Divergence from, and Convergence to, Uniformity of Probability Density Quantiles

TL;DR

This paper studies how probability density quantiles evolve under repeated transformations, establishing convergence to uniformity and linking divergence measures to statistical tests for shape and tail behavior.

Contribution

It introduces new fixed point theorems for pdQ convergence and connects divergence measures to optimal uniformity tests.

Findings

Repeated pdQ applications converge to uniform distribution.

Kullback-Leibler divergence informs power of uniformity tests.

New fixed point theorems for pdQ mappings.

Abstract

The probability density quantile (pdQ) carries essential information regarding shape and tail behavior of a location-scale family. Convergence of repeated applications of the pdQ mapping to the uniform distribution is investigated and new fixed point theorems are established. The Kullback-Leibler divergences from uniformity of these pdQs are mapped and found to be ingredients in power functions of optimal tests for uniformity against alternative shapes.

Figures

Tables

Table 1. Table 1: Quantiles of some distributions, their pdQs  and quantities relevant to the asymptotic power function ( 10 ). In general, we denote x u = Q ​ ( u ) = F − 1 ​ ( u ) subscript 𝑥 𝑢 𝑄 𝑢 superscript 𝐹 1 𝑢 x_{u}=Q(u)=F^{-1}(u) , but for the normal F = Φ 𝐹 Φ F=\Phi with density φ 𝜑 \varphi , we use z u = Φ − 1 ​ ( u ) subscript 𝑧 𝑢 superscript Φ 1 𝑢 z_{u}=\Phi^{-1}(u) . The logistic quantile function is only defined for u ≤ 0.5 𝑢 0.5 u\leq 0.5 but it is symmetric about u = 0.5 . 𝑢 0.5 u=0.5. LN represents the standard lognormal distribution. The quantile function for the Pareto is for the Type II distribution with shape a = 1 𝑎 1 a=1 , and the pdQ   is the same for Type I and Type II Pareto models.

			$I(𝒰:f^{\ast })$	$I(f^{\ast }:𝒰)$			$J(𝒰,f^{\ast })$
	$Q\left(u\right)$	$f^{\ast }\left(u\right)$	$-{\mu }_0$	${\mu }_1$	${\sigma }_0$	${\sigma }_1$	${\mu }_1-{\mu }_0$	$\frac{{\mu }_1-{\mu }_0}{{\sigma }_1}$	$\frac{{\sigma }_0}{{\sigma }_1}$
Normal	$z_u$	$2\sqrt{\pi }\phi \left(z_u\right)$	0.153	0.097	0.707	0.354	0.250	0.707	2.000
Logistic	$\ln \left(u/\left(1-u\right)\right)$	$6u\left(1-u\right)$	0.208	0.125	0.843	0.393	0.333	0.848	2.143
Laplace	$\ln \left(2u\right),u\le 0.5$	$2\text{min}\{u,1-u\}$	0.307	0.193	1.000	0.500	0.500	1.000	2.000
$t_2$	$\frac{2u-1}{{\left\{2u\left(1-u\right)\right\}}^{1/2}}$	$\frac{2^7{\left\{u\left(1-u\right)\right\}}^{3/2}}{3\pi }$	0.391	0.200	1.264	0.463	0.591	1.276	2.728
Cauchy	$\tan \left\{\pi \left(u-0.5\right)\right\}$	$2{\sin }^2\left(\pi u\right)$	0.693	0.307	1.814	0.538	1.000	1.857	3.369
Exp.	$-\ln \left(1-u\right)$	$2\left(1-u\right)$	0.307	0.193	1.000	0.500	0.500	1.000	2.000
Gumbel	$-\ln \left(-\ln \left(u\right)\right)$	$-4u\ln \left(u\right)$	0.191	0.116	0.803	0.381	0.307	0.806	2.109
LN	$e^{z_u}$	$\frac{2\sqrt{\pi }}{e^{1/4}}\phi \left(z_u\right)e^{-z_u}$	0.403	0.222	1.225	0.500	0.625	1.250	2.449
Pareto	${\left(1-u\right)}^{-1}$	$3{\left(1-u\right)}^2$	0.901	0.432	2.000	0.667	1.333	2.000	3.000

Equations

$${\nu }_{n,i}:=\int \{f^{n\ast }(x){\}}^{i}dx=\frac{1}{{\kappa }_{n-1}^i}{\nu }_{n-1,i+1},$$

$${\kappa }_n=\frac{1}{{\prod }_{j=0}^{n-1}{\kappa }_j^{n+1-j}}{\mu }_{n+2}.$$

$${\mu }_j$$

$${\int }_0^1|f^{2\ast }(u)-1{|}^{r}du=\frac{1}{{\kappa }_1^r}{\int }_0^1|f^{\ast }(x)-{\kappa }_1{|}^r{f}^{\ast }(x)dx.$$

$${\int }_0^1\{f^{n\ast }(x)-1{\}}^{2}dx={\kappa }_n-1,$$

$${\int }_0^1|f^{n\ast }(x)-1{|}^{r}dx\le {\int }_0^1(f^{n\ast }(x)-1{)}^2(f^{n\ast }(x{)}^{r-2}+1)dx={\nu }_{n,r}-2{\nu }_{n,r-1}+{\nu }_{n,r-2}+{\kappa }_n-1,$$

$${\nu }_{n,r}=\frac{{\mu }_{n+r}}{{\kappa }_{n-1}^r{\kappa }_{n-2}^{r+1}\dots {\kappa }_0^{n+r-1}}$$

$${\mu }_{n+r}={\kappa }_{n+r-2}{\kappa }_{n+r-3}^2\dots {\kappa }_0^{n+r-1},$$

$${\nu }_{n,r}={\kappa }_{n+r-2}{\kappa }_{n+r-3}^2\dots {\kappa }_n^{r-1}\to 1$$

$$\frac{{\mu }_{n_k+2}}{{\mu }_{n_k+1}}=\prod _{i=0}^{n_k}{\kappa }_i\ge (1+\delta {)}^k\to \infty \text{mbox}ask\to \infty .$$

$${\int }_0^1\left|f^{(n_{k_i}+1)\ast }(x)-1\right|1_{\left\{\left|f^{(n_{k_i}+1)\ast }(x)-1\right|\le \delta \right\}}dx$$

$$=\frac{1}{{\kappa }_{n_{k_i}}}{\int }_0^1\left|f^{(n_{k_i})\ast }(x)-{\kappa }_{n_{k_i}}\right|f^{(n_{k_i})\ast }(x)1_{\left\{\left|f^{n_{k_i}\ast }(x)-{\kappa }_{n_{k_i}}\right|\le \delta {\kappa }_{n_{k_i}}\right\}}dx.$$

$$\left|f^{(n_{k_i}+1)\ast }-1\right|\stackrel{\mathbb{P}}{⟶}0,f^{n_{k_i}\ast }\left|f^{n_{k_i}\ast }-{\kappa }_{n_{k_i}}\right|\stackrel{\mathbb{P}}{⟶}|1-b|,1_{\left\{\left|f^{n_{k_i}\ast }(x)-{\kappa }_{n_{k_i}}\right|\le \delta {\kappa }_{n_{k_i}}\right\}}\stackrel{\mathbb{P}}{⟶}1$$

$$l_{\mathbf{x}}=\sum _1^m\ln (f_1^{\ast }(x_i))\ge c_{m,\alpha },$$

$$(l_{\mathbf{x}}/\sqrt{m}-\sqrt{m}{\mu }_0)/{\sigma }_0\ge z_{1-\alpha }.$$

$${\Pi }_m(f_1^{\ast })=\Phi \left(\sqrt{m}\frac{({\mu }_1-{\mu }_0)}{{\sigma }_1}+z_{\alpha }\frac{{\sigma }_0}{{\sigma }_1}\right).$$

Full text

Keywords: convergence in $L_{r}$ norm; fixed point theorem; Kullback-Leibler divergence; relative entropy; semi-metric; uniformity test

1 Introduction

1.1 Background and summary

For each continuous location-scale family of distributions with square-integrable density there is a probability density quantile (pdQ) which is an absolutely continuous distribution on the unit interval. Members of the class of such pdQs differ only in shape, and the asymmetry of their shapes can be partially ordered by their Hellinger distances or Kullback-Leibler divergences from the class of symmetric distributions on this interval. In addition, the tail behaviour of the original family can be described in terms of the boundary derivatives of its pdQ. Empirical estimators of the pdQs enable one to carry out inference, such as fitting shape parameter families to data; details are in Staudte, (2017).

The Kullback-Leibler directed divergence and symmetrized divergence (KLD) of a pdQ with respect to the uniform distribution on [0,1] is investigated in Section 1.2, with remarkably simple numerical results, and a map of these divergences for some standard location-scale families is constructed. The ‘shapeless’ uniform distribution is the center of the pdQ universe, as is explained in Section 2, where it is found to be a fixed point. We then investigate the convergence to uniformity of repeated applications of the pdQ transformation, by means of fixed point theorems for a semi-metric. In Section 3 power functions of hypothesis tests of uniformity against specified alternative shapes are shown to be dependent on the symmetrized Kullback-Leibler divergence. Further ideas are discussed in Section 4.

1.2 Definitions and divergence map

Let $\mathcal{F}\,$ denote the class of cumulative distribution functions (cdf s) on the real line and for each $F\in\mathcal{F}\,$ define the associated quantile function of $F$ by $Q(u)=\inf\{x:\ F(x)\geq u\}$, for $0<u<1.$ When the random variable $X$ has cdf  $F$, we write $X\sim F$. When the density function $f=F^{\prime}$ exists, we also write $X\sim f$ or $f\sim F$. We only discuss $F$ absolutely continuous with respect to Lebesgue measure, but the results can be extended to the discrete and mixture cases using suitable dominating measures.

Definition 1.1

Let $\mathcal{F}\,^{\prime}=\{F\in\mathcal{F}\,:\;f=F^{\prime}\text{ exists and is positive}\}$. For each $F\in\mathcal{F}\,^{\prime}$ we follow Parzen, (1979) and define the quantile density function $q(u)=Q^{\prime}(u)=1/f(Q(u))$. Parzen called its reciprocal function $fQ(u)=f(Q(u))$ the density quantile function. For $F\in\mathcal{F}\,^{\prime}$, and $U$ uniformly distributed on [0,1], assume $\kappa={\mathbb{E}}[fQ(U)]=\int f^{2}(x)\,dx$ is finite; that is, $f$ is square integrable. Then we can define the continuous pdQ of $F$ by $f^{*}(u)=fQ(u)/\kappa$, $0<u<1$. Let $\mathcal{F}\,^{\prime*}\subset\mathcal{F}\,^{\prime}$ denote the class of all such $F$.

Not all $f$ are square-integrable, and this requirement for the mapping $f\to f^{*}$ means that $\mathcal{F}\,^{\prime*}$ is a proper subset of $\mathcal{F}\,^{\prime}.$ The advantages of working with $f^{*}$s over $f$s are that they are free of location and scale parameters, they ignore flat spots in $F$ and have a common bounded support. Moreover, $f^{*}$ often has a simpler formula than $f$; see Table 1 for examples.

Next we evaluate and plot the Kullback & Leibler, (1951) divergences from uniformity. The Kullback & Leibler, (1951) divergence of density $f_{1}$ from density $f_{2}$, when both have domain [0,1], is defined as $I(f_{1}:f_{2}):=\int_{0}^{1}\ln(f_{1}(u)/f_{2}(u))\,f_{1}(u)\,du={\mathbb{E}}[\ln(f_{1}(U)/f_{2}(U))\,f_{1}(U)],$ where $U$ denotes a random variable with the uniform distribution ${\cal U}$ on [0,1]. The divergences from uniformity are easily computed through $I({\cal U}:f^{*})=-\int_{0}^{1}\ln(f^{*}(u))\,du=-{\mathbb{E}}[\ln(f^{*}(U))]$ and $I(f^{*}:{\cal U})=\int_{0}^{1}\ln(f^{*}(u))\,f^{*}(u)\,du={\mathbb{E}}[\ln(f^{*}(U))\,f^{*}(U)].$ (Kullback,, 1968, p. 6) interprets $I(f^{*}:{\cal U})$ as the mean evidence in one observation $V\sim f^{*}$ for $f^{*}$ over ${\cal U}$; it is also known as the relative entropy of $f^{*}$ with respect to ${\cal U}$. In Table 1 are shown the quantile functions of some standard distributions, along with their pdQs, associated divergences $I({\cal U}:f^{*}),I(f^{*}:{\cal U})$ and symmetrized divergence (KLD)  defined by $J({\cal U},f^{*}):=I({\cal U}:f^{*})+I(f^{*}:{\cal U})$.

Definition 1.2

Given pdQs  $f_{1}^{*}$, $f_{2}^{*}$, let $d(f_{1}^{*},f_{2}^{*}):=\sqrt{I(f^{*}_{1}:f^{*}_{2})+I(f_{2}^{*}:f_{1}^{*})}$. Then $d$ is a semi-metric on the space of pdQs; i.e., $d$ satisfies all requirements of a metric except the triangle inequality. Introducing the coordinates $(s_{1},s_{2})=(\sqrt{I({\cal U}:f^{*})}\,,\sqrt{I(f^{*}:{\cal U})})$, we can define the distance from uniformity of any $f^{*}$ by the Euclidean distance of $(s_{1},s_{2})$ from the origin $(0,0)$, namely $d({\cal U},f^{*})$.

Remark This $d$ does not satisfy the triangle inequality: for example, if $\cal U,$ $\cal N$ and $\cal C$ denote the uniform, normal and Cauchy pdQs, then $d({\cal U},{\cal N})=0.5,$ $d({\cal N},{\cal C})=0.4681$ but $d({\cal U},{\cal C})=1.$ However, $d$ can provide an informative measure of distance from uniformity.

In Figure 1 are shown the loci of points $(s_{1},s_{2})$ for some continuous shape families. The light dotted arcs with radii 1/2, 1 and 2 are a guide to these distances from uniformity. The large discs in purple, red and black correspond to $\cal U,$ $\cal N$ and $\cal C$. The blue cross at distance $1/\sqrt{2}$ from the origin corresponds to the exponential distribution. Nearby is the standard lognormal point marked by a red cross. The lower red curve is nearly straight and is the locus of points corresponding to the lognormal shape family.

The Chi-squared($\nu$), $\nu>1$, family also appears as a red curve; it passes through the blue cross when $\nu=2$, as expected, and heads toward the normal disc as $\nu\to\infty.$ The Gamma family has the same locus of points as the Chi-squared family. The curve for the Weibull($\beta$) family, for $0.5<\beta<3$, is shown in blue; it crosses the exponential blue cross when $\beta=1$. The Pareto($a$) curve is shown in black. As $a$ increases from 0, this line crosses the arcs distant 2 and 1 from the origin for $a=(2\sqrt{2}\,+1)/7\approx 0.547$ and $a=(\sqrt{5}\,-1)/2\approx 1.618$, respectively, and approaches the exponential blue cross as $a\to\infty$.

The Power($b$) or Beta($b,1$) for $b>1/2$ family is represented by the magenta curve of points moving toward the origin as $b$ increases from 1/2 to 1, and then moving out towards the exponential blue cross as $b\to\infty$. For each choice of $\alpha>0.5,$ $\beta>0.5$ the locus of the Beta($\alpha,\beta$) pdQ divergences lies above the chi-squared red curve and mostly below the power($b$) magenta curve; however, the U-shaped Beta distributions have loci above it.

The lower green line near the Pareto black curve gives the loci of root-divergences from uniformity of the Tukey($\lambda$) with $\lambda<1$, while the upper green curve corresponds to $\lambda\geq 1$. It is known that the Tukey($\lambda$) distributions, with $\lambda<1/7$, are good approximations to Student’s $t$ distributions for $\nu>0$ provided $\lambda$ is chosen properly. The same is true for their corresponding pdQs (Staudte,, 2017, Sec.3.2). For example, the pdQ of $t_{\nu}$ with $\nu=0.24$ degrees of freedom is well approximated by the choice $\lambda=-4.063.$ Its location is marked by the small black disk in Figure 1; it is distant 2 from uniformity. The generalized Tukey distributions of Freimer et al., (1988) with two shape parameters also fill a large funnel shaped region (not marked on the map) emanating from the origin and just including the region bounded by the green curves of the Tukey symmetric distributions.

The larger the value of $d({\cal U},f^{*})$, the easier it should be to discriminate between $\cal U$ and $f^{*}$ based on data, as we will see when calculating power functions of tests in Section 3.1. First, however, we show that $\cal U$ plays a unique role in the space of pdQs.

2 Convergence of density shapes to uniformity via fixed point theorems

The transformation $f\to f^{*}$ of Definition 1.1 is quite powerful, removing location and scale and moving the distribution from the support of $f$ to the unit interval. Examples suggest that another application of the transformation $f^{2*}:=(f^{*})^{*}$ leaves less information about $f$ in $f^{2*}$ and hence it is closer to the uniform density. Further, with $n$ iterations $f^{(n+1)*}:=(f^{n*})^{*}$ for $n\geq 2$, we would expect that $f^{n*}$ converges to the uniform density as $n\to\infty$. An R script Team, (2008) for finding repeated $*$-iterates of a given pdQ  is available as Supplementary Online Material.

2.1 Conditions for convergence to uniformity

Definition 2.1

Given $f\in\mathcal{F}\,^{\prime}$, we say that $f$ is of $*$-order $n$ if $f^{*},f^{2*},\dots,f^{n*}$ exist but $f^{(n+1)*}$ does not. When the infinite sequence $\{f^{n*}\}_{n\geq 1}$ exists, it is said to be of infinite $*$-order.

For example, the Power($3/4$) family is of $*$-order $2$, while the Power($2$) family is of infinite $*$-order. The $\chi^{2}_{\nu}$ distribution is of finite $*$-order for $1<\nu<2$ and infinite $*$-order for $\nu\geq 2.$ The normal distribution is of infinite $*$-order.

We write $\mu_{n}:=\int_{-\infty}^{\infty}\{f(y)\}^{n}\,dy$, $\kappa_{n}=\int_{0}^{1}\{f^{n*}(x)\}^{2}\,dx$, $n\geq 1$, and $\kappa_{0}=\int_{-\infty}^{\infty}\{f(x)\}^{2}\,dx$. The next proposition characterises the property of infinite $*$-order.

Proposition 2.2

For $f\in\mathcal{F}\,^{\prime}$ and $m\geq 1$, the following statements are equivalent:

(i) $\mu_{m+2}<\infty$;

(ii) $\mu_{j}<\infty$ for all $1\leq j\leq m+2$;

(iii) $\kappa_{j}<\infty$ and $\kappa_{j}=\frac{\mu_{j}\mu_{j+2}}{\mu_{j+1}^{2}}$ for all $1\leq j\leq m$.

In particular, $f$ is of infinite $*$-order if and only if $\mu_{n}<\infty$, $n\geq 1$.

**Proof of Proposition 2.2: ** For each $i,n\geq 1$, provided all terms below are finite, we have the following recursive formula

[TABLE]

giving

[TABLE]

(i) $\Rightarrow$ (ii) For $1\leq j\leq m+2$,

[TABLE]

(ii) $\Rightarrow$ (iii) Use (2) and proceed with induction for $1\leq n\leq m$.

(iii) $\Rightarrow$ (i) By Definition 1.1, $\kappa_{1}<\infty$ means that $\kappa_{0}<\infty$. Hence (i) follows from (2) with $n=m$.

Next we investigate the involutionary nature of the $*$-transformation.

Proposition 2.3

Let $f^{*}$ be a pdQ  and assume $f^{2*}$ exists. Then $f^{*}\sim{\cal U}$ if and only if $f^{2*}\sim{\cal U}$.

**Proof of Proposition 2.3: **For $r>0$, we have

[TABLE]

If $f^{*}(u)\sim{\cal U}$, then $\kappa_{1}=1$ and (3) ensures $\int_{0}^{1}|f^{2*}(u)-1|^{r}du=0$, so $f^{2*}(u)\sim{\cal U}$.

Conversely, if $f^{2*}(u)\sim{\cal U}$, then using (3) again gives $\int_{0}^{1}|f^{*}(x)-\kappa_{1}|^{r}f^{*}(x)\,dx=0$. Since $f^{*}(x)>0$ a.s., we have $f^{*}(x)=\kappa_{1}$ a.s. and this can only happen when $\kappa_{1}=1$. Thus $f^{*}\sim{\cal U}$, as required.

.

Proposition 2.3 shows that the uniform distribution is a fixed point in the Banach space of integrable functions on [0,1] with the $L_{r}$ norm for any $r>0$. It remains to show $f^{n*}$ has a limit and that the limit is the uniform distribution. It was hoped the classical machinery for convergence in Banach spaces (Luenberger,, 1969, Ch.10) would prove useful in this regard, but the *-mapping is not a contraction. For this reason, although there are many studies of fixed point theory in metric and semi-metric spaces (see, e.g., Bessenyei & Páles, (2017) and references therein), the fixed point Theorems 2.4, 2.5 and 2.6 shown below do not seem to be covered in these general studies. For simplicity, we use \stackrel{{\scriptstyle\mbox{\scriptsize{L_{r}}}}}{{\longrightarrow}} to stand for the convergence in $L_{r}$ norm and $\stackrel{{\scriptstyle{\scriptsize{{\mathbb{P}}}}}}{{\longrightarrow}}$ for convergence in probability as $n\to\infty$.

Theorem 2.4

For $f\in\mathcal{F}\,^{\prime}$ with infinite $*$-order, the following statements are equivalent:

(i) f^{n*}\stackrel{{\scriptstyle\mbox{\scriptsize{L_{2}}}}}{{\longrightarrow}}1;

(ii) For all $r>0$, f^{n*}\stackrel{{\scriptstyle\mbox{\scriptsize{L_{r}}}}}{{\longrightarrow}}1;

(iii) $\frac{\mu_{n}\mu_{n+2}}{\mu_{n+1}^{2}}\to 1$ as $n\to\infty$.

Remark Notice that $\mu_{n}={\mathbb{E}}\left\{f^{*}(U)^{n-1}\right\}$, $n\geq 1$, are the moments of the random variable $f^{*}(U)$ with $U\sim{\cal U}$, Theorem 2.4 says that the convergence of $\{f^{n*}:\ n\geq 1\}$ is purely determined by the moments of $f^{*}(U)$. This is rather puzzling because it is well known that the moments do not uniquely determine the distribution (Feller,, 1971, p. 227), meaning that different distributions with the same moments have the same converging behaviour. However, if $f$ is bounded, then $f^{*}(U)$ is a bounded random variable so its moments uniquely specify its distribution (Feller,, 1971, pp. 225–226), leading to stronger results in Theorem 2.5.

**Proof of Theorem 2.4: ** It is obvious that (ii) implies (i).

(i) $\Rightarrow$ (iii): By Proposition 2.2, $\kappa_{n}=\frac{\mu_{n}\mu_{n+2}}{\mu_{n+1}^{2}}$. Now

[TABLE]

so (iii) follows immediately.

(iii) $\Rightarrow$ (ii): It suffices to show that f^{n*}\stackrel{{\scriptstyle\mbox{\scriptsize{L_{r}}}}}{{\longrightarrow}}1 for any integer $r\geq 4$. To this end, since for $a,b\geq 0$, $|a-b|^{r-2}\leq a^{r-2}+b^{r-2}$, we have from (4) that

[TABLE]

where, as before, $\nu_{n,r}=\int_{0}^{1}\{f^{n*}(x)\}^{r}dx$. However, applying (1) gives

[TABLE]

and (2) ensures

[TABLE]

which imply

[TABLE]

as $n\to\infty$. Hence, it follows from (5) that $\int_{0}^{1}|f^{n*}(x)-1|^{r}dx\to 0$ as $n\to\infty$, completing the proof.

We write $\|g\|=\sup_{x}|g(x)|$ for each bounded function $g$.

Theorem 2.5

If $f$ is bounded, then

(i) for all $n\geq 0$, $\|f^{(n+1)*}\|\leq\|f^{n*}\|$ and the inequality becomes equality if and only if $f^{n*}\sim{\cal U}$;

(ii) f^{n*}\stackrel{{\scriptstyle\mbox{\scriptsize{L_{r}}}}}{{\longrightarrow}}1 for all $r>0$.

**Proof of Theorem 2.5: ** It follows from (4) that $\kappa_{n}\geq 1$ and the inequality becomes equality if and only if $f^{n\ast}\sim{\cal U}$.

(i) Let $Q^{n\ast}$ be the inverse of the cumulative distribution function of $f^{n\ast}$, then $f^{(n+1)\ast}(u)=\frac{f^{n*}(Q^{n*}(u))}{\kappa_{n}}\leq\frac{\|f^{n*}\|}{\kappa_{n}}$, giving $\|f^{(n+1)\ast}\|\leq\frac{\|f^{n*}\|}{\kappa_{n}}{\leq\|f^{n*}\|}$. If $f^{n\ast}\sim{\cal U}$, then Proposition 2.3 ensures that $f^{(n+1)\ast}\sim{\cal U}$, so $\|f^{(n+1)\ast}\|=\|f^{n*}\|$. Conversely, if $\|f^{(n+1)*}\|=\|f^{n*}\|$, then $\kappa_{n}=1$, so $f^{n\ast}\sim{\cal U}$.

(ii) It remains to show that $\kappa_{n}\to 1$ as $n\to\infty$. In fact, if $\kappa_{n}\not\to 1$, since $\kappa_{n}\geq 1$, there exist a $\delta>0$ and a subsequence $\{n_{k}\}$ such that $\kappa_{n_{k}}\geq 1+\delta$, which implies

[TABLE]

However, $\frac{\mu_{n_{k}+2}}{\mu_{n_{k}+1}}\leq\|f\|<\infty$, which contradicts (6).

Theorem 2.6

For $f\in\mathcal{F}\,^{\prime}$ with infinite $*$-order such that $\{\mu_{n}\mu_{n+2}\mu_{n+1}^{-2}:\ n\geq 1\}$ is a bounded sequence, then the following statements are equivalent:

(i) $f^{n*}\stackrel{{\scriptstyle{\scriptsize{{\mathbb{P}}}}}}{{\longrightarrow}}1$;*

(ii) For all $r>0$, f^{n*}\stackrel{{\scriptstyle\mbox{\scriptsize{L_{r}}}}}{{\longrightarrow}}1;

(iii) $\mu_{n}\mu_{n+2}\mu_{n+1}^{-2}\to 1$ as $n\to\infty$.

*Proof of Theorem 2.6: ** It suffices to show that (i) implies (iii). Recall that $\kappa_{n}=\mu_{n}\mu_{n+2}\mu_{n+1}^{-2}$, for each subsequence $\{\kappa_{n_{k}}\}$, there exists a converging sub-subsequence $\{\kappa_{n_{k_{i}}}\}$ such that $\kappa_{n_{k_{i}}}\to b$ as $i\to\infty$. It remains to show that $b=1$. To this end, for $\delta>1$, we have

[TABLE]

(i*) ensures that

[TABLE]

as $i\to\infty$, so applying the bounded convergence theorem to both sides of (7) to get $0=|1/b-1|$, i.e., $b=1$.

2.2 Examples of convergence to uniformity

The main results in section 2.1 cover all the standard distributions with infinite $*$-order in Johnson et al., (1994), Johnson et al., (1995). In fact, as observed in the Remark after Theorem 2.4 that the convergence to uniformity is purely determined by the moments of $f^{*}(U)$ with $U\sim{\cal U}$, we have failed to construct a density such that $\{f^{n*}:\ n\geq 1\}$ does not converge to the uniform distribution. Here we give a few examples to show that the main results in section 2.1 are indeed very convenient to use.

Example 1: Power function family.

From Table 1 the Power$(b)$ family has density $f_{b}(x)=bx^{b-1},~{}0<x<1$, so it is of infinite $*$-order if and only if $b\geq 1$. As $f_{b}$ is bounded for $b\geq 1$, Theorem 2.5 ensures that $f^{n*}_{b}$ converges to the uniform in $L_{r}$ for any $r>0$.

Example 2: Exponential distribution.

Suppose $f(x)=e^{x},\;~{}x<0$. Then $f^{*}(u)=2u$, $0<u<1$, which belongs to the Power(2) distribution; and so by Example 1, $f$ is bounded, so Theorem 2.5 says that $f^{n*}$ converges to the uniform distribution as $n\to\infty.$ By symmetry, the same result holds for $f(x)=e^{-x},\;~{}x>0$.

Example 3: Pareto distribution.

The Pareto($a$) family, with $a>0$, has $f_{a}(x)=ax^{-a-1}$ for $x>1$, which is bounded, so an application of Theorem 2.5 yields that the sequence $\{f^{n*}_{a}\}_{n\geq 1}$ converges to the uniform distribution as $n\to\infty.$

Example 4: Cauchy distribution.

The pdQ of the Cauchy density is given by $f^{*}(u)=2\sin^{2}(\pi u)$, $0<u<1$, see Table 1; it retains the bell shape of $f$. It follows that $F^{*}(t)=t-\sin(2\pi t)/(2\pi),$ for $0<t<1$. It seems impossible to obtain an analytical form of $f^{n*}$ for $n\geq 2$. However, as $f$ is bounded, using Theorem 2.5, we can conclude that $f^{n*}$ converges to the uniform distribution as $n\to\infty$.

Example 5: Normal distribution.

Although it is possible to obtain $\{f^{n*}\}$ by induction and then derive directly that $f^{n*}$ converges to the uniform distribution as $n\to\infty$, one can easily see that the pdf is bounded and so Theorem 2.5 can be employed to get the same conclusion.

Example 6:

Let $f(x)=-\ln x$, $x\in(0,1)$, then $\mu_{n}=n!$ and $\kappa_{n}=\frac{n+2}{n+1}\to 1$ as $n\to\infty$, so we have from Theorem 2.4 that for any $r>0$, $f^{n*}$ converges in $L_{r}$ norm to constant 1 as $n\to\infty$.

3 Testing for uniformity

The larger the value of $d({\cal U},f^{*})$, the easier it should be to discriminate between $\cal U$ and $f^{*}$. This is indeed the case, as we now demonstrate.

3.1 Power of tests for detecting non-uniform shapes

The connection between Kullback-Leibler divergences and the Neyman-Pearson Lemma is well-known, see text and references in Eguchia & Copas, (2006). The following material on power functions for tests for uniformity, while contextual to comparing shapes, may prove useful in other situations.

Given a random sample of $m$ independent, identically distributed (i.i.d.) variables, each from a distribution with density $f$, it is feasible to carry out a nonparametric test of uniformity by estimating the pdQ with a kernel density estimator $\widehat{f}_{m}^{*}$ and comparing it with the uniform density on [0,1] using any one of a number of metrics. Consistent estimators $\widehat{f}_{m}^{*}$ for $f^{*}$ based on normalized reciprocals of the quantile density estimators derived in Prendergast & Staudte, (2016) are available and described in Staudte, (2017, Section 2). However an investigation into such omnibus nonparametric testing procedures, and comparison with other kernel density based techniques in Bowman, (1992); Fan, (1994) and Pavia, (2015), is beyond the scope of this work.

Given the large number of tests for uniformity that are available, see text and references in Stephens, (2006), one may well ask why introduce new ones? The doyen of goodness-of-fit testing Stephens, (2006) provides the answer:

Since transformations are often used to produce a set of uniforms, it might be appropriate to conclude with some cautionary remarks on when uniformity is not to be expected. This will be so, for example, when the U set is derived from the PIT and when some parameters, unknown in the distribution, are replaced by estimates. In this situation, even when the estimates are efficient, the U set will be superuniform, giving much lower values of, say, EDF statistics, than if the set were uniform; this remains so even as the sample size grows bigger.

In practice this means that if a test for uniformity is preceded by a probability integral transformation (PIT) including parameter estimates, then the actual levels of such tests will not be those nominated unless (often complicated and model specific) adjustments are made. Examples are in Lockhart et al., (1986) and Schader & Schmid, (1997).

In this section we study the simpler problem of testing the null hypothesis of uniformity $H_{0}:f^{*}(u)=1$ for all $0<u<1$ (denoted $\cal U$) against a specified alternative $H_{1}:f^{*}=f^{*}_{1}$. This test will give us a standard by which to judge the power of any future nonparametric test when the specific alternative holds. Given a vector of $\mathbf{X}=(X_{1},\dots,X_{m})$ of i.i.d. variables from $f^{*}$, and realized values $\mathbf{x}=(x_{1},\dots,x_{m})$ the Neyman-Pearson (NP) test rejects $H_{0}$ in favor of $H_{1}$ when the product $\prod_{1}^{m}f_{1}^{*}(x_{i})$ is large, or equivalently, when

[TABLE]

where $c_{m,\alpha}$ is chosen to achieve level $\alpha$. In general this critical point will be difficult to determine, but for the normal pdQ  alternative $f_{1}^{*}(u)=2\sqrt{\pi}\,\varphi(z_{u})$ we have: $l_{\mathbf{x}}=m\ln(\sqrt{2})-\frac{1}{2}\,\sum_{1}^{m}\{\Phi^{-1}(x_{i})\}^{2}$, where $\varphi$ is the density of the standard normal. Under the null hypothesis $\sum_{1}^{m}\{\Phi^{-1}(U_{i})\}^{2}\sim\chi^{2}_{m}$, so an equivalent test to (8) would reject the null hypothesis of uniformity in favor of normality when $\sum_{1}^{m}\{\Phi^{-1}(x_{i})\}^{2}\leq\chi^{2}_{m}(\alpha).$ This is the most powerful level-$\alpha$ test of these simple hypotheses based on $m$ observations.

Returning to a general alternative pdQ  we can find asymptotically most powerful level-$\alpha$ tests based on the fact that $l_{\mathbf{X}}$ is a sum of i.i.d. random variables with common mean $\mu_{0}={\mathbb{E}}_{0}[\ln(f_{1}^{*}(U))]$ and variance $\sigma^{2}_{0}=\hbox{Var}_{0}[\ln(f_{1}^{*}(U))]$, which we assume are finite. By virtue of a CLT, the large-sample NP test rejects $H_{0}$ at asymptotic level $\alpha$ when

[TABLE]

To obtain an expression for the large sample power of this test, let $\mu_{1}={\mathbb{E}}_{1}[\ln(f_{1}^{*}(X))]$ and $\sigma^{2}_{1}=\hbox{Var}_{1}[\ln(f_{1}^{*}(X))]$, again assumed to be finite; then the asymptotic power of the test (9) against $f^{*}$ based on $m$ observations is readily found to be:

[TABLE]

We need $\alpha$, $m$, $l_{\mathbf{x}}$, $\mu_{0}$, $\sigma^{2}_{0}$ to carry out the test (9); and we also need $\mu_{1}$, $\sigma^{2}_{1}$ to compute the asymptotic power (10). Notice that the distances from the origin in Figure 1 of Section 1.2 are based on the directed divergences $I({\cal U}:f_{1}^{*})=-{\mathbb{E}}_{0}[\ln(f_{1}^{*}(X))]=-\mu_{0}$ and $I(f_{1}^{*}:{\cal U})={\mathbb{E}}_{1}[\ln(f_{1}^{*}(X))]=\mu_{1}$, so the symmetrized divergence KLD is $J({\cal U},f_{1}^{*})=\mu_{1}-\mu_{0}.$ Thus the power function (10) is non-decreasing in the KLD or its square root, the distance $d({\cal U},f^{*}_{1})$ between null and alternative.

Some examples of $\mu_{0}$, $\mu_{1}$, $\mu_{1}-\mu_{0}$, $(\mu_{1}-\mu_{0})/\sigma_{1}$ and $\sigma_{0}/\sigma_{1}$ are given in Table 1. Note the particularly simple values for the symmetrized divergence $J({\cal U},f_{1}^{*})=\mu_{1}-\mu_{0}.$ Distributions with shapes ‘visually far’  from uniformity have large values in the two right-most columns, so that the test will more easily detect them. For the normal alternative, the asymptotic power (10) at level $\alpha$ is $\Pi_{m}(f_{1}^{*})=\Phi(\sqrt{m/2}+2\,z_{\alpha}),$ which exceeds $\alpha$ when $m>2z_{1-\alpha}^{2}.$

There are other situations where the asymptotic power functions are monotone increasing functions of the KLD with many one-sample examples in Kulinskaya et al., (2008, 2010, 2014), one-parameter exponential families Morgenthaler & Staudte, (2012), two-sample binomial tests Prendergast & Staudte, (2014) and non-central chi-squared and non-central F families arising in tests for equivalence Morgenthaler & Staudte, (2016). This reveals a general phenomenon but no meta-theorem containing these results is yet available.

3.2 Examples of power functions for shape families

The power functions of testing uniformity against the pdQs  of four shape families, are shown in Figure 2. The first two models, power function model Beta($b,1$) and the symmetric Beta($b,b$) model for $b>0.5$ contain the null hypothesis. Their respective power functions (10) for a level 0.05 test of uniformity based on $m=25$ and 100 observations are shown in the top two plots. Similar plots for alternatives Lognormal($\sigma$) and Pareto($a$) families are also shown for much smaller sample sizes 9 and 16 indicating that small samples will likely detect these alternative shapes.

The plots in Figure 2 require the null and alternative means and variances of the test statistic, and were obtained by numerical integration. In the case of the Beta($b$,1) model exact results are derived as follows. the quantile function is $Q(b)=u^{1/b}$ and for $b>1/2$ its density is square integrable, leading to the pdQ  $f_{1}^{*}(u)=(2-\frac{1}{b})\,u^{1-\frac{1}{b}}.$ The log-likelihood for one observation $X=x$ required in (9) is $l_{x}(b)=\ln(2-1/b)+(1-1/b)\ln(x)$. Thus $\mu_{0}(b)={\mathbb{E}}_{0}[l_{\mathbf{X}}(b)]=\ln(2-1/b)+1/b-1$ and $\mu_{1}(b)-\mu_{0}(b)=(1-1/b)^{2}/(2-1/b).$ Further $\sigma_{0}(b)=\{\hbox{Var}_{0}[l_{\mathbf{X}}(b)]\}^{1/2}=|1-1/b|$ and $\sigma_{1}(b)=\{\hbox{Var}_{1}[l_{\mathbf{X}}(b)]\}^{1/2}=|1-1/b|/(2-1/b).$ Hence $(\mu_{1}(b)-\mu_{0}(b))/\sigma_{1}(b)=|1-1/b|$ and $\sigma_{0}(b)/\sigma_{1}(b)=2-1/b.$ The asymptotic power function (10) is therefore $\Pi_{m}(f_{1}^{*}(b))=\Phi(\sqrt{m}\;|1-1/b|+(2-1/b)\,z_{\alpha})$ for $b>1/2.$

4 Summary and Discussion

The pdQ  transformation from a density function $f$ to $f^{*}$ extracts the important information of $f$ such as its asymmetry and tail behaviour and ignores the less critical information such as gaps, location and scale and thus provides a powerful tool in studying the shapes of density functions.

We found the directed divergences from uniformity of the pdQs of many standard location-scale families and used them to make a map locating each shape family relative to others and giving its distance from uniformity. We also found the most powerful tests of uniformity against alternative shapes and showed that their power functions are monotone increasing in the distances from the origin on the map.

In terms of the limiting behaviour of repeated applications of the pdQ mapping, when the density function $f$ is bounded, we showed that each application lowers its modal height and hence the resulting density function $f^{*}$ is closer to the uniform density than $f$. Furthermore, we established a necessary and sufficient condition for $f^{n*}$ converging in $L_{2}$ norm to the uniform density, giving a positive answer to a conjecture raised in Staudte, (2017). In particular, if $f$ is bounded, we proved that $f^{n*}$ converges in $L_{r}$ norm to the uniform density for any $r>0$. The fixed point theorems can be interpreted as follows. As we repeatedly apply the pdQ transformation, we keep losing information about the shape of the original $f$ and will eventually exhaust the information, leaving nothing in the limit, as represented by the uniform density, which means no points carry more information than other points. Thus the pdQ transformation plays a similar role to the difference operator in time series analysis where repeated applications of the difference operator to a time series with polynomial component lead to a white noise with a constant power spectral density (Brockwell & Davis,, 2009, p. 19).

We conjecture that every almost surely positive density $g$ on $[0,1]$ is a pdQ of a density function, hence uniquely represents a location-scale family. This is equivalent to saying that there exists a density function $f$ such that $g=f^{*}$. When $g$ satisfies $\int_{0}^{1}\frac{1}{g(t)}dt<\infty$, one can show that the cdf $F$ of $f$ can be uniquely (up to location-scale parameters) represented as $F(x)=H^{-1}(H(1)x)$, where $H(x)=\int_{0}^{x}\frac{1}{g(t)}dt$ (Professor A.D. Barbour, personal communication). The condition $\int_{0}^{1}\frac{1}{g(t)}dt<\infty$ is equivalent to saying that $f$ has bounded support and it is certainly not necessary, e.g., $g(x)=2x$ for $x\in[0,1]$ and $f(x)=e^{x}$ for $x<0$ (see Example 2 in Section 2.2).

In summary, the study of shapes of probability densities is facilitated by composing them with their own quantile functions, which puts them on the same finite support where they are absolutely continuous with respect to Lebesgue measure, and thus amenable to metric and semi-metric comparisons. In addition, we showed that further applications of this transformation, which intuitively reduces information and increases the relative entropy, is generally valid but requires a non-standard approach for proof. Similar results are likely to be obtainable in the multivariate case. Further research could investigate the relationship between relative entropy and tail-weight or distance from the class of symmetric pdQs.

Acknowledgments:  The authors thank Professor P.J. Brockwell for helpful commentary on an earlier version of this manuscript. The research by Professor Aihua Xia is supported by ARC Discovery Grant DP150101459