampling design used to select the sample $s$. The sample size is $n_{s}=D_{1}+\cdots+D_{N}$. In the sequel we will only consider fixed size sampling designs, such that $n_{s}\equiv n$.

The first order inclusion probabilities are frequently chosen to be proportional to an auxiliary variable ${\mathcal{X}}$. In symbols: $\pi_{i}\propto x_{i}$, where $x_{i}$ is the value of ${\mathcal{X}}$ for unit $i$ ($i=1,\,\dots,\,N$). The rationale of this choice is simple: if the values of the variable of interest are positively correlated with (or, even better, approximately proportional to) the values of the auxiliary variable, then the Horvitz-Thompson estimator of the population mean will be highly efficient.

For each unit $i$, let $p_{i}$ be a positive number, with $p_{1}+\cdots+p_{N}=n$. The Poisson sampling design ($Po$, for short) with parameters $p_{1}$, $\dots$, $p_{N}$ is characterized by the independence of the r.v.s $D_{i}$s, with $Pr_{Po}(D_{i}=1)=p_{i}$. In symbols

[TABLE]

The rejective sampling, or normalized conditional Poisson sampling (cfr. [21], [36]) is obtained from the Poisson sampling by conditioning w.r.t. $n_{s}=n$. Using the suffix $R$ to denote the rejective sampling design, $E_{R}[D_{i}\,|n_{s}=n]$ is not generally equal to $p_{i}$, although they are asymptotically equivalent, as $N$ and $n$ increase ([21]). In [12] an algorithm is proposed to compute $p_{i}$s in terms of $\pi_{i}$s for the conditional Poisson sampling.

The rejective sampling design is characterized by a fundamental property: it possesses maximum entropy among all sampling designs of fixed size and fixed first order inclusion probabilities (as shown in [22]), where the entropy of a sampling design $P$ is

[TABLE]

The Hellinger distance between a sampling design $P$ and the rejective design is defined as

[TABLE]

From now on, the character of interest is denoted by $\mathcal{Y}$, and its value for unit $i$ by $y_{i}$. $\mathcal{T}_{1},\,\dots,\,\mathcal{T}_{L}$ are the design variables, and $t_{i1},\,\dots,\,t_{iL}$ are their values for unit $i$. The design variables may include strata indicators, as well as variables measuring cluster and unit characteristics (cfr. [29]). They are used to construct the sampling design, and to compute the sampling weights, i.e. the reciprocals of the first order inclusion probabilities.

The basic assumptions on which the present paper relies are listed below.

• A1.

$({\mathcal{U}}_{N};\;N\geq 1)$ is a sequence of finite populations of increasing size $N$.

• A2.

For each $N$, $(y_{i},\,t_{i1},\,\dots,\,t_{iL})$, $i=1,\,\dots,\,N$ are realizations of a superpopulation model $\{(Y_{i},\,T_{i1},\,\dots,\,T_{iL}),\;i=1,\,\dots,\,N\}$ composed by i.i.d. $(L+1)$-dimensional r.v.s. The symbol $\mathbb{P}$ denotes the (superpopulation) probability distribution of r.v.s $(Y_{i},\,T_{i1},\,\dots,\,T_{iL})$s, and $\mathbb{E}$, $\mathbb{V}$ are the corresponding mean and variance, respectively.

• A3.

For each population ${\mathcal{U}}_{N}$, sample units are selected according to a sample design with positive first order inclusion probabilities $\pi_{1}$, $\dots$, $\pi_{N}$, and fixed sample size $n=\pi_{1}+\cdots+\pi_{N}$. The first order inclusion probabilities are taken proportional to $x_{i}=h(t_{i1},\,\dots,\,t_{iL})$, $h(\cdot)$ being an arbitrary positive function. To avoid complications in the notation, we will assume that $\pi_{i}=nx_{i}/\sum_{i=1}^{N}x_{i}$ for each unit $i$.

Although the sample size $n$, the inclusion probabilities $\pi_{i}$s, and the r.v.s $D_{i}$s, as well, depend on $N$, in order to use a simple notation the symbols $n$, $\pi_{i}$, $D_{i}$ are used, instead of the more complete $n_{N}$, $\pi_{i,N}$, $D_{i,N}$. It is also assumed that

[TABLE]

• A4.

The sample size $n$ increases as the population size $N$ does, with

[TABLE]

• A5.

For each population $({\mathcal{U}}_{N};\;N\geq 1)$, let $P_{R}$ be the rejective sampling design with inclusion probabilities $\pi_{1}$, $\dots$, $\pi_{N}$, and let $P$ be the actual sampling design (with the same inclusion probabilities). Then

[TABLE]

• A6.

$\mathbb{E}[X_{1}^{2}]<\infty$, so that the quantity in $(\ref{eq:quant_d})$ is equal to:

[TABLE]

Assumptions A2, A3 allow one to take into account the possible dependence between the design variables and the study variable. Of course, this is a key motivation for using non-simple, probability-proportional-to-size designs (dubbed $\pi$ps sampling designs), where the dependence between $X_{i}$s and $Y_{i}$s is important for the efficiency of the estimation of the population mean (and other population parameters, as well). Notice that assumptions A2, A3 do not limit the kind of dependence between $X_{i}$s and $Y_{i}$s, that can be completely general.

An obvious example of sampling designs satisfying A3 are $\pi$ps sampling designs, where the first order inclusion probability of unit $i$ is proportional to the value of a size measure. Another elementary example is the stratified design. Assume that the population is subdivided into $L$ strata, composed by $N_{1}$, $\dots$, $N_{L}$ units, respectively ($N_{1}+\cdots+N_{L}=N$). Let further $w_{l}=N_{l}/N$, and let $p_{1}$, $\dots$, $p_{L}$ be arbitrary positive numbers such that $p_{1}+\cdots+p_{L}=1$. The stratified design drawing (by simple random sampling) $n_{l}=np_{l}$ units from stratum $l$ ($=1,\,\dots,\,L)$ can be considered as a special $\pi$ps sampling design where the first order inclusion probability for unit $i$ is taken proportional to an auxiliary variable (acting as a size measure) $x_{i}$ defined as

[TABLE]

In fact, from $(\ref{eq:example_strata})$ it easily follows that

[TABLE]

In particular, if $p_{l}=w_{l}$, then the sampling design reduces to stratified proportional sampling.

As discussed in [13], assumption A5 implies that the Kullback-Leibler divergence of the actual sampling design $P$ w.r.t. the rejective design

[TABLE]

tends to zero as both $n$, $N$ increase. Hence, the sampling designs satisfying assumption A5 are essentially “high entropy” designs. The importance of the high entropy property of sampling designs is discussed in [8], [19] and references therein. Examples of sampling designs satisfying A5, as shown in [3] and [4], are simple random sampling, successive sampling, Rao-Sampford design, Chao design, stratified design, two-stage design.

The population distribution function (p.d.f., for short) is:

[TABLE]

where the indicator function $I_{(y_{i}\leq y)}$ is equal to $1$ if $y_{i}\leq y$, and is equal to [math] otherwise.

A finite population parameter is a functional (not necessarily real-valued) of the p.d.f.:

[TABLE]

The simplest (and widely used, as well) approach to estimate a finite population parameter of the form $(\ref{eq:popul_param})$ consists in estimating first the p.d.f. (6), and then in replacing $F_{N}$ in $(\ref{eq:popul_param})$ by such an estimate. As an estimator of the p.d.f. $(\ref{eq:popul_df})$ we consider here the Hájek estimator:

[TABLE]

which is a proper distribution function. It can be considered as the “finite population version” of the empirical distribution function, that plays a fundamental role in nonparametric statistics. The finite population parameter $(\ref{eq:popul_param})$ is then estimated by

[TABLE]

In a sense, $(\ref{eq:hajek_param})$ is the “finite population version” of statistical functionals.

The main task of Sections 3, 4 is to study the asymptotic properties of $(\ref{eq:dfhajek})$, $(\ref{eq:hajek_param})$, respectively. In the sequel, the joint superpopulation d.f. of $(Y_{i},\,X_{i})$ will be denoted by

[TABLE]

and the marginal superpopulation d.f.s of $Y_{i}$ and $X_{i}$ by

[TABLE]

respectively. Furthermore, the notation

[TABLE]

will be used. Note that $K_{\alpha}(+\infty)={\mathbb{E}}[X_{1}^{\alpha}]$.

3 Estimating population distribution function

The goal of the present section is to derive the limiting distribution of the Hájek estimator $(\ref{eq:dfhajek})$, as the sample size and the population size increase. To this purpose, consider the stochastic process $W^{H}_{N}=(W^{H}_{N}(y);\;y\in\mathbb{R})$, where

[TABLE]

It can be viewed as the finite population sampling version of the well-known empirical process. The main result of the present section is Proposition 1, that establishes the weak convergence of $W^{H}_{N}$ to a Gaussian limiting process. Proposition 1 is in spirit similar to the main result in [13], but with fundamental differences that will be stressed in the sequel.

Before stating Proposition 1, we stress that in our asymptotic approach the actual population $y_{i}$s and $x_{i}$s values are considered as fixed. The only source of variability is the sampling design, namely \mbox{\boldmathD}_{N}. If we let the population size $N$ go to infinity, we must also consider corresponding sequences \mbox{\boldmathy}_{\infty}=(y_{1},\,y_{2},\dots), \mbox{\boldmathx}_{\infty}=(x_{1},\,x_{2},\dots) of $y_{i}$s and $x_{i}$s values. The actual \mbox{\boldmathy}_{N}=(y_{1},\,\dots,\,y_{N}), \mbox{\boldmathx}_{N}=(x_{1},\,\dots,\,x_{N}) are the segments of the first $N$ $y_{i}$s, $x_{i}$s in the sequences \mbox{\boldmathy}_{\infty}, \mbox{\boldmathx}_{\infty}, respectively. As $N$ increases, \mbox{\boldmathy}_{N} tends to \mbox{\boldmathy}_{\infty} and \mbox{\boldmathx}_{N} tends to \mbox{\boldmathx}_{\infty}. By A2, \mbox{\boldmathy}_{\infty}, \mbox{\boldmathx}_{\infty} live in a probability space $((\mathbb{R}^{2})^{\infty},\,\mathcal{B}(\mathbb{R}^{2})^{\infty},\,\mathbb{P}^{\infty})$, where $\mathcal{B}(\mathbb{R}^{2})^{\infty}$ is the product Borel $\sigma$-field over $(\mathbb{R}^{2})^{\infty}$, and $\mathbb{P}^{\infty}$ is the product measure on $(\mathbb{R}^{\infty},\,\mathcal{B}(\mathbb{R})^{\infty})$ generated by $\mathbb{P}$. The probability statements we consider are of the form Pr_{P}(\cdot|\mbox{\boldmathy}_{N},\,\mbox{\boldmathx}_{N}), with $N$ going to infinity. Conditioning w.r.t. \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N} means that $y_{i}$s and $x_{i}$s are considered as fixed (although produced by a superpopulation model). The suffix $P$ means that the probability refers to the sampling design. The results we will obtain hold for “almost all” sequences \mbox{\boldmathy}_{\infty}, \mbox{\boldmathx}_{\infty} that the superpopulation model in A2 can produce, i.e. for a set of sequences having $\mathbb{P}^{\infty}$-probability 1. With a slight lack of precision, but more simply and intuitively, in the sequel we will use the expression “for almost all $y_{i}$s, $x_{i}$s values”.

Proposition 1**.**

If the sampling design $P$ satisfies assumptions A1-A6, with $\mathbb{P}$-probability $1$, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N} the sequence $(W_{N}^{H};\;N\geq 1)$, converges weakly, in $D[-\infty,\>+\infty]$ equipped with the Skorokhod topology, to a Gaussian process $W^{H}=(W^{H}(y);\;y\in\mathbb{R})$ with zero mean function and covariance kernel

[TABLE]

with $d$ given by $(\ref{eq:val_d})$.

When $X_{i}$ and $Y_{i}$ are independent, the covariance kernel $(\ref{eq:cov_ker_hajek})$ reduces to

[TABLE]

where

[TABLE]

is, with $\mathbb{P}$-probability 1, the limit of

[TABLE]

as $N$ goes to infinity. Taking into account that $u\wedge v-uv$ is the covariance kernel of a Brownian bridge $B=(B(t);\;0\leq t\leq 1)$ (i.e. a Wiener process tied down at 1), we have thus proved the following corollary of Proposition 1.

Corollary 1**.**

If the sampling design $P$ satisfies assumptions A1-A6, and if $X_{i}$ and $Y_{i}$ are independent, with $\mathbb{P}$-probability $1$, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N} the sequence $(W_{N}^{H};\;N\geq 1)$, converges weakly, in $D[-\infty,\>+\infty]$ equipped with the Skorokhod topology, to a Gaussian process that can be represented in the form

[TABLE]

as $N$ goes to infinity, where $B$ is a Brownian bridge and $A$ is given by $(\ref{eq:defin_A})$.

Corollary 1 essentially coincides with Proposition 2 in [13]. Proposition 1 is new. Due to the choice of the inclusion probabilities in A3, i.e. $\pi_{i}\propto x_{i}$, and due to the possible dependence between $X_{i}$ and $Y_{i}$ (that usually comes true in practice), the limiting Gaussian process is not proportional to a Brownian bridge. Proposition 1 shows how the dependence between variable of interest and design variables affects the covariance kernel of the Gaussian limiting law of $W_{N}^{H}$. If compared to Proposition 2 in [13], its main consequence is that, whenever there is some kind of dependence between the design variables (or, equivalently, the sampling weights) and the variable of interest, the empirical process $(\ref{eq:hajek_process})$ does not converge weakly to a Brownian bridge, but to a Gaussian process with a covariance kernel having a complicate form, depending on the relationships between the character of interest and the design variables. The form of such a relationship is usually unknown.

From the proof of Proposition 1 it is clear that the assumption of independence and identical distribution of r.v.s $(Y_{i},\,X_{i})$ is far from being necessary. It can be replaced by forms of dependence that admit the strong law of large numbers.

Before ending the present section we note, in passim, that Proposition 1 implies that, with $\mathbb{P}$-probability $1$, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}:

[TABLE]

where the symbol $\stackrel{{\scriptstyle p}}{{\rightarrow}}$ denotes the convergence in probability w.r.t. the sampling design (or better, w.r.t. the sequence of sampling designs in A3). Using the same arguments as the proof of the Glivenko-Cantelli theorem, it is not difficult to prove the following further result.

Proposition 2**.**

If the sampling design $P$ satisfies assumptions A1-A6, with $\mathbb{P}$-probability $1$, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, $\sup_{y}|\widehat{F}_{H}(y)-F_{N}(y)|$ converges to [math] in probability w.r.t. the sampling design.

Remark. Propositions 1, 2, also hold when the inclusion probabilities $\pi_{i}$s depend on $y_{i}$s, i.e. when the sampling design is informative. This is true, in particular, when, for units in the sample, $\pi_{i}$s only depend on $y_{i}$s of sample units, i.e. for adaptive designs. Even if this would be a point of separate interest, we do not pursue in this direction.

4 Estimating finite population parameters

The goal of the present section is to study the large sample distribution of estimators of the finite population parameters that are functions of p.d.f. $F_{N}(\cdot)$. In particular, we concentrate on estimators of the form $(\ref{eq:hajek_param})$. In a sense, the results of the present section can be viewed as a finite population version of the theory of statistical functionals, that mainly refers to the case of i.i.d. observations (cfr. [37], Ch. 20).

The appropriate tool to study asymptotic properties of statistical functionals is the notion of Hadamard-differentiability. Let $\theta(\cdot):\>l^{\infty}[-\infty,\,+\infty]\rightarrow E$ be a map having as domain the normed space $l^{\infty}[-\infty,\,+\infty]$ (endowed with the sup-norm), and taking values on an appropriate normed space $E$ with norm $\|\cdot\|_{E}$. The map $\theta(\cdot)$ is Hadamard-differentiable at $F$ if there exists a continuous linear mapping $\theta^{\prime}_{F}:\>l^{\infty}[-\infty,\,+\infty]\rightarrow E$ such that

[TABLE]

The quantity $\theta^{\prime}_{F}(\cdot)$ is the Hadamard derivative of $\theta$ at $F$. Let us consider the (sequence of) stochastic process

[TABLE]

In view of Theorem 20.8 in [37] and Proposition 1, the following result holds.

Proposition 3**.**

Suppose that $\theta(\cdot)$ is (continuously) Hadamard-differentiable at $F$, with Hadamard derivative $\theta^{\prime}_{F}(\cdot)$. Under assumptions A1-A6, with $\mathbb{P}$-probability $1$, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, the sequence $(T^{H}_{N};\;N\geq 1)$ converges weakly to $\theta^{\prime}_{F}(W^{H})$, as $N$ increases.

Proposition 3 essentially provides, under mild conditions, an asymptotic approximation for the sampling distribution of $T^{H}_{N}$. In particular, if $\theta$ is real-valued, since $\theta^{\prime}_{F}(\cdot)$ is linear and $W^{H}$ is a Gaussian process, the law of $\theta^{\prime}_{F}(W^{H})$ is normal with mean zero and variance

[TABLE]

5 A class of resampling procedure and its basic properties

The goal of this section is to introduce a unified class of resampling procedures working under the sampling designs considered in Section 2, and that provides an approximation of the sampling distribution of estimators of the form $(\ref{eq:hajek_param})$.

The main theoretical justification we will provide is based on asymptotic arguments: the probability distribution of the estimator $\theta(\widehat{F}_{H})$ and its approximation based on resampling both converge to the same limit. This is actually the main argument in favour of the classical (nonparametric) bootstrap for i.i.d. data: cfr., for instance, [5]. The results of the present section can be viewed as an attempt to reconciliate the arguments used in sampling finite populations with those used in classical nonparametric statistics.

The first attempt to define a resampling technique for finite populations based on asymptotic distribution theory is in [10] for simple random sampling, and in [14] for general designs. In the latter paper, a technique based on rescaling classical bootstrap is proposed, and its properties are studied. However, two points have to be stressed. The first one is that the technique developed in [14] is specifically designed to estimate quantiles. The second one is that it is fully justified from an asymptotic point of view only when there are no relationships between $\pi_{i}$s (and hence $x_{i}$s) and $y_{i}$s. In other words, the rescaled bootstrap proposed in [14] does not work when the dependence between $y_{i}$s and $x_{i}$s cannot be neglected.

In view of the above remarks, in this section a new resampling algorithm for finite population is introduced, that works

• $(i)$

for general estimators $\theta(\widehat{F}_{H})$ of general population parameters $\theta(F_{N})$;

• $(ii)$

when $x_{i}$s (i.e. the design variables) and $y_{i}$s (i.e. the variable of interest) are related by some kind of dependence. No special assumption is made on the relationship between $x_{i}$s and $y_{i}$s, apart from its (possible) existence.

As already said in the introduction, the class of resampling techniques rests on a two-phase procedure. In the first phase, on the basis of the sampling data a pseudo-population, i.e. a prediction of the actual population is constructed. In the second phase, a sample of size $n$ (the same as the “original” one) is drawn from such a pseudo-population, according to a $\pi ps$ sample design $P^{*}$ (the resampling design) with inclusion probabilities appropriately chosen and satisfying the entropy condition A5. The resampling design $P^{*}$ is not assumed to coincide with the sampling design $P$ used to collect data from the actual population.

From now on, the term sampling design $P$ denotes the sampling procedure drawing $n$ units from the “original” population $\mathcal{U}_{N}$. The resampling design $P^{*}$ is the sampling procedure drawing $n$ units from the predicted (pseudo-)population $\mathcal{U}^{*}_{N^{*}}$. Details of the two phases on which the resampling procedure relies are in Sections 5.1, 5.2.

5.1 Phase 1: Pseudo-population

A pseudo-population, i.e. a design-based population predictor of the population, is

[TABLE]

where $N^{*}_{i}$s are integer-valued r.v.s, with (joint) probability distribution $P_{pred}$. In practice, $(\ref{eq:predict_des})$ means that $N^{*}_{i}D_{i}$ population units are predicted to have $y$-value equal to $y_{i}$ and $x$-value equal to $x_{i}$, for each sample unit $i$. In the sequel, the familiar bootstrap symbols $y^{*}_{k}$, $x^{*}_{k}$ will be used to denote the $y$-value and $x$-value of unit $k$ of the predicted population, respectively. Of course $N^{*}_{i}$ units of the predicted population satisfy the relationships $y^{*}_{k}=y_{i}$, $x^{*}_{k}=x_{i}$, i\in\mbox{\boldmaths}. The d.f. of the pseudo-population is equal to

[TABLE]

where

[TABLE]

is the total number of units of the pseudo-population.

As far as the terms $N^{*}_{i}$ are concerned, we will make the following assumptions on expectations, variances, covariances w.r.t. $P_{pred}$.

• P1.

E[N^{*}_{i}\,|\mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N}]=\pi_{i}^{-1}D_{i}K_{1N}(\mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N})

• P2.

V(N^{*}_{i}\,|\mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N})\leq\pi_{i}^{-1}D_{i}K_{2N}(\mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N})

• P3.

\left|C(N^{*}_{i},\,N^{*}_{h}\,|\mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N})\right|\leq\frac{c}{N}\pi_{i}^{-1}\pi_{h}^{-1}D_{i}D_{h}K_{3N}(\mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N}) $i\neq h$

$c$ being an appropriate constant, with

[TABLE]

and K_{jN}(\mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N}), $j=2,\,3$ are bounded in probability, conditionally on \mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N}, as $N$ increases. The symbol $\rightarrow$ in $(\ref{eq:cond_conv})$ denotes convergence in probability w.r.t. \mbox{\boldmathD}_{N} and for almost all $y_{i}$s, $x_{i}$s.

5.2 Phase 2: Resampling design from the pseudo-population

In phase 2 a sample \mbox{\boldmaths}^{*} of size $n$ (the same as the original sample) is selected from the predicted population according to a resampling design $P^{*}$ with first order inclusion probabilities $\pi_{k}^{*}=nx^{*}_{k}/\sum_{h=1}^{N^{*}}x^{*}_{h}$ and satisfying the entropy assumption A5. The Hájek estimator of the d.f. of the predicted population $F^{*}_{N^{*}}(y)$ is equal to

[TABLE]

where $D^{*}_{k}=1$ if the unit $k$ of the predicted population is drawn, and $D^{*}_{k}=0$ otherwise.

Proposition 4**.**

Under assumptions A1-A6, P1-P3, for almost all $y_{i}$s, $x_{i}$s values, and in probability w.r.t. \mbox{\boldmathD}_{N},

[TABLE]

as $N$ goes to infinity.

The statement “in probability w.r.t. \mbox{\boldmathD}_{N}” means that the set of \mbox{\boldmathD}_{N}s values, for which Lemma 7 in Appendix holds, possesses a probability tending to 1 as $N$ increases.

Define now the “resampled version” of the processes $W_{N}^{H}$ $(\ref{eq:hajek_process})$ and $T^{H}_{N}$ $(\ref{eq:tndef})$, namely

[TABLE]

Proposition 5 contains the main result of the present section and it can be proved essentially with the same technique as Propositions 1, 3, respectively.

Proposition 5**.**

Suppose that the sampling design $P$ and the resampling design $P^{*}$ both satisfy assumptions A1-A6, and that conditions P1-P3 are fulfilled. Conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}, $(D_{1}N^{*}_{1},\,\dots,\,D_{N}N^{*}_{N})$, the following statements hold.

• R$1$.

The sequence $(W_{N}^{H*};\;N\geq 1)$ converges weakly, in $D[-\infty,\>+\infty]$ equipped with the Skorokhod topology, to a Gaussian process $W^{H}$ with zero mean function and covariance kernel $(\ref{eq:cov_ker_hajek})$.

• R$2$.

If $\theta(\cdot)$ is continuously Hadamard differentiable at $F$, then $(T_{N}^{H*};\;N\geq 1)$ converges weakly to $\theta^{\prime}_{F}(W^{H})$, as $N$ increases.

In both $R1$, $R2$ weak convergence takes place for a set of $y_{i}$s, $x_{i}$s having $\mathbb{P}$-probability 1, and for a set of \mbox{\boldmathD}_{N}s and $(N^{*}_{1},\,\dots,\,N^{*}_{N})$ of probability tending to $1$.

Proposition 5 shows that the resampled process $W_{N}^{H*}$ ($T_{N}^{H*}$) possesses the same limiting law as the “original” process $W_{N}^{H}$ ($T_{N}^{H}$) in Proposition 1 (3). In other words, the proposed resampling procedure asymptotically recovers the probability law of $W_{N}^{H}(\cdot)$ and $T_{N}^{H}(\cdot)$, respectively .

Proposition 5 does not require that the resampling design coincides with the original sampling design, as in [23]. The essential required conditions are two: $(i)$ the predicted population is constructed as in phase 1; $(ii)$ the first order inclusion probabilities of the resampling design are proportional to the corresponding $x_{i}$ values, exactly as the original sampling design. Intuitively speaking, this happens because both the original sampling design and the resampling design possess high entropy, and in this case their limiting behaviour essentially depends on the first order inclusion probabilities.

In Proposition 5 the probability distribution of $W_{N}^{H*}$ ( $T_{N}^{H*}$) is considered conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}, $(N^{*}_{1},\,\dots,\,N^{*}_{N})$. In other terms, the predicted population is considered as fixed (as well as \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}), and the only source of variability is the resampling design from the predicted population. Using Lemmas $1.1$, $1.2$ in [16], it is possible to see that the same result also holds when one considers the distribution of $W_{N}^{H*}$ ( $T_{N}^{H*}$) conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}. In this case only \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N} are considered as fixed, and there are two sources of variability: $(i)$ the variability of the process generating the predicted population and $(ii)$ the variability of the resampling design from the predicted population. More precisely, the following proposition (that can be proved with the same reasoning as in [16], based on Lemmas $1.1$, $1.2$ in the above paper) holds true.

Proposition 6**.**

Suppose the sampling design $P$ and the resampling design $P^{*}$ satisfy assumptions A1-A6. Conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}, the following statements hold.

• U$1$.

The sequence $(W_{N}^{H*};\;N\geq 1)$ converges weakly, in $D[-\infty,\>+\infty]$ equipped with the Skorokhod topology, to a Gaussian process $W^{H}$ with zero mean function and covariance kernel $(\ref{eq:cov_ker_hajek})$.

• U$2$.

If $\theta(\cdot)$ is continuously Hadamard differentiable at $F$, then $(T_{N}^{H*};\;N\geq 1)$ converges weakly to $\theta^{\prime}_{F}(W^{H})$, as $N$ increases.

In both $U1$, $U2$ weak convergence takes place for a set of $y_{i}$s, $x_{i}$s having $\mathbb{P}$-probability 1, and for a set of \mbox{\boldmathD}_{N}s of probability tending to $1$.

The main consequence of Propositions 5, 6 is that in generating the bootstrap samples two different approaches can be followed:

• 1.1

Conditional Approach: construct a predicted population and generate $M$ bootstrap samples \mbox{\boldmaths}^{*} from it;

• 1.2

Unconditional approach: construct $M$ predicted populations and generate one bootstrap sample \mbox{\boldmaths}^{*} from each of them.

Clearly, the unconditional approach is more computationally intensive and time consuming than the conditional one.

The basic steps of the resampling procedure are described below. To simplify the notation, in the sequel we will assume that $\theta(\cdot)$ is real-valued, i.e. we will consider the case of scalar population parameters.

• Step 1

Generate $M$ independent bootstrap samples \mbox{\boldmaths}^{*} of size $n$ on the basis of the two-phase procedure described above.

• Step 2

For each bootstrap sample, compute the corresponding Hájek estimator $(\ref{eq:hajek_resampled})$. They will be denoted by $\widehat{F}^{*}_{H,m}(y)$, $m=1,\,\dots,\,M$.

• Step 3

Compute the corresponding estimates of $\theta(\cdot)$:

[TABLE]

• Step 4

Compute the $M$ quantities

[TABLE]

• Step 5

Compute the variance of $(\ref{eq:def_z})$:

[TABLE]

where

[TABLE]

Denote further by

[TABLE]

the empirical distribution function of $Z^{*}_{n,m}$s, and by

[TABLE]

the corresponding $p$th quantile.

The empirical d.f. $(\ref{eq:resampling_edf})$ is essentially an approximation of the (resampling) distribution of $T^{H*}_{N}$ as defined by equation (28). In Proposition 7 it is shown that it converges to the same limit as the d.f. of $T^{H*}_{N}$, and that a similar result holds for the quantiles $(\ref{eq:resampling_quantiles})$.

Proposition 7**.**

Suppose that assumptions A1-A6 are satisfied, let $\sigma^{2}_{\theta}$ be defined as in $(\ref{eq:asympt_var_theta})$, let $\Phi_{0,\sigma^{2}_{\theta}}$ be a normal distribution function with expectation [math] and variance $\sigma^{2}_{\theta}$, and let $\Phi_{0,\sigma^{2}_{\theta}}^{-1}(p)$ be the $p$-quantile of $\Phi_{0,\sigma^{2}_{\theta}}$ (i.e. the unique solution of $\Phi_{0,\sigma^{2}_{\theta}}(z)=p$), $0<p<1$.

For almost all $y_{i}$s, $x_{i}$s values, and in probability w.r.t. \mbox{\boldmathD}_{N}, $(N^{*}_{1},\,\dots,\,N^{*}_{N})$, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}, $(N^{*}_{1},\,\dots,\,N^{*}_{N})$, the following results hold:

[TABLE]

as $M$, $N$ go to infinity.

In addition, if the sequence $\left(Z^{*}_{m}-\overline{Z}^{*}_{M}\right)^{2}$ is dominated by a r.v. $U$ with finite expectation, i.e. $\left(Z^{*}_{m}-\overline{Z}^{*}_{M}\right)^{2}\leq U$ for each $n$, $N$ and $M$, then in probability w.r.t. \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}, $(N^{*}_{1},\,\dots,\,N^{*}_{N})$, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}, $(N^{*}_{1},\,\dots,\,N^{*}_{N})$ it yields

[TABLE]

where convergence in $(\ref{eq:risult_3})$ is in probability w.r.t. resampling replications.

The main consequences of Proposition 7 are two. First of all, the estimator $\widehat{S}^{2*}$ is a consistent estimator of the variance of $\theta(\widehat{F}_{H})$. In the second place, the confidence intervals

[TABLE]

both possess asymptotic confidence level $1-\alpha$ as $N$ and $M$ increase.

6 Some strategies for constructing pseudo-populations

In view of Proposition 5, all techniques to construct a pseudo-population are asymptotically equivalent, provided that they satisfy conditions $P1$-$P3$ of Section 5.1. In this sense in the present paper a unified approach for resampling based on pseudo-populations is given. However in practical applications, i.e. for finite $n$, a crucial aspect that would potentially affect the performance of resampling, is how the pseudo-population is constructed. The idea behind pseudo-populations is simple: as the sample and population sizes increase, the pseudo-population tends to be “similar” to the real finite population. Hence, it would be intuitive to use a pseudo-population that is as similar as possible to the actual population. In a sense, the pseudo-population should be somehow calibrated w.r.t. the population. Such an intuition can be put into practice in several ways. In the present section some proposals based on different calibration approaches are illustrated, which lead to different pseudo-populations.

6.1 Horvitz-Thompson pseudo-population

Following the popular Horvitz-Thompson (HT) approach to $\pi$ps sampling and estimation, each unit i\in\mbox{\boldmaths}, should be “predicted” in ${\ \mathcal{U}^{*}_{N^{*}}}$ a number of times equal to its design weight $\pi_{i}^{-1}$, assumed all integers. For the general non-integer case the following strategy has been proposed since the 90s ([23]). Let $r_{i}=\pi_{i}^{-1}-\lfloor\pi_{i}^{-1}\rfloor$, and consider independent Bernoulli r.v.s $\epsilon_{i}$s with Pr(\epsilon_{i}=1\,|\mbox{\boldmathD}_{N},\,\mbox{\boldmathY}_{N},\,\mbox{\boldmathX}_{N})=r_{i}. A HT pseudo-population is constructed by replicating every sampled unit i\in\mbox{\boldmaths} $N^{*HT}_{i}=\lfloor\pi_{i}^{-1}\rfloor+\epsilon_{i}$ times, with corresponding values $y_{i}$, $x_{i}$. The size of a HT pseudo-population $N^{*HT}=\sum_{i=1}^{N}N^{*HT}_{i}D_{i}$ is in general not equal to $N$. However, the ratio $N^{*HT}/N$ tends in probability to $1$ as $N$, $n$ increase. Furthermore, and more strongly, it is easy to see that HT pseudo-population satisfies the regularity conditions $P1$-$P3$, and hence the resampling distribution of $\sqrt{n}(\theta(\widehat{F}^{*}_{H})-\theta(F^{*}_{N^{*}}))$ tends to the same limit as the sampling distribution of $\sqrt{n}(\theta(\widehat{F}_{H})-\theta(F_{N}))$.

6.2 Multinomial pseudo-population

For $k=1,\,\dots,\,N$, perform independent trials consisting in choosing a unit from the original sample, where each unit $i$ is selected with probability

[TABLE]

If at trial $k$ unit $i$ is selected, unit $k$ of the pseudo-population will take values $y^{*}_{k}=y_{i}$ and $x^{*}_{k}=x_{i}$. If $N^{*MUL}_{i}$, i\in\mbox{\boldmaths}, is the number of replications of unit $i$ in the pseudo-population, then (conditionally on \mbox{\boldmathD}_{N}, \mbox{\boldmathY}_{N}, \mbox{\boldmathX}_{N}) the r.v. (N^{*}_{i};\;i\in\mbox{\boldmaths}) possesses a multinomial distribution, with

[TABLE]

This approach goes essentially back to [30] and guarantees by construction a pseudo-population calibrated w.r.t. the population size. Again, the multinomial pseudo-population satisfies conditions $P1$-$P3$, so that the resampling distribution of $\sqrt{n}(\theta(\widehat{F}^{*}_{H})-\theta(F^{*}_{N^{*}}))$ tends to the same limit as the sampling distribution of $\sqrt{n}(\theta(\widehat{F}_{H})-\theta(F_{N}))$.

6.3 Conditional Poisson pseudo-population

The HT scheme in Section 6.1 is essentially based on drawing a Poisson sample from $s$, where unit i\in\mbox{\boldmaths} does have inclusion probability $r_{i}$. A simple idea to calibrate such scheme in order to produce a pseudo-population of exactly $N$ units, consists in defining the quantities

[TABLE]

and in drawing from $s$ a sample \mbox{\boldmaths}_{0} of

[TABLE]

units, according to a conditional Poisson sampling design with first order inclusion probabilities $\tau_{i}$s.

For each unit i\in\mbox{\boldmaths}, let $\epsilon_{i}$ be equal to $1$ iff $i$ is in \mbox{\boldmaths}_{0}, and $\epsilon_{i}=0$ otherwise. Each unit $i$ of the original sample is replicated in the pseudo-population exactly

[TABLE]

times.

Proposition 8**.**

The conditional Poisson pseudo-population satisfies conditions $P1$-$P3$.

As a consequence of Proposition 8, the resampling distribution of $\sqrt{n}(\theta(\widehat{F}^{*}_{H})-\theta(F^{*}_{N^{*}}))$ tends to the same limit as the sampling distribution of $\sqrt{n}(\theta(\widehat{F}_{H})-\theta(F_{N}))$.

6.4 Double-Calibrated pseudo-population

The conditional Poisson pseudo-population illustrated in Subsection 6.3 is calibrated w.r.t. the population size $N$, but not w.r.t. the mean of the auxiliary variable ${\mathcal{X}}$. A natural idea would consist in modifying $N^{*CPP}_{i}$ defined by equation (41) in order to satisfy a further constraint: the mean of ${\mathcal{X}}$ in the pseudo-population is equal to the mean of ${\mathcal{X}}$ in the actual population.

Take $N^{*CPP}_{i}$, i\in\mbox{\boldmaths} as an “initial” solution for replicates of sample units in the pseudo-population, and let further

[TABLE]

The basic idea is to choose pseudo-population replicates that satisfy both constraints on population size and mean of ${\mathcal{X}}$, and that are as close as possible to the initial $N^{*CPP}_{i}$s. More formally, the pseudo-population replicates are taken equal to $N^{*DCal}_{i}$s, the solution of the following quadratic problem:

[TABLE]

Proposition 9**.**

The calibrated pseudo-population with replicates $N^{*DCal}_{i}$ that solves the optimization problem $(\ref{eq:probl_min})$ possesses the following property:

[TABLE]

Intuitively speaking, Proposition 9 tells us that as $N$, $n$ increase, the solution of the optimization problem ($\ref{eq:probl_min})$ tends to coincide with $N^{*CPP}_{i}$. Hence, for “very large” population and sample size, $N^{*CPP}_{i}$s can be taken as a good approximation of the actual solution of the optimization problem $(\ref{eq:probl_min})$. Of course, this is only an asymptotic result, and for the use of “not too large” $n$, $N$, the use of $N^{*DCal}_{i}$ instead of $N^{*CPP}_{i}$ could produce considerably different results in the resampling procedure.

The values $N^{*DCal}_{i}$s obtained by solving $(\ref{eq:probl_min})$ are not necessarily integer-valued. In order to obtain integer values, it is enough to apply to $N^{*DCal}_{i}$s a randomization device similar to that of CPP pseudo-population described in Section 6.3.

6.5 Hot-deck pseudo-population

The basic idea of the calibrated pseudo-population introduced in Subsection 6.4 consists in constructing a pseudo-population that is “similar” for some characteristics of the auxiliary variable ${\mathcal{X}}$ w.r.t. the original population. This idea is pursued by taking only the sample $x_{i}$s values. Although in many practical cases this is true (for instance, when the data user is different from the sample design planner, and only sample weights $\pi_{i}$s are available), in some cases $x_{i}$s are available for all population units. When all $x_{i}$s are available, the notion of (finite) population predictor can be extended by considering predictors of the form $\{(x^{*}_{i},\,y^{*}_{i}),\;k=1,\,\dots,\,N\}$, where $x^{*}_{i}=x_{i}$ for every unit $i=1,\,\dots,\,N$ and $y^{*}_{i}=\widehat{y}_{i}=$ imputed value for $y_{i}$, according to hot-deck imputation. In detail, the hot-deck pseudo-population is composed by $N$ units, i.e. $\mathcal{U}^{*}_{N^{*}}={\mathcal{U}}^{*}_{N}$. A pair of values $(x^{*}_{i},\,y^{*}_{i})$ corresponds to each unit $i\in{\mathcal{U}}^{*}_{N}$, with

[TABLE]

In other terms, for each unit $i\in{\mathcal{U}}^{*}_{N}$ a donor unit $j(i)$ is chosen, such that

[TABLE]

The values $x^{*}_{i}$, $y^{*}_{i}$ for unit $i$ are then taken equal to those of its donor, leading to a pseudo-population which is calibrated by construction w.r.t. both population size $N$ and the entire distribution of the auxiliary variable ${\mathcal{X}}$.

Proposition 10**.**

If the pseudo-population is constructed via hot-deck imputation of $y$s values, then, as $n$, $N$ increase, the resampling distribution of $\sqrt{n}(\theta(\widehat{F}^{*}_{H})-\theta(F^{*}_{N^{*}}))$ tends to the same limit as the sampling distribution of $\sqrt{n}(\theta(\widehat{F}_{H})-\theta(F_{N}))$.

7 Simulation Study

Main goal of the simulation is to empirically evaluate the effects that different choices for constructing the pseudo-population ${\mathcal{U}}^{*}_{N^{*}}$ (where resampling is actually performed) may have upon the accuracy of the resulting inference in practical applications. The simulation has been designed by focusing three key points:

• $a)$

exploration of small to moderate $n$ and $N$ in order to highlight differences due to finite sizes as well as to evaluate approximations based on asymptotic arguments as given in the first part of the present paper;

• $b)$

analysis of specific features of the pseudo-population ${\mathcal{U}}^{*}_{N^{*}}$ due to different construction choices;

• $c)$

investigation of the statistical properties of the final estimates provided by resampling into such different pseudo-populations.

The simulated scenarios, parameters and estimators are summarized in Table 1. For the sake of comparisons, beside the five strategies proposed in Section 6, the direct bootstrap ([1]) is also simulated, since it is a recent competitor based on a non-predictive resampling approach. The variates ${\mathcal{Y}}$, ${\mathcal{X}}$ have been simulated under the same model as in [1].

Samples have been simulated under two different fixed size $\pi$ps designs of increasing entropy: Pareto sampling and (normalized) conditional Poisson sampling (CPS for short), this latter already mentioned in Section 2 as a maximum entropy design. Notice that Pareto design is high entropy, although not yet proved asymptotically maximum entropy; however it is heuristically recognized to be very close to the asymptotically maximum entropy Rao-Sampford design ([6]). Moreover, unlike the CPS design, the Pareto sampling is very simple to implement, and can be used in simple acceptance-rejection rules to produce CPS samples with a significant reduction of computational burden. Simulation has been implemented partly in Mathematica code and partly in the R environment. $1000$ Monte Carlo ($MC$) runs, simulating the sample space, have been combined with $M=1000$ resampling runs from each generated sample. The $MC$ error deriving from these choices has been controlled via the empirical bias of the (unbiased) Horvitz-Thompson estimator $\hat{\bar{Y}}_{HT}$, and it has been kept under 1% (relative to the true population mean $\bar{Y}$).

Simulation results are gathered in Tables 2-5 where the simulated methods to construct the pseudo-population are indicated by the following acronyms: $HT$ illustrated in subsection 6.1; $MUL$ for the Multinomial pseudo-population in 6.2; $CPP$ for the conditional Poisson pseudo-population in 6.3; $DCal$ for the double-calibrated pseudo-population in 6.4; $HD$ for the hot-deck pseudo-population in 6.5; and $Dir$ for the (non-predictive, non pseudo-population based) direct bootstrap ([1]).

*****************TABLE 2 ABOUT HERE ************************

Results in Table 2 offer indications about the ability of the pseudo-population ${\mathcal{U}}^{*}_{N^{*}}$ as a *predictor *of the actual population ${\mathcal{U}}_{N}$, according to key point $b)$ above. Except for the direct bootstrap involving no pseudo-population, it has been checked in two respects: $i)$ the pseudo-population size $N^{*}$ and mean of the auxiliary variable $\bar{X}^{*}$ as predictors of (known) population $N$ and $\bar{X}_{N}$ respectively, as measured via empirical (relative) bias $RB\left[N^{*};N\right]=100\times\left[E_{MC}(N^{*})-N\right]/N$ (where $E_{MC}$ indicates the average over all the Monte Carlo runs and $RB\left[\bar{X}^{*};\bar{X}_{N}\right]$ follows accordingly); and $ii)$ how able the pseudo-population is to reproduce the actual p.d.f. as measured by the maximal MC value of the Kolmogorov statistic $\max_{MC}\sup_{y}\left|F^{*}_{N^{*}}(y)-F_{N}(y)\right|,\;y\in\mathbb{R}.$

A clear connection appears between the conservation of both $N$ and $\bar{X}$ and the ability of reproducing the entire population d.f.: indeed $HD$ and $DCal$ pseudo-populations, which account for such a conservation to the largest extent, emerge as the best performers, uniformly in all the simulated scenarios. Also, this reflects on the ability of the resampling algorithm based on such pseudo-populations, to reproduce the estimator distribution.

*****************TABLEs 3 & 4 ABOUT HERE ************************

According to key point $c)$ above, both kinds of confidence intervals (CI) ilustrated in section 5 have been simulated. Table 3 concerns CI (36) which basically correspond to bootstrap percentile method, and Table 4 refers to CI (37). Performances at (nominal) confidence level $95$% has been investigated via empirical coverage (Cov), with respect to the true population parameter, and average length (AL). Notice that although the percentile method is the crudest available for producing CI via resampling, we rate it appropriate for the goals of the present simulation because it allows the evaluation of the ability of the resampling algorithm to produce $p$-values, and ultimately to reproduce the estimator sampling distribution particularly in its tails. In Table 3 all the methods investigated for constructing ${\mathcal{U}}^{*}_{N^{*}}$ provide acceptable levels of empirical coverage based on the $0.025$ and $0.975$ percentiles of the resampling distribution. Moreover they all tend to improve for increasing sizes $N$ and $n$, as expected according to asymptotic results in Section 5. However $HD$ and $DCal$, which provide the best predictor of ${\mathcal{U}}_{N}$, also give the best coverage probabilities, uniformly in all scenarios simulated for both linear and non linear estimators. Notably, $HD$ shows the largest average lengths in addition to the largest empirical coverages, which suggests a tendency to supply conservative CI.

A similar behaviour can be observed in Table 4, although the resampling plays here a minor role, limited to the (point) bootstrap estimate (30) for the estimator variance then coupled with standard normal distribution percentiles. Notice that this is also the method for interval estimation suggested for the non-predictive direct bootstrap. However, $Dir$ exhibits lower empirical coverage probabilities than the predictive pseudo-population based methods, seemingly due to systematic smaller lengths. The notable exception of $DCal$ may be explained by its weaker ability to produce accurate point bootstrap estimates than the other predictive methods simulated. Still $HD$ emerges as the best performer for uniformly giving the larger empirical coverages in all scenarios simulated and for maintaining its conservative peculiarity.

Finally and as a desirable feature of a resampling algorithm applying to complex sampling from finite populations, it has been investigated a popular property of the classic i.i.d. Efron’s bootstrap: the ability of the resampled distribution of an estimator of the population mean to match the (original) sample mean as its empirical first moment. Such property, dubbed bootstrap unbiasedness, has been measured by the (percentage) relative bias $RB\left[\hat{\theta}^{*}_{m};\hat{\theta}\right]=100\times E_{MC}\left\{\left[E^{*}(\hat{\theta}^{*}_{m})-\hat{\theta}\right]/\hat{\theta}\right\}$ where $E^{*}$ indicates the empirical average over the $M$ resampling runs and by taking $\hat{\theta}=\bar{Y}$ and $\hat{\theta}^{*}_{m},m=1\cdots M$ as its resampled distribution. Table 5 reports simulation results with respect to both Horvitz-Thompson and Hájek estimation of population mean. Empirical evidence confirms $HT$ and $Dir$ as algorithms purposively constructed under the conventional Horvitz-Thompson paradigm for linear parameters. All the other proposed strategies for producing the pseudo-population perform well under the Hájek approach to estimation. Again $HD$ appears as ensuring bootstrap-unbiasedness to the largest extent.

*****************TABLE 5 ABOUT HERE ************************

As a final remark concerning the actual implementation of specific algorithms, note that all the simulated populations have been checked to ensure $\pi_{i}<1,\quad i=1,\,\dots,\,N$. However, for $MUL$ it may still occur $\pi^{*}_{k}\geq 1$ for one or more (sampled) unit $k$ included in the pseudo-population. This empirically appears to be often the case as the number of MC runs increases. As a consequence, an ad hoc routine has to be implemented on top of the resampling algorithm, aiming at including such units in each bootstrap sample and sequentially recomputing the resampling inclusion probability until they are all strictly smaller than 1, and by simultaneously reducing the (re)sample size accordingly (see, for instance, [36] for details).

8 Conclusions

In this paper a new class of resampling methods applying to non-i.i.d. finite population sampling is proposed under a principled predictive approach. The proposed resampling unifies any method based on pseudo-populations, i.e. according to the plug-in principle upon which the original Efron’s bootstrap is based. A large sample theory is derived for the predictive resampling, in the Hájek finite population asymptotic setup, and in the same spirit of the classical asymptotics for i.i.d. bootstrap by [5]. It is also proved that all techniques to produce the pseudo-population are asymptotically equivalent, under mild regularity conditions.

In addition, five strategies have been illustrated for constructing the pseudo-population in practice. Two of them go back to results already appeared in the literature and the remaining three are new proposals with improved performance, as shown in a simulation study. Empirical evidence confirms that how to construct the pseudo-population is a crucial choice for small to moderate population and sample sizes, under general sampling designs such as $\pi$ps designs. As a general recommendation such choice should be guided by enforcing the ability of the pseudo-population to be a good predictor of the actual population. The simulation study indicates the pseudo-population based on hot-deck imputation ($HD$) as the soundest method, provided that auxiliary $x_{i}$s values are available for all population units. When $x_{i}$s are known only for sample units, as it might be the case in applications, good results are offered by a pseudo-population calibrated w.r.t. both the population size and the mean (total) of the auxiliary variable ($DCal$), when combined with percentile confidence intervals.

Appendix

Lemma 1**.**

Let $d_{N}=\sum_{i=1}^{N}\pi_{i}(1-\pi_{i})$. Then, as $N$ increases,

[TABLE]

Proof of Lemma 1.

Taking into account that $\pi_{i}=f_{N}x_{i}/\overline{X}_{N}$, with $f_{N}=n/N$ and $\overline{X}_{N}=\sum_{i=1}^{N}x_{i}/N$, it is enough to observe that

[TABLE]

and to apply the strong law of large numbers. ∎

Lemma 2**.**

Consider the quantity $K_{\alpha}(y)$ in $(\ref{eq:def_K})$. The following results hold:

[TABLE]

Proof of Lemma 2.

Using the same notation as in Lemma 1, from $\pi_{i}=f_{N}x_{i}/\overline{X}_{N}$ it follows that

[TABLE]

by the strong law of large numbers. Proof of $(\ref{eq:limit_a})$ is completed by observing that

[TABLE]

Proof of $(\ref{eq:limit_b})$ is similar. ∎

Lemma 3**.**

Define the quantities

[TABLE]

Then, as $N$ goes to infinity, a.s.-$\mathbb{P}$, the following results hold

[TABLE]

Proof of Lemma 3.

Relationship $(\ref{eq:limite_zin})$ is an immediate consequence of Lemmas 1, 2. As far as $(\ref{eq:limite_s2n})$ is concerned, observe first that

[TABLE]

where

[TABLE]

Next, it is not difficult to see that

[TABLE]

with

[TABLE]

as $N$ tends to infinity, a.s.-$\mathbb{P}$. From $(\ref{eq:limite_b1na})$-$(\ref{eq:limite_b1nc})$, it follows that

[TABLE]

In the same way, using Lemmas 1, 2, it is possible to see that

[TABLE]

From $(\ref{eq:limite_b1})$-$(\ref{eq:limite_b3})$, result $(\ref{eq:limite_s2n})$ easily follows. ∎

Lemma 4**.**

For every positive $\epsilon$, with $\mathbb{P}$-probability $1$ there exists an integer $N_{\epsilon}$ such that

[TABLE]

Proof of Lemma 4.

Let $((y_{i},\,x_{i});\;i\geq 1)$ be a sequence satisfying Lemmas 1-3 (the set of such sequences does have $\mathbb{P}$-probability $1$), and let $\epsilon>0$ “small”. Then, there exists $N_{\epsilon}\geq 1$ (depending on the whole sequence $((y_{i},\,x_{i});\;i\geq 1)$) such that

[TABLE]

From $(\ref{eq:ineq4-1})$ the inequalities

[TABLE]

hold, with $0<\gamma<1/2$ and for every $N\geq N_{\epsilon}$. Inequalities $(\ref{eq:ineq4-2})$ and $(\ref{eq:ineq4-3})$ prove $(\ref{eq:ineq_lemma4})$. ∎

Lemma 5**.**

Let $\epsilon$ be a positive number, and let

[TABLE]

Then

[TABLE]

Proof of Lemma 5.

Immediate consequence of Lemmas 3, 4. ∎

Lemma 6**.**

Conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, as $N$ increases the r.v.

[TABLE]

tends in probability to $1$, a.s.-$\mathbb{P}$.

Proof of Lemma 6.

The expectation of $(\ref{eq:rv-weaklaw})$ w.r.t. the sampling design, and conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N} is equal to $1$. As far as the variance is concerned, we have first

[TABLE]

As an easy consequence of Lemma 2, the r.v. $N^{-1}\sum\pi_{i}^{-1}$ converges a.s.$\mathbb{P}$. Furthermore, the assumption of maximal asymptotic entropy of the sampling design implies (cfr. [22], Th. 7.4) that

[TABLE]

$C$ being an absolute constant. This proves the lemma. ∎

Proof of Proposition 1.

The proof is based on Lemmas 1-5, and it rests on the same ideas as the proof of Proposition 1 in [13]. For this reason, it is only sketched. First of all, it is not difficult to see that the limiting law of the process $(W_{N}^{H}(\cdot);\;N\geq 1)$ coincides with the limiting law of $(\sqrt{f}\widetilde{W}_{N}^{H}(\cdot);\;N\geq 1)$, where

[TABLE]

Hence, using Lemmas 1-5 and [21] (see also Section 2 of [38] and Theorem 1 in [3]), it is seen that the asymptotic distribution of $\widetilde{W}_{N}^{H}(y)$ is normal with mean zero and variance $f^{-1}C^{H}(y,\,y)$. The same kind of result holds for all finite-dimensional distributions of $\widetilde{W}_{N}^{HT}(\cdot)$, as a consequence of the Cramér-Wold device.

As far as the tightness is concerned, using the same reasoning as in [13] we can confine ourselves to the conditional Poisson sampling design. We have

[TABLE]

with $\mathbb{P}$-probability 1, $C$, $Q$ being appropriate constants. Finally, using the Glivenko-Cantelli theorem and the right continuity of $F$, from $(\ref{eq:tight_A1})$ it follows that

[TABLE]

with $\mathbb{P}$-probability 1, $R$ being an appropriate constant. Inequality $(\ref{eq:tight_A2})$ proves the tightness part, and this completes the proof. ∎

Proof of Proposition 4.

To prove $(\ref{eq:equiv_N*})$, observe first that

[TABLE]

as $N$ increases, in probability w.r.t. \mbox{\boldmathD}_{N} and for almost all $y_{i}$s, $x_{i}$s. In the second place:

[TABLE]

From P1-P3 and $(\ref{eq:cond_conv})$ it is simple to see that

[TABLE]

in probability w.r.t. \mbox{\boldmathD}_{N} and for almost all $y_{i}$s, $x_{i}$s, from which $(\ref{eq:equiv_N*})$ follows. ∎

Lemma 7**.**

Under assumptions A1-A6, P1-P3, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}, as $N$ increases the statements of Lemmas 1-5 hold true for the predicted population, and for almost all $y_{i}$s, $x_{i}$s values, and in probability w.r.t. \mbox{\boldmathD}_{N} and $P_{pred}$.

Proof of Lemma 7.

Consider the quantity:

[TABLE]

First of all, using the symbols already introduced, it is not difficult to see that

[TABLE]

Furthermore, from $(\ref{eq:mean_miltinom})$-$(\ref{eq:cov_miltinom})$, it follows that

[TABLE]

and

[TABLE]

Using exactly the same arguments as in Propositions 1, 2, it is now seen that, with $\mathbb{P}$-probability 1 and in probability w.r.t. the sampling design (i.e. w.r.t. \mbox{\boldmathD}_{N}),

[TABLE]

as $N$ increases. From these results it follows that

[TABLE]

as $N$ increases, again for almost all $y_{i}$s, $x_{i}$s values, and in probability w.r.t. \mbox{\boldmathD}_{N}. From $(\ref{eq:converge1})$, $(\ref{eq:converge2})$, and Lemma 7, it is not difficult to conclude that, conditionally on \mbox{\boldmathD}_{N}, \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, for almost all $y_{i}$s, $x_{i}$s values, and in probability w.r.t. \mbox{\boldmathD}_{N}, $P_{pred}$,

[TABLE]

that coincides with $(\ref{eq:val_d})$.

The same arguments can be used to show that Lemmas 2-5 still hold when the actual population is replaced by the predicted population, conditionally on \mbox{\boldmathy}_{N}, \mbox{\boldmathx}_{N}, \mbox{\boldmathD}_{N}, for a set of $y_{i}$s, $x_{i}$s having $\mathbb{P}$-probability 1 and for a set of \mbox{\boldmathD}_{N}s of (design) probability tending to $1$ as $N$, increases, and where convergence is in probability w.r.t. the (random) mechanism generating the predicted population, $P_{pred}$. This ends the proof. ∎

Proof of Proposition 7.

Let

[TABLE]

be the (resampling) d.f. of $Z^{*}_{n,m}$ $(\ref{eq:def_z})$. By Dvoretzky-Kiefer-Wolfowitz inequality (cfr. [26]), we have first

[TABLE]

Using the Borel-Cantelli first lemma, and taking into account that $R^{*}_{n}(z)$ converges uniformly to $\Phi_{0,\sigma^{2}_{\theta}}(z)$, $(\ref{eq:risult_1})$ immediately follows. Statement $(\ref{eq:risult_2})$ follows from $(\ref{eq:risult_1})$ and the strict monotonicity of $\Phi_{0,\sigma^{2}_{\theta}}(z)$. Finally, $(\ref{eq:risult_3})$ is a consequence of Theorem 2.5.5. in [33] (pp. 90-91). ∎

Proof of Proposition 8.

First of all, it is immediate to see that

[TABLE]

Furthermore, from

[TABLE]

conditions $P1$ and $P2$ are fulfilled. Finally, if $\tau_{ih}$ are the second order inclusion probabilities of the conditional Poisson sampling, from Th. 7.4 in [22] it follows that

[TABLE]

$c$ being an absolute constant, and hence also condition $P3$ is satisfied. ∎

Proof of Proposition 9.

Take a positive real $\epsilon$, and consider the relaxed optimization problem:

[TABLE]

As a consequence of already seen results, the r.v.

[TABLE]

tends in distribution to a normal variate with zero mean and positive variance, so that

[TABLE]

for each positive $\epsilon$. In other terms, up to a term $O_{p}(n^{-1/2})$ the relaxed problem $(\ref{eq:probl_minrelax})$ possesses solution $N^{*}_{i}=N^{*CPP}_{i}$. Since the optimum of the relaxed problem $(\ref{eq:probl_minrelax})$ is continuous in $\epsilon$ (cfr. [35], Sect. 9.2.2), by letting $\epsilon$ go to [math] slowly enough as $N$, $n$ increase, it is seen that up to a term $O_{p}(n^{-1/2})$ the solution of the optimization problem $(\ref{eq:probl_min})$ is $N^{*}_{i}=N^{*CPP}_{i}$. Result $(\ref{eq:conv_nstar})$ now follows by observing that

[TABLE]

∎

Proof of Proposition 10.

As a consequence of Proposition 4 in [15], it is possible to see that the joint d.f. of $X$ and $Y$ in the actual finite population

[TABLE]

and the joint d.f. of $X$ and $Y$ in the pseudo-population

[TABLE]

tend in probability to the same limiting d.f., as $n$, $N$ increase. Hence, the joint d.f. of the hot-deck pseudo-population, $H^{HD}_{N}(x,\,y)$, tends to coincide with the actual $H_{N}(x,\,y)$. This is enough, in its turn, to conclude that the resampling distribution of $\sqrt{n}(\theta(\widehat{F}^{*}_{H})-\theta(F^{*}_{N^{*}}))$ tends to the same limit as the sampling distribution of $\sqrt{n}(\theta(\widehat{F}_{H})-\theta(F_{N}))$ ∎