On Some Properties of Beta-Generated Distributions

Abstract

This note is concerned with a broad family of univariate distributions, namely the class of the beta-generated distributions which is created on the basis of the beta distribution by incorporating on this classic model a parent distribution function F with respective density f. The class of beta-generated distributions have received a great attention in the recent literature on distribution theory. The aim of this note is to provide with some properties and characterizations of the beta-generated distributions which are already known from the theory of order statistics.

It is a great honor to present this work to the memory of my beloved friend Pedro Gil, an exceptional scientist and an outstanding human personality.

5 Appendix: Proof of Proposition 2.2

$\left( \Longrightarrow \right)$ Suppose that $r_{\theta }(x)= \frac{f_{\theta }(x)}{1-F_{\theta }(x)}=w(\theta ),\theta \in \varTheta \subseteq \mathbb {R}$. Then,

$$\begin{aligned} \begin{array}{ll} \mathscr {H}_{Sh}(f_{\theta }) &{} =-\int _{\mathbb {R}} f_{\theta }(x)\ln f_{\theta }(x)dx \\ &{} =-\int _{\mathbb {R}} f_{\theta }(x)\ln \left[ r_{\theta }(x)\left( 1-F_{\theta }(x)\right) \right] dx \\ &{} =-\ln w(\theta )-\int _{\mathbb {R}} f_{\theta }(x)\ln \left( 1-F_{\theta }(x)\right) dx, \end{array} \end{aligned}$$

or

$$\begin{aligned} \mathscr {H}_{Sh}(f_{\theta })=-\ln w(\theta )-\int _{\mathbb {R}} B(1,1)g_{F_{\theta }}^{1,1}(x)\ln \left( 1-F_{\theta }(x)\right) dx. \end{aligned}$$

Taking into account that $B(1,1)=1$ and using Lemma 1(b) of Zografos and Balakrishnan [19], we get

$$\begin{aligned} \mathscr {H}_{Sh}(f_{\theta })=-\ln w(\theta )-\left[ \varPsi (1)-\varPsi (2)\right] . \end{aligned}$$

Based on

$$\begin{aligned} \varPsi (x+1)-\varPsi (x)=\frac{1}{x}, \end{aligned}$$

(7)

(cf. Gradshteyn and Ryzhik [11, §. 365]), it is obtained that,

$$\begin{aligned} \mathscr {H}_{Sh}(f_{\theta })=-\ln w(\theta )+1. \end{aligned}$$

(8)

On the other hand,

$$\begin{aligned} \begin{array}{ll} \mathscr {H}_{Sh}\left( g_{F_{\theta }}^{1,\beta }\right) &{} =-\int _{\mathbb {R}} g_{F_{\theta }}^{1,\beta }(x)\ln g_{F_{\theta }}^{1,\beta }(x)dx \\ &{} =-\int _{\mathbb {R}} g_{F_{\theta }}^{1,\beta }(x)\ln \left[ \frac{1}{B(1,\beta )} f_{\theta }(x)\left( 1-F_{\theta }(x)\right) ^{\beta -1}\right] dx \\ &{} =-\int _{\mathbb {R}} g_{F_{\theta }}^{1,\beta }(x)\ln \left[ \frac{1}{B(1,\beta )} r_{\theta }(x)\left( 1-F_{\theta }(x)\right) ^{\beta }\right] dx \end{array} \end{aligned}$$

(9)

and taking into account that $r_{\theta }(x)=w(\theta )$ and $B(1,\beta )=1/\beta$ it is obtained that

$$\begin{aligned} \mathscr {H}_{Sh}\left( g_{F_{\theta }}^{1,\beta }\right) =-\ln \beta -\ln w(\theta )-\beta \int _{\mathbb {R}} g_{F_{\theta }}^{1,\beta }(x)\ln \left( 1-F_{\theta }(x)\right) dx. \end{aligned}$$

Using again Lemma 1(b) of Zografos and Balakrishnan [19] and Eq. (7) above,

$$\begin{aligned} \begin{array}{ll} \mathscr {H}_{Sh}\left( g_{F_{\theta }}^{1,\beta }\right) &{} =-\ln \beta -\ln w(\theta )-\beta \left[ \varPsi (\beta )-\varPsi (\beta +1)\right] \\ &{} =-\ln \beta -\ln w(\theta )+1. \end{array} \end{aligned}$$

(10)

Equations (8) and (10) lead now to the desired result.

$\left( \Longleftarrow \right)$ Suppose that $\mathscr {H} _{Sh}(f_{\theta })-\mathscr {H}_{Sh}(g_{F_{\theta }}^{1,\beta })=\ln \beta$, for $\beta \ge 1.$ Based on (9) and on the identity $B(1,\beta )=1/\beta ,$

$$\begin{aligned} \mathscr {H}_{Sh}\left( g_{F_{\theta }}^{1,\beta }\right) =-\ln \beta -\beta \int _{\mathbb {R}} g_{F_{\theta }}^{1,\beta }(x)\ln \left( 1-F_{\theta }(x)\right) dx-\int _{\mathbb {R}} g_{F_{\theta }}^{1,\beta }(x)\ln r_{\theta }(x)dx. \end{aligned}$$

Using Lemma 1(b) of Zografos and Balakrishnan [19] and Eq. (7), we obtain

$$\begin{aligned} \int _{\mathbb {R}} g_{F_{\theta }}^{1,\beta }(x)\ln \left( 1-F_{\theta }(x)\right) dx=\varPsi (\beta )-\varPsi (\beta +1)=-\frac{1}{\beta }. \end{aligned}$$

Then, the above formula for $\mathscr {H}_{Sh}\left( g_{F_{\theta }}^{1,\beta }\right)$ is simplified as follows,

$$\begin{aligned} \begin{array}{ll} \mathscr {H}_{Sh}\left( g_{F_{\theta }}^{1,\beta }\right) &{} =-\ln \beta +1-\int _{\mathbb {R}} g_{F_{\theta }}^{1,\beta }(x)\ln r_{\theta }(x)dx \\ &{} =-\ln \beta +1-\int _{\mathbb {R}} \frac{1}{B(1,\beta )}f_{\theta }(x)\left( 1-F_{\theta }(x)\right) ^{\beta -1}\ln r_{\theta }(x)dx \\ &{} =-\ln \beta +1-\beta \int _{\mathbb {R}} f_{\theta }(x)\left( 1-F_{\theta }(x)\right) ^{\beta -1}\ln r_{\theta }(x)dx. \end{array} \end{aligned}$$

Because, by hypothesis, $\mathscr {H}_{Sh}(f_{\theta })-\mathscr {H} _{Sh}(g_{F_{\theta }}^{1,\beta })=\ln \beta$, using the previous expression for $\mathscr {H}_{Sh}\left( g_{F_{\theta }}^{1,\beta }\right)$, it is obtained

$$\begin{aligned} \mathscr {H}_{Sh}(f_{\theta })-1+\beta \int _{\mathbb {R}} f_{\theta }(x)\left( 1-F_{\theta }(x)\right) ^{\beta -1}\ln r_{\theta }(x)dx=0. \end{aligned}$$

(11)

Applying the transformation $u=1-F_{\theta }(x)$ to the last integral

$$\begin{aligned} \int _{\mathbb {R}} f_{\theta }(x)\left( 1-F_{\theta }(x)\right) ^{\beta -1}\ln r_{\theta }(x)dx=\int _{0}^{1} u^{\beta -1}\ln r_{\theta }\left( F_{\theta }^{-1}(1-u)\right) du. \end{aligned}$$

(12)

Equations (11) and (12) lead to

$$\begin{aligned} \mathscr {H}_{Sh}(f_{\theta })-1+\beta \int _{0}^{1} u^{\beta -1}\ln r_{\theta }\left( F_{\theta }^{-1}(1-u)\right) du=0 \end{aligned}$$

and taking into account that $\mathscr {H}_{Sh}(f_{\theta })$ does not depend on x or u,

$$\begin{aligned} \beta \int _{0}^{1} u^{\beta -1}\left\{ \mathscr {H}_{Sh}(f_{\theta })-1+\ln r_{\theta }\left( F_{\theta }^{-1}(1-u)\right) \right\} du=0. \end{aligned}$$

In view of Exercise 4 of Aliprantis and Burkinshaw (see [4, p. 90]) for $n=\beta -1, n \ge 0,$ it is obtained that

$$\begin{aligned} \ln r_{\theta }\left( F_{\theta }^{-1}(1-u)\right) +\mathscr {H} _{Sh}(f_{\theta })-1=0, \ \text { for }\beta \ge 1. \end{aligned}$$

The result now follows by using a similar argument as that of the last two lines in the proof of Theorem 2.1 of Barapour et al. [7].    $\square$