Extremal behaviour of a periodically controlled sequence with imputed values

TL;DR

This paper studies the extremal behavior of periodically controlled sequences with imputed missing values, establishing dependence conditions, extremal index relations, and proposing an estimator, with applications to Markovian sequences.

Contribution

It introduces a new model for sequences with periodic control and imputation, analyzes their extremal properties, and proposes a consistent estimator for the missing data parameter.

Findings

Sequences are T-periodic under stationarity.

Dependence conditions for extremal analysis are verified.

An estimator for the missing value control parameter is proposed.

Abstract

Extreme events are a major concern in statistical modeling. Ran\-dom missing data can constitute a problem when modeling such rare events. Imputation is crucial in these situations and therefore models that describe different imputation functions enhance possible applications and enlarge the few known families of models which cover these situations. In this paper we consider a family of models $\{Y_n\},$ $n\geq 1,$ which can be associated to automatic systems which have a periodic control, in the sense that it is guaranteed that at instants multiple of $T,$ $T\geq 2,$ no value is lost. Random missing values are here replaced by the biggest of the previous observations up to the one surely registered. We prove that when the underlying sequence is stationary, $\{Y_n\}$ is $T-$periodic and if it also verifies some local dependence conditions then $\{Y_n\}$ verifies one of the well known $D^{(s)}_T(u_n),$ $s\geq 1,$ dependence conditions for $T-$periodic sequences. We also obtain the extremal index of $\{Y_n\}$ and relate it to the extremal index of the underlying sequence. A consistent estimator for the parameter that "controls" the missing values is here proposed and its finite sample properties are analysed. The obtained results are illustrated with Markovian sequences of recognized interest in applications.

Tables

Table 1. Table 1: Various statistical results for the estimation of p ∈ ] 0 , 1 [ p\in]0,1[ in model ( 1.2 )

$p$	n	$\hat{\mu }\left({\hat{p}}_{n,2}\right)$	$BIAS\left({\hat{p}}_{n,2}\right)$	$\hat{\sigma }\left({\hat{p}}_{n,2}\right)$	$RMSE\left({\hat{p}}_{n,2}\right)$
0.10	$n=250$	0.0989	-0.0011	0.0267	0.0267
	$n=1000$	0.1008	0.0008	0.0137	0.0137
	$n=5000$	0.1001	0.0001	0.0059	0.0059
0.25	$n=250$	0.2492	-0.0008	0.0395	0.0395
	$n=1000$	0.2494	-0.0006	0.0198	0.0198
	$n=5000$	0.2495	-0.0005	0.0083	0.0083
0.50	$n=250$	0.5025	0.0025	0.0461	0.0461
	$n=1000$	0.4997	-0.0003	0.0222	0.0223
	$n=5000$	0.4999	0.0001	0.0096	0.0096
0.75	$n=250$	0.7510	0.0010	0.0373	0.0373
	$n=1000$	0.7506	0.0006	0.0188	0.0188
	$n=5000$	0.7498	-0.0002	0.0085	0.0085
0.90	$n=250$	0.9000	0.0000	0.0263	0.0263
	$n=1000$	0.9001	0.0001	0.0129	0.0129
	$n=5000$	0.9001	0.0001	0.0060	0.0060

Equations

$$Y_n=U_n{X}_n+(1-U_n)\underset{i=n-1}{\overset{\left[\frac{n-1}{T}\right]T}{\bigvee }}{U}_i{X}_{i}n\ge 1,T\ge 2,$$

$$Y_n=U_n{X}_n+(1-U_n)U_{n-1}{X}_{n-1},n\ge 1,$$

$$F_{Y_{kT+j}}(x)=\{\begin{array}{ll}F(x),& j\in \{0,1\},T\ge 2,\\ pF(x)+(1-p)G_j(x),& j\in \{2,\dots ,T-1\},T\ge 3,\end{array}$$

$$G_j(x)=\sum _{\varnothing \subseteq S\subseteq \{kT+1,\dots ,kT+j-1\}}{p}^{|S|}(1-p{)}^{j-1-|S|}\times$$

$$P\left(\bigcap _{s=1}^p\{Y_{i_s+T}\le x_s\}\right)$$

$$P\left(\bigcap _{j=1}^{T-1}\{Y_{sT+j}=Y_{sT}\}\right)=(1-p{)}^{T-1},s\ge 1,T\ge 2.$$

$$P\left(\bigcap _{j=1}^{T-1}\{Y_{sT+j}=Y_{sT}\}\right)$$

$${\hat{p}}_{n,T}=1-{\left(\frac{1}{\left[\frac{n+1}{T}\right]-1}\sum _{s=1}^{\left[\frac{n+1}{T}\right]-1}{1\mathrm{I}}_{A_s}\right)}^{\frac{1}{T-1}},T\ge 2,$$

$${\alpha }_{n,{\ell }_n}$$

$$P\left(M^{(\text{Y})}(A_p)\le u_n,M^{(\text{Y})}(B_q)\le u_n\right)$$

$$E\left(P\left(M^{(\text{X})}(A_p^{(\text{U})})\le u_n\mid \text{U}\right)P\left(M^{(\text{X})}(B_q^{(\text{U})})\le u_n\mid \text{U}\right)\right)+\mathcal{O}({\alpha }_{n,{\ell }_n-T})$$

$$=E\left(P\left(M^{(\text{X})}(A_p^{(\text{U})})\le u_n\mid \text{U}\right)\right)E\left(P\left(M^{(\text{X})}(B_q^{(\text{U})})\le u_n\mid \text{U}\right)\right)+\mathcal{O}({\alpha }_{n,{\ell }_n-T})$$

$$|P\left(M^{(Y)}(A_p)\le u_n,M^{(Y)}(B_q)\le u_n\right)-$$

$$\underset{n\to +\infty }{\lim }n\frac{1}{T}\sum _{i=1}^{T}P\left(Y_i>u_n\ge M_{i+1,i+s-1}^{(\mathbf{Y})},M_{i+s,\left[\frac{n}{k_{n}T}\right]T}^{(\mathbf{Y})}>u_n\right)=0,$$

$$\underset{n\to +\infty }{\lim }n\frac{1}{T}\sum _{i=1}^T\sum _{j=i+s}^{[\frac{n}{k_{n}T}]T}P(Y_i>u_n,Y_{j-1}\le u_n<Y_j)=0,$$

$$\tau =\underset{n\to +\infty }{\lim }n\frac{1}{T}\sum _{j=1}^T{\overline{F}}_j(u_n)=\underset{n\to +\infty }{\lim }n\left(1-\frac{1}{T}\sum _{j=1}^T{F}_j(u_n)\right),$$

$${\theta }_{\mathbf{Y}}=\frac{1}{\tau }\underset{n\to +\infty }{\lim }n\frac{1}{T}\sum _{i=1}^{T}P(Y_i>u_n\ge M_{i+1,i+s-1}^{(\mathbf{Y})}).$$

$$n\overline{F}(u_n)\underset{n\to +\infty }{\to }{\tau }_{\mathbf{X}}\text{and}n{\overline{G}}_j(u_n)\underset{n\to +\infty }{\to }{\tau }_j.$$

$$n\frac{1}{T}\sum _{j=1}^T{\overline{F}}_j(u_n)=$$

$$=$$

$$nP\left(\underset{j=0}{\overset{T}{\bigvee }}{X}_j>u_n\ge X_{T+1},M_{T+2,\left[\frac{n}{k_{n}T}\right]T+T}^{(\mathbf{X})}>u_n\right)\underset{n\to +\infty }{\to }0$$

$$n\frac{1}{T}\sum _{i=1}^{T}P\left(Y_i>u_n\ge M_{i+1,i+T}^{(\mathbf{Y})},M_{i+T+1,\left[\frac{n}{k_{n}T}\right]T}^{(\mathbf{Y})}>u_n\right)$$

$$\displaystyle P\left(Y_{i}>u_{n}\geq Y_{i+1},\ M^{(\bf{X})}_{i+2,\left[\frac{n}{k_{n}T}\right]T}>u_{n}\ \biggl{|}\ U_{i}=1,U_{i+1}=1\right)$$

$$\displaystyle P\left(Y_{i}>u_{n}\geq Y_{i+1},\ M^{(\bf{X})}_{i+2,\left[\frac{n}{k_{n}T}\right]T}>u_{n}\ \biggl{|}\ U_{i}=0,U_{i+1}=1\right)$$

$$\displaystyle P\left(Y_{i}>u_{n}\geq Y_{i+1},\ M^{(\bf{X})}_{i+2,\left[\frac{n}{k_{n}T}\right]T}>u_{n}\ \biggl{|}\ U_{i}=1,U_{i+1}=0\right)$$

$$\displaystyle P\left(Y_{i}>u_{n}\geq Y_{i+1},\ M^{(\bf{X})}_{i+2,\left[\frac{n}{k_{n}T}\right]T}>u_{n}\ \biggl{|}\ U_{i}=0,U_{i+1}=0\right)$$

$$P\left(Y_{T-1}>u_n\ge Y_T,M_{T+1,\left[\frac{n}{k_{n}T}\right]T}^{(\mathbf{X})}>u_n\right)$$

$$P\left(Y_T>u_n\ge Y_{T+1},M_{T+2,\left[\frac{n}{k_{n}T}\right]T}^{(\mathbf{X})}>u_n\right)$$

$$n\frac{1}{T}\sum _{i=1}^{T}P\left(Y_i>u_n\ge M_{i+1,i+T}^{(\mathbf{Y})},M_{i+T+1,\left[\frac{n}{k_{n}T}\right]T}^{(\mathbf{Y})}>u_n\right)$$

$$nP\left(M_{0,2}^{(\mathbf{X})}>u_n\ge X_3,M_{4,\left[\frac{n}{2k_n}\right]2+2}^{(\mathbf{X})}>u_n\right)$$

$$n\sum _{j=0}^T\sum _{k=T+2}^{\left[\frac{n}{k_{n}T}\right]T+T}P(X_j>u_n,X_{k-1}\le u_n<X_k)\underset{n\to +\infty }{\to }0.$$

Full text

Extremal behaviour of a periodically controlled sequence with imputed values

Helena Ferreira1, Ana Paula Martins1, Maria da Graça Temido2

1Departamento de Matemática, Centro de Matemática e Aplicações (CMA-UBI),

Universidade da Beira Interior, Portugal

2 CMUC, Departamento de Matemática, Universidade de Coimbra, Portugal

helenaf@ubi.pt, amartins@ubi.pt, mgtm@mat.uc.pt

Abstract: Extreme events are a major concern in statistical modeling. Random missing data can constitute a problem when modeling such rare events. Imputation is crucial in these situations and therefore models that describe different imputation functions enhance possible applications and enlarge the few known families of models which cover these situations. In this paper we consider a family of models $\{Y_{n}\},$ $n\geq 1,$ which can be associated to automatic systems which have a periodic control, in the sense that it is guaranteed that at instants multiple of $T,$ $T\geq 2,$ no value is lost. Random missing values are here replaced by the biggest of the previous observations up to the one surely registered. We prove that when the underlying sequence is stationary, $\{Y_{n}\}$ is $T-$periodic and if it also verifies some local dependence conditions then $\{Y_{n}\}$ verifies one of the well known $D^{(s)}_{T}(u_{n}),$ $s\geq 1,$ dependence conditions for $T-$periodic sequences. We also obtain the extremal index of $\{Y_{n}\}$ and relate it to the extremal index of the underlying sequence. A consistent estimator for the parameter that “controls” the missing values is here proposed and its finite sample properties are analysed. The obtained results are illustrated with Markovian sequences of recognized interest in applications.

Resumé: Les événements extrêmes sont une préoccupation majeure dans la modélisation statistique. Les données absentes de façon aléatoire peuvent constituer un problème lors de la modélisation de ces événements rares. Les techniques d’imputation sont cruciaux dans ces situations et, par conséquent, les modèles qui décrivent différentes fonctions d’imputation améliorent les applications possibles et élargissent les quelques familles connues de modèles qui couvrent ces situations. Dans cet article, nous considérons une famille de modèles $\{Y_{n}\},$ $n\geq 1,$ qui peuvent être associés à des systèmes automatiques qui ont un contrôle périodique, en ce sens qu’il est garanti qu’à des instants multiples de $T,$ $T\geq 2,$ aucune valeur n’est perdue. Les valeurs absentes de façon aléatoire sont ici remplacées par la plus grande des observations précédentes jusqu’à celle sûrement enregistrée. Nous prouvons que lorsque la séquence sous-jacente est stationnaire, $\{Y_{n}\}$ est $T-$ périodique et si elle vérifie encore certaines conditions de dépendance locales, alors $\{Y_{n}\}$ vérifie l’une des conditions de dépendence $D^{(s)}_{T}(u_{n}),$ $s\geq 1,$ pour des séquences $T-$ périodiques. Nous obtenons également l’indice extrémal de $\{Y_{n}\}$ que nous relions à l’indice extrémal de la séquence sous-jacente. Un estimateur convergent pour le paramètre qui “contrôle” les valeurs absentes est ici proposé et ses propriétés à distance fini sont analysées. Les résultats obtenus sont illustrés en utilisant des séquences markoviennes qui ont un intérêt reconnu dans les applications.

Key Words: Missing values, Periodic sequence, Local dependence conditions, Extremal index

1 Introduction and preliminary results

Data collection is prevalent in everyday life and is used in several domains, such as finance, climate observation, computer science, etc. The main goal of any data collection effort is to compile quality data, but issues with missing data oftenly occur when data is measured and recorded. The data unavailability may be caused by the failure of some system, such as a reading device, or simply by lack of retention due to the intrinsic properties of the data, e.g. financial or environmental data only reported at certain time instants (Hall and Hüsler (2006); Hall and Scotto (2008) and references therein).

Many analysis methods require for the use of imputation, i.e. missing values to be replaced with reasonable values up-front. An overview about univariate time series imputation can be found in Moritz et al. (2015) and an introduction to R’s package imputeTS, which is solely dedicated to univariate time series imputation is presented in Moritz and Bartz-Beielstein (2017).

Falk et al. (2011) summarize the several strategies that are usually applied when missing data values occur in time series: (i) the missing value is replaced by a predefined value $x_{0}$ which can be sometimes 999 (if one is interested in small values and no such large values occur) or -1 (if one is interested in large values and no negative values occur); (ii) the data is completely lost and the time series is sub-sampled with a smaller (and random) sample size; (iii) an automatic measurement device is used to replace the missing data by a proxy value.

The extremal properties of sequences with random missing values replaced by 0 were studied by Falk et al. (2011). The sub-sample referred in strategy (ii) above, may result from missing values that occur according to some deterministic pattern or occur randomly. The effect of deterministic missing values on the properties of strictly stationary (stationary) sequences has been studied by Ferreira and Martins (2003), Martins and Ferreira (2004), Scotto et al. (2003), among others. Random missing values have been considered by Weissman and Cohen (1995) for the case of constant failure probability and independent failures, and their results were generalized for situations where the failure pattern has a weak dependence structure by Hall and Hüsler (2006). This was pursued by Hall and Scotto (2008), when the underlying process is represented as a moving average driven by heavy-tailed innovations and the sub-sampling process is strongly mixing.

When the missing values are replaced by an automatic measurement device the resulting sample will be a mixture of two original samples. This case was considered by ulting sample will be a mixture of two original samples. This case was considered by Hall and Hüsler (2006) and later by Hall and Temido (2009) in the context of max-semistability. There they discussed the extremal properties of a model where missing values are replaced by independent replicas of the original values. Investigating the extremal properties of models that describe other imputation functions enhances possible applications in situations where it may be of interest to avoid the occurrence of missing values and an automatic replacement of a device or machine may be available. This situation motivated the model we consider in this paper and that we describe in what follows.

Let us consider a system with a periodic control, in the sense that it is guaranteed that at instants multiple of $T,$ $T\geq 2,$ no value is lost. If for some reason there are missing values, it is then natural to use the observations that are for surely registered, due to the periodic control, in the replacement of these values. A model that translates this idea, where a missing value is replaced by the biggest of the previous observations up to the one at an instant multiple of $T,$ which is surely registered, is defined by

[TABLE]

where $[a]$ denotes the integer part of $a\in{\rm I\kern-1.99997ptR},$ $\{U_{n}\}_{n\geq 0}$ is a sequence of independent variables, such that, for all $k\geq 0,$ $U_{kT}=1$ almost surely, and $U_{n}$ follows a Bernoulli distribution with parameter $p\in]0,1[,$ for all $n\neq kT.$ $\{X_{n}\}_{n\geq 0}$ denotes a positive stationary sequence, independent of $U_{n},$ $n\geq 0,$ with marginal continuous distribution function (d.f.) $F.$ Here and throughout $\bigvee_{j=s}^{t}W_{j}\equiv\bigvee_{j=t}^{s}W_{j},$ for any $s,t\in\mathbb{Z},$ $s<t,$ and random variables $W_{j}.$

Model (1.1) can be associated to automatic systems which have a manual periodic verification. As we can see, at each instant $n\neq kT,\,k\geq 0,$ we can observe $X_{n}$ or in the case that it is not observed it is replaced by the maximum of the previous observations up to the last one that was surely registered. The registration of the observations is periodically controlled, with the guarantee that at instants $kT,$ $k\geq 0,$ no observations are lost and therefore in the period $\left\{\left[\frac{n-1}{T}\right]T,\ldots,n-1\right\}$ at least the observation with index $\left[\frac{n-1}{T}\right]T$ is available. The case $T=1$, if considered, would correspond to the non occurrence of missing data.

To better understand model (1.1) let us consider the following illustrative example.

Example 1.1

Let $\{Z_{n}\}_{n\geq-1}$ be a sequence of independent and identically distributed (i.i.d.) random variables with unit Fréchet marginal d.f. $F_{Z}$. With this sequence we define the moving maxima $X_{n}=\frac{1}{2}Z_{n}\vee\frac{1}{2}Z_{n-1},\ n\geq 0,$ which is stationary and also has unit Fréchet margins.

A model with the characteristics of (1.1) is given by

[TABLE]

where $\{U_{n}\}_{n\geq 0}$ is an independent Bernoulli sequence with parameter $p$ for all $n\neq 2k$ and $U_{2k}=1,$ $k\geq 0,$ almost surely.

Here $\{Y_{n}\}$ is controlled at instants which are multiples of 2 ($T=2$), so at these instants we always have the guarantee that an observation of the moving maxima was retained. At all the other instants we can have a moving maxima value, whenever it is observed, or the maximum of the previous observations up to the last observation “controlled”, which in this case corresponds only to the previous observation because $T=2.$ In Figure 1 this becomes clear with 100 observation of (1.2), since newly imputed observations are marked differently than the rest of the series and the instants $2s,$ $s\geq 1,$ are highlighted.

We shall return to this simple example, throughout the work, to illustrate several of the results presented.**

Our main goal is to characterize the extremal behaviour of $\{Y_{n}\}_{n\geq 1}$ given in (1.1). In order to achieve this, in Section 3, lets start by noting that, for all $k\geq 0$ and for all $x\in{\rm I\kern-1.99997ptR},$ the marginal d.f.’s of $\{Y_{n}\}_{n\geq 1}$ satisfy

[TABLE]

with $kT+j\neq 0$ and

[TABLE]

where $\delta_{\bullet}(B)$ denotes the Dirac measure on $B,$ $F_{i_{1},\ldots,i_{p}}$ denotes the d.f. of the vector $(X_{i_{1}},\ldots,X_{i_{p}})$ and we conventione that $x\times\frac{1}{0}=+\infty.$

We also prove that the sequence $\{Y_{n}\}_{n\geq 1}$ defined in (1.1) is a $T$ periodic sequence. Indeed, for any choice of integers $1\leq i_{1}<i_{2}<\ldots<i_{p},$ we have

[TABLE]

since $\{U_{n}\}_{n\geq 1}$ is a $T-$periodic sequence, $\{X_{n}\}_{n\geq 0}$ is a stationary sequence and they are independent. From now on we use the notation $F_{j}(x)\mathrel{\mathop{\mathchar 58\relax}}=F_{Y_{j}}(x)$, for any $j\geq 1$, for which holds $F_{j}(x)\mathrel{\mathop{\mathchar 58\relax}}=F_{j+kT}(x),$ for all $k\geq 1$ and $T\geq 2$.

Extreme value theory known for periodic sequences can then be applied to this periodically controlled sequence with imputed values $\{Y_{n}\}_{n\geq 1},$ since it is also a $T-$periodic sequence. Alpuim (1988) showed that under Leadbetter’s global mixing condition $D(u_{n}),$ the only possible limit laws for the normalized maxima of a $T-$periodic sequence are the three extreme value distributions and generalized, as well, the definition of extremal index for such sequences. Under local mixing conditions $D_{T}^{(s)}(u_{n}),\ s=1,2,$ Ferreira (1994) studied the extremal behaviour of periodic sequences and under the weaker local mixing conditions $D_{T}^{(s)},\ s\geq 3,$ Ferreira and Martins (2003) obtained the expression for the extremal index of a $T$-periodic sequence from the joint distribution of $s$ consecutive variables of the sequence.

In Section 3 we obtain necessary conditions, that rely on the underlying sequence $\{X_{n}\}_{n\geq 0},$ for sequence $\{Y_{n}\}_{n\geq 1}$ to satisfy Leadbetter’s $D(u_{n})$ condition, as well as some local dependence condition $D_{T}^{(s)},$ $s\geq 1,$ for $T-$periodic sequences. The validation of these conditions will permit the determination of its extremal index expression. The results here obtained are illustrated with examples of recognized interest in applications, such as Markovian sequences.

The next section is devoted to the estimation of the model parameter $p\in]0,1[.$ We propose a consistent estimator for this parameter and analyse its finite sample behavior.

2 Model parameter estimation

The proposed model (1.1) depends on an unknown parameter $p\in]0,1[,$ that “controls” the number of missing values, and on an underlying stationary sequence with unknown marginal d.f. $F.$ The estimation of $p$ and $F$ is therefore essential for practical applications of this model.

The next result, that characterizes the probabilities $P\left(\bigcap_{j=1}^{T-1}\{Y_{sT+j}=Y_{sT}\}\right),$ $s\geq 1,$ $T\geq 2,$ gives a simple procedure to estimate the parameter $p$ involved in the definition of model (1.1).

Theorem 2.1

For the sequence $\{Y_{n}\}_{n\geq 1}$ defined in (1.1) it holds

[TABLE]

• Proof:

For all $s\geq 1,$ $Y_{sT}=X_{sT}$ almost surely and, if $U_{sT+j}=1$ for some $j\in\{1,\ldots,T-1\}$, we have $P\left(\bigcap_{j=1}^{T-1}\{Y_{sT+j}=Y_{sT}\}\right)=0,$ because the underlying d.f.’s are continuous. Therefore we can write

[TABLE]

Now since $U_{sT+j}=0,$ for all $j=1,\ldots,T-1,$ the several maxima in (2.4) are all equal to the variable $X_{sT}$ and the result follows immediately. $\square$

The way to estimate parameter $p$ of model (1.1) becomes clear from the previous result. So, if $(Y_{1},\ldots,Y_{n})$ is a random sample of $\{Y_{n}\}_{n\geq 1}$ an estimator for $p\in]0,1[$ is given by

[TABLE]

with $A_{s}=\displaystyle{\bigcap_{j=1}^{T-1}\{Y_{sT+j}=Y_{sT}\}}.$

From the weak law of large numbers and the fact that $E[{1\kern-1.99997pt\rm I}_{\{\bigcap_{j=1}^{T-1}\{Y_{sT+j}=Y_{sT}\}\}}]=P\left(\bigcap_{j=1}^{T-1}\{Y_{sT+j}=Y_{sT}\}\right),$ we can state that estimator (2.5) is a consistent estimator for $p.$

The d.f. $F$ can be estimated from the observations $Y_{sT}\equiv X_{sT},$ $s\geq 1,$ with the empirical d.f. or a kernel estimator. A review of these estimators and an explanation on their functionality and applicability in R can be found in Quintela-del-Río and Estévez-Pérez (2012).

2.1 Simulation results

We now analyze the finite sample properties of the estimator given in (2.5) with simulated data from model (1.2) given in Example 1.1. Each simulated data set consists of 1000 independent copies of $n$ realizations of a random sample $(Y_{1},\ldots,Y_{n})$ of (1.2) having one particular value of $p\in]0,1[$ out of five, for $T=2.$ Three different sample sizes are considered for each data set. The sample means $\widehat{\mu}(\widehat{p}_{n,2})$ and the sample standard deviations $\widehat{\sigma}(\widehat{p}_{n,2})$ of the estimates $\widehat{p}_{n,2,i},$ $i=1,\ldots,1000,$ depending on the sample size $n,$ were computed. The bias and the root mean squared errors ($RMSE(\widehat{p}_{n,2})$) were also determined. Table 1 summarizes the estimation results obtained. The estimator has a good behavior even for small sample sizes.

3 Extremal behaviour

The study of the extremal behavior of stationary or periodic sequences most often relies upon the verification of appropriate dependence conditions which assure that the limiting distribution of the maximum term is of the same type as the limiting distribution of the maximum of i.i.d. random variables. Usual conditions used in the literature for stationary and periodic sequences are Leadbetter’s $D(u_{n})$ condition (Leadbetter 1974), conditions $D^{(s)}(u_{n}),$ $s\geq 1,$ of Chernick et al. (1991) and conditions $D^{(s)}_{T}(u_{n}),$ $s,T\geq 1,$ of Ferreira (1994) and Ferreira and Martins (2003).

For any sequence of real numbers $\{u_{n}\}_{n\geq 1},$ the dependence condition $D(u_{n}),$ for the sequence $\{Y_{n}\}_{n\geq 1},$ of Leadbetter, states that $\alpha_{n,\ell_{n}}\to 0,$ as $n\to+\infty$, for some sequence $\ell_{n}=o(n)$, where

[TABLE]

and $M^{(\textbf{Y})}_{s,s+t}$ denotes $\displaystyle{\bigvee_{i=s}^{s+t}Y_{i}}\equiv\displaystyle{\bigvee_{i=s+t}^{s}Y_{i}}.$

In the next result we show that if condition $D(u_{n})$ holds for the stationary sequence $\{X_{n}\}_{n\geq 0}$ then it also holds for the $T-$ periodic sequence $\{Y_{n}\}_{n\geq 1}.$

Theorem 3.1

If, for any positive sequence $\{u_{n}\}_{n\geq 1},$ condition $D(u_{n})$ holds for the stationary sequence $\{X_{n}\}_{n\geq 0}$ then it also holds for the sequence $\{Y_{n}\}_{n\geq 1}$ defined in (1.1).

• Proof:

Consider the sets of consecutive integers $A_{p}=\left\{i_{1},\cdots,i_{1}+p\right\}$ and $B_{q}=\left\{j_{1},\cdots,j_{1}+q\right\}$, with $j_{1}-(i_{1}+p)>\ell_{n}.$

The definition of $\{Y_{n}\}_{n\geq 1}$ induces over $\{X_{n}\}_{n\geq 0}$ the corresponding sets of integers $A^{{}^{\prime}}_{p}=\left\{\left[\frac{i_{1}-1}{T}\right]T,\cdots,i_{1}+p\right\}$ and $B^{{}^{\prime}}_{q}=\left\{\left[\frac{j_{1}-1}{T}\right]T,\cdots,j_{1}+q\right\}$, such that $\left[\frac{j_{1}-1}{T}\right]T-(i_{p}+p)>\ell_{n}-T.$ Hence, each realization $u$ of $\textbf{U}\mathrel{\mathop{\mathchar 58\relax}}=\left\{U_{i},\ i\in A_{p}^{{}^{\prime}}\cup B_{q}^{{}^{\prime}}\right\}$ gives rise to another pair of subsets of positive integers, say $A_{p}^{(\textbf{U})}$ and $B_{q}^{(\textbf{U})}$, and therefore

[TABLE]

Due to the fact that $\{X_{n}\}_{n\geq 0}$ satisfies $D(u_{n})$ condition, the last average becomes

[TABLE]

because $\{U_{n}\}_{n\geq 0}$ is a sequence of independent variables.

Returning to the sequence $\{Y_{n}\}_{n\geq 1}$ we deduce

[TABLE]

with $\beta_{n,\ell_{n}}=\alpha_{n,\ell_{n}-T}$ and for $\ell^{\prime}_{n}=\ell_{n}+T=o(n)$ we have $\beta_{n,\ell^{\prime}_{n}}\xrightarrow[n\to+\infty]{}0,$ as required. $\square$

If the $T$-periodic sequence $\{Y_{n}\}_{n\geq 1}$ satisfies condition $D(u_{n}),$ for all $\{u_{n}\}_{n\geq 1},$ then we say that it also satisfies condition $D_{T}^{(s)}(u_{n}),$ $T,s\geq 1,$ when there exists a sequence of integers $\{k_{n}\}_{n\geq 1}$ such that $k_{n}\to+\infty,$ $k_{n}\frac{\ell_{n}}{n}\to 0,$ $k_{n}\alpha_{n,\ell_{n}}\to 0,$ as $n\to+\infty,$ and

[TABLE]

where $M^{(\bf{Y})}_{i,j}=-\infty,$ for $i>j,$ and $M^{(\bf{Y})}_{i,i}=Y_{i}.$ These local dependence conditions were first defined in Ferreira (1994), for $s=1$ and $s=2.$ This family was later enlarged by Ferreira and Martins (2003) with values of $s\geq 3.$

Observe that, when $s\geq 2,$ condition (3.6) is implied by

[TABLE]

which limits the distance between exceedances of level $u_{n},$ i.e., in each interval there can only occur more than one exceedance of $u_{n}$ if separated by less than $s-1$ non-exceedances of $u_{n}.$ Consequently, the local dependence conditions $D^{(s)}_{T},\ s\geq 1,$ become weaker as the value of $s$ increases and thereby enhance the number of processes to which our results apply. Condition (3.6), when $T=1,$ coincides with the one considered in $D^{(s)}(u_{n})$ of Chernick et al. (1991) for stationary sequences.

Under some local dependence condition $D_{T}^{(s)}(u_{n}),$ $s\geq 1,$ the extremal index of $\{Y_{n}\}_{n\geq 1}$, $\theta_{\bf{Y}},$ is defined by $-\frac{1}{\tau}\displaystyle{\log(\lim_{n\to\infty}P(M^{(\bf{Y})}_{1,n}\leq u_{n}))},$ where $\{u_{n}\}_{n\geq 1}$ is a sequence of positive numbers such that

[TABLE]

with $\overline{F}_{j}(x)=1-F_{j}(x)$ the tail functions of $Y_{j}$ for $j=1,...,T.$ Hence, it can be computed from

[TABLE]

In order to apply the previous results we shall impose the following two conditions on the tail of the d.f.’s $F$ and $G_{j},$ $j=2,\ldots,T-1,\,T\geq 3,$ where $G_{j}$ denotes the d.f. of $\bigvee_{i=j-1}^{\left[\frac{j-1}{T}\right]T}U_{j}X_{j}$, defined by (1). Namely

[TABLE]

The particular case $T=2$ leads to $\tau=\tau_{\bf{X}}$. Under such conditions, for $T\geq 3$, we have

[TABLE]

[TABLE]

We derive now a relation between a slightly stronger condition than $D^{(T+1)}(u_{n})$ of Chernick et al. (1991) for the underlying stationary sequence $\{X_{n}\}_{n\geq 0}$ and condition $D^{(T+1)}_{T}(u_{n})$ for the $T$-periodic sequence $\{Y_{n}\}_{n\geq 1}$.

Theorem 3.2

If for any positive sequence $\{u_{n}\}_{n\geq 1},$ the stationary sequence $\{X_{n}\}_{n\geq 0}$ satisfies

[TABLE]

then condition $D_{T}^{(T+1)}(u_{n})$ holds for the sequence $\{Y_{n}\}_{n\geq 1}.$

• Proof:

Observe first that

[TABLE]

In what concerns the probability involved in this last sum, for$i\,\in\,\{1,2,\ldots,T-2\}$, we have

[TABLE]

as well as

[TABLE]

where we used the stationarity of $\{X_{n}\}$. The remaining probabilities are equal to zero, since

[TABLE]

and

[TABLE]

For $i=T-1$ and $i=T$ we respectively have

[TABLE]

and

[TABLE]

Due to (3.11) with these last two upper bounds, we can now write

[TABLE]

Thus, condition $D_{T}^{(T+1)}(u_{n})$ holds for $\{Y_{n}\}_{n\geq 1}$ since (3.12) goes to zero as $n\to+\infty,$ form (3.10). $\square$

Note that condition (3.10) holds for $\{X_{n}\}_{n\geq 0}$ if it is a simple moving maxima sequence as defined in Example 1.1. In fact, $\{X_{n}\}_{n\geq 0}$ satisfies condition $D(u_{n}),$ for any sequence of real numbers $\{u_{n}\}_{n\geq 1},$ with $\alpha_{n,\ell_{n}}=0$ for $\ell_{n}\geq 3,$ since it is 2-dependent, and if we consider $r^{\prime}_{n}=\left[\frac{n}{2k_{n}}\right],$ $u_{n}=nx,\,x>0,$ with $\tau=1/x,$ it holds

[TABLE]

We may then conclude that $\{Y_{n}\}_{n\geq 1},$ given in (1.2) of Example 1.1, satisfies condition $D_{2}^{(3)}(u_{n}).$

Remark 1

If the underlying stationary sequence $\{X_{n}\}_{n\geq 0}$ is an $m-$dependent sequences, i.e., $(X_{i_{1}},\ldots,X_{i_{s}})$ is independent of $(X_{j_{1}},\ldots,X_{j_{t}}),$ for all positive integers $s,t$ and $i_{1}<\ldots<i_{s}<j_{1}<\ldots<j_{t}$ with $j_{1}-i_{s}\geq m+1,$ then condition (3.10) holds for some $T\leq m+1$ and $\{u_{n}\}_{n\geq 1}$ such that $n\overline{F}(u_{n})\xrightarrow[n\to+\infty]{}\tau_{X}.$

Remark 2

Condition (3.10) is implied by condition

[TABLE]

Under condition (3.10) for $\{X_{n}\}_{n\geq 0}$ we shall see that it is possible to obtain a relation between the extremal index of $\{Y_{n}\}_{n\geq 1},$ $\theta_{{\bf Y}},$ and the extremal index of $\{X_{n}\}_{n\geq 0},$ $\theta_{{\bf X}}.$ Before we present a required lemma.

Lemma 3.1

**

1. a)

For $i\in\{1,\ldots,T-1\}$ it holds

[TABLE]

with $S_{i}=\{i+1,\ldots,T\},$ $\overline{S}_{i}=\{T+1,\ldots,T+i\}$ and $\displaystyle{A_{i}\mathrel{\mathop{\mathchar 58\relax}}=\bigcap_{j\in S_{i}}\{U_{j}=1\}}.$ 2. b)

$P\left(Y_{T}>u_{n},M^{(\bf{Y})}_{T+1,2T}\leq u_{n}\right)=P\left(X_{T}>u_{n},M^{(\bf{X})}_{T+1,2T}\leq u_{n}\right)p^{T-1}.$**

• Proof:

a) We trivially have, for all $i\in\{1,\ldots,T-1\},$

[TABLE]

Consider now that $i\in\{1,\ldots,T-2\}.$ In order to deal with $M^{(\bf{Y})}(S_{i})$, observe that if there is some $m\in\{i+1,\ldots,T-1\}$ such that $U_{m}=0$, then the probability (a)) involves the following probabilities

[TABLE]

and

[TABLE]

since $m\geq i+1$. The proof follows immediately bacause $M^{(\bf{Y})}(S_{i})$ leads directly to $M^{(\bf{X})}(S_{i})$ under the occurrence of $A_{i}$. For $i=T-1$ the result follows as well because $S_{i}=\{T\}$ and $M^{(\bf{Y})}(S_{i})=X_{T}.$

b) Straightforward from the previously used arguments and the fact that $U_{kT}=1,$ $k\geq 0,$ $T\geq 1,$ almost surely. $\square$

Theorem 3.3

If $\{X_{n}\}_{n\geq 0}$ satisfies condition (3.10) for some $T\geq 2$ and $\{u_{n}\}_{n\geq 1}$ satisfying (3.8), then

[TABLE]

with

[TABLE]

where $J_{1}=\emptyset$, $S_{i}=\{i+1,\ldots,T\},\,\overline{S}_{i}=\{T+1,\ldots,T+i\}$ for $i\in\{1,\ldots,T-1\}$ and, for $T\geq 3$ and $i\in\{2,\ldots,T-1\}$,  $\underline{S}_{i}=\{1,\ldots,i-1\}$.

• Proof:

Since $\{Y_{n}\}_{n\geq 1}$ satisfies $D_{T}^{(T+1)}(u_{n})$, we deduce $\theta_{{\bf Y}}$ from (3.7). For $i=T$ we have, from Lemma 3.1 b),

[TABLE]

For any $i\in\{1,\ldots,T-1\},$ taking into account Lemma 3.1 a), where $P(A_{i})=p^{T-i-1},$ we can write

[TABLE]

[TABLE]

Now, due to the fact that condition (3.10) implies condition $D^{(T+1)}(u_{n})$ of [3] and $\{X_{n}\}_{n\geq 0}$ is a stationary sequence, the extremal index of $\{X_{n}\}_{n\geq 0}$ is given by

[TABLE]

Hence

[TABLE]

which concludes the proof. $\square$

We shall now apply the previous result in the computation of the extremal index of $\{Y_{n}\}_{n\geq 1},$ given in (1.1), in two different scenarios. First, we shall consider a periodic control at instants multiple of $T=2,$ i.e. $\{Y_{n}\}_{n\geq 1}$ 2-periodic, and that the underlying stationary sequences $\{X_{n}\}_{n\geq 0}$ is the moving maxima of Example 1.1. Second, we consider a 3-periodic sequence $\{Y_{n}\}_{n\geq 1}$ with an underlying sequence $\{X_{n}\}_{n\geq 0}$ a max-autoregressive sequence (ARMAX) of order one.

Example 3.1** ($\{X_{n}\}_{n\geq 0}$ moving maxima)**

As previously noted, sequence $\{Y_{n}\}_{n\geq 1}$ of Example 1.1 satisfies condition $D_{2}^{(3)}(u_{n})$ with $u_{n}=nx,\,x>0,$ therefore its extremal index, $\theta_{\bf Y},$ can be computed from (3.7). This yields $\theta_{\bf Y}=\frac{1}{2}$ which is equal to $\theta_{\bf X}$ as expected.

Since $\{X_{n}\}_{n\geq 0}$ satisfies condition (3.10) for $u_{n}=nx,$ $x>0,$ the extremal index $\theta_{\bf Y}$ can also be calculated from (3.14) of Theorem 3.3. Note that in this case $\tau_{\bf X}=\tau$. Consequently

[TABLE]

with

[TABLE]

For the probabilities in $P_{1,2}$ we have

[TABLE]

Taking this all into account, we finally obtain from (3.15) that $\theta_{\bf Y}=\frac{1}{2}.$**

Example 3.2** ($\{X_{n}\}_{n\geq 0}$ ARMAX of order one)**

If we have a solar thermal energy storage system where the temperature level in a tank is periodically controlled and eventually, for some reason, temperatures at certain time points are not retained, our model (1.1) can be used to describe the temperature in such a situation. According to Haslett (1979), the model defined by

[TABLE]

can be used to describe the temperature level in a tank. The extremal behaviour of this first order ARMAX storage model, for the particular case $\alpha=0,$ was studied by Alpuim (1989) and in its multivariate version by Ferreira and Ferreira (2013).

We shall now consider in (1.1) $\{Y_{n}\}_{n\geq 1}$ to be a 3-periodic positive sequence and the underlying sequence $\{X_{n}\}_{n\geq 0}$ the first order ARMAX process of Alpuim (1989),

[TABLE]

where $t\in]0,1[$ is a constant, $X_{0}$ is a positive random variable with d.f. $H_{0},$ independent of the sequence of i.i.d. positive random variables $\{W_{n}\}_{n\geq 1}$ with d.f. $L.$

Let us assume that the Markovian sequence $\{X_{n}\}_{n\geq 1}$ is stationary, i.e., there exists $x>0$ such that $L(x/t)>0$ and

[TABLE]

as proved in Alpuim (1989). Therefore, the non-degenerate d.f. $H$ of $X_{n},$ $n\geq 0,$ satisfies the following equation

[TABLE]

where $\alpha(H)=\inf\{x\mathrel{\mathop{\mathchar 58\relax}}H(x)>0\}\geq 0.$

It can be easily verified that, for $n\geq 1,$

[TABLE]

Sequence $\{X_{n}\}_{n\geq 1}$ satisfies the condition $D(u_{n}),$ for any sequence $\{u_{n}\}_{n\geq 1},$ because it is strong-mixing (see Alpuim 1988).

If $H$ belongs to the max-domain of attraction of the Fréchet d.f with parameter $\alpha>0,$ then the normalized levels $\{u_{n}\}_{n\geq 1}$ for $\{X_{n}\}_{n\geq 1},$ ${\it{i.e.}},$ such that $n(1-H(u_{n}))\xrightarrow[n\to+\infty]{}\tau_{{\bf{X}}}\geq 0,$ satisfy

[TABLE]

In this case $\{X_{n}\}_{n\geq 1}$ has extremal index $\theta_{{\bf{X}}}=1-t^{\alpha}$ and

[TABLE]

(see Ferreira and Ferreira (2013) for further details). Similarly, we establish that

[TABLE]

Condition (3.10) holds for $\{X_{n}\}_{n\geq 0}$, for the normalized levels $\{u_{n}\}_{n\geq 1}$ and any $T\geq 2,$ since

[TABLE]

[TABLE]

for any positive integer sequence $\{k_{n}\}$ such that $k_{n}\xrightarrow[n\to+\infty]{}+\infty$ and$k_{n}/n\xrightarrow[n\to+\infty]{}0.$

The validation of condition (3.10) for $\{X_{n}\}_{n\geq 0}$ guarantees that condition $D^{(4)}_{3}(u_{n})$ holds for $\{Y_{n}\}_{n\geq 1}$ (Theorem 3.2) and therefore its extremal index can be computed from the expression given in Theorem 3.3. Indeed, since all the factors with products containing $L^{j}\left(u_{n}/t^{m}\right)$ tend to 1, for all $j,\,m\geq 1$, we have

[TABLE]

Furthermore, $G_{1}(x)=H(x)$ and $G_{2}(x)=(1-p)H(x)-pP(X_{1}\leq x,\,X_{0}\leq x),$ leading (3.8) to $\tau_{1}=\tau_{\bf{X}}$ and $\tau_{2}=\tau_{\textbf{X}}+p\theta_{\textbf{X}}\tau_{\textbf{X}}$. Hence, by (3.9), it holds

[TABLE]

As a consequence, (3.14) becomes

[TABLE]

Figure 2 shows the effect that the extremal index of the underlying sequence $\theta_{\textbf{X}}\in]0,1]$ and the parameter $p\in]0,1[$ have on the extremal index of $\{Y_{n}\}_{n\geq 1},$ $\theta_{{\bf Y}}.$ As we can see, when $p$ is very close to one we get $\theta_{{\bf Y}}\simeq\theta_{\textbf{X}},$ since in this case there are almost no missing values and so $Y_{n}=X_{n},$ $n\geq 1,$ almost surely. When $p$ is very close to zero, almost all values of the underlying sequence are missing, except for the values at instants multiple of three, since $T=3.$ In this case, sequence $\{Y_{n}\}_{n\geq 1}$ will have the following form $X_{0},X_{0},X_{3},X_{3},X_{3},X_{6},X_{6},X_{6},...,X_{sT},X_{sT},X_{sT}\ldots,$ and so if $\theta_{\textbf{X}}\simeq 1$ ($\theta_{\textbf{X}}=1$ occurs, for example, for i.i.d sequences), exceedances of high levels will form clusters of mean size approximately $T=3,$ yielding an extremal index approximately equal to $1/3$ as observed in Figure 2.

Moreover, considering that the periodic control took place at instants multiple of two, $T=2,$ then, with $P_{1,2}$ given by (3.1) we would obtain, from Theorem 3.3,

[TABLE]

since it was proved that condition (3.10) also holds for the underlying ARMAX sequence of order one when $T=2.$**

Imposing a stronger condition on the behaviour of the underlying stationary sequence $\{X_{n}\}_{n\geq 0}$ than condition (3.10), we obtain, with similar arguments as used to prove Theorem 3.2, the validation of condition $D_{T}^{(T)}(u_{n})$ for $\{Y_{n}\}_{n\geq 1}$ and consequently also of $D_{T}^{(T+1)}(u_{n}),$ as stated in the next result.

Theorem 3.4

If for any sequence $\{u_{n}\}_{n\geq 1},$ the sequence $\{X_{n}\}_{n\geq 0}$ satisfies

[TABLE]

then condition $D_{T}^{(T)}(u_{n})$ holds for the sequence $\{Y_{n}\}_{n\geq 1}.$

Condition (3.17) is indeed more demanding than condition (3.10) as we can verify with the moving maxima sequence $\{X_{n}\}_{n\geq 0}$ defined in Example 1.1. In this case condition (3.17) with $T=2$ does not hold since

[TABLE]

Furthermore, the associated sequence $\{Y_{n}\}_{n\geq 1},$ also does not satisfy condition $D^{(2)}_{2}(u_{n})$ since

[TABLE]

Remark 3

Condition (3.17) is implied by the following condition

[TABLE]

We now present an analogous result to Theorem 3.3, relating $\theta_{\bf{Y}}$ and $\theta_{\bf{X}}$ under the hypothesis that $\{X_{n}\}_{n\geq 0}$ satisfies condition (3.17) and consequently $\{Y_{n}\}_{n\geq 1}$ satisfies condition $D_{T}^{(T)}(u_{n}).$ The main difference can be found in the first term of the expression for $\theta_{\bf{Y}}$ and in the definition of $\overline{S}_{i}.$

Theorem 3.5

If $\{X_{n}\}_{n\geq 0}$ satisfies condition (3.17), for some $T\geq 2$ and $\{u_{n}\}_{n\geq 1}$ satisfying (3.8), then $\{Y_{n}\}_{n\geq 1}$ satisfies condition $D_{T}^{(T)}(u_{n})$ and

[TABLE]

with

[TABLE]

where $I_{1}=J_{1}=\emptyset,\,$ $S_{i}=\{i+1,\ldots,T\}$ for any $i\,\in\,\{1,...,T-1\}$ and, for $T\geq 3$ and any $i\,\in\,\{2,...,T-1\}$ $\overline{S}_{i}=\{T+1,\ldots,T+i-1\}$ and $\underline{S}_{i}=\{1,...,i-1\}$.

We observe that any i.i.d. positive sequence $\{X_{n}\}$ trivially satisfies (3.17) for normalized levels $u_{n}$, and therefore sequence $\{Y_{n}\}$ satisfies condition $D_{T}^{(T)}(u_{n})$. Then, considering for instance $T=3$ and proceeding as before, we get $\tau_{2}=\tau_{\textbf{X}}(1+p)$, $\,\,\tau=\frac{1}{3}\tau_{\textbf{X}}(3+p-p^{2})$ and

[TABLE]

Hence

[TABLE]