Characterizations through generalized and dual generalized order statistics, with an application to statistical prediction problem

Abstract

Some distribution functions have been characterized based on two non-adjacent generalized and dual generalized order statistics. Moreover, we show that these characterization properties provide a beneficial strategy to predict future events, which are based on past or current events and on an arbitrary distribution function.

Introduction

Kamps (1995) introduced the concept of generalized order statistics (GOS) as a unification of several models of ascendingly ordered random variables (RVs) with different interpretations. Ordinary order statistics (OOS), $k-$records (ordinary upper record values, when $k=1$), sequential order statistics (SOS), ordering via truncated distributions and censoring schemes can be discussed as they are special cases of the GOS. Ever since the advent of GOS, the use of such a model has been steadily growing along the years because it is a more capable of describing different reliability models.

The concept of dual generalized order statistics (DGOS) was introduced by Burkschat et al. (2003) to enable a common approach to descendingly ordered RVs like reversed OOS and lower records models.

In this work, we consider a wide subclasses of GOS and DGOS, known as $m$-generalized order statistics ($m-$GOS) and $m-$dual generalized order statistics ($m-$DGOS), which contain many important models of ordered RVs such as OOS, $k-$records, SOS and type II censored OOS. Let $F(.)$ be an arbitrary continuous distribution function (DF), with the probability density function (PDF) $f(.)$ and survival function $\overline{F}(.)=1-F(.)$. Then the RVs $X_{1,n;m,k}\le X_{2,n;m,k}\le \cdots \le X_{n,n;m,k}$ ($k>0,m\ge -1$) are said to be $m$-GOS, if their joint probability density function (JPDF) is given by (cf. Kamps, 1995)

$$f_{1,2,\dots ,n:n}^{(m,k)}(x_1,x_2,\dots ,x_n)=\left(\prod _{j=1}^n{\gamma }_j^{\left(n\right)}\right)\left(\prod _{j=1}^{n-1}{\overline{F}}^m\left(x_j\right)f\left(x_j\right)\right){\overline{F}}^{k-1}\left(x_n\right)f\left(x_n\right),$$

where $\underline{r}\left(F\right)\le x_1\le \cdots \le x_n\le \overline{r}\left(F\right)$, $\underline{r}\left(F\right)=inf\{x:F\left(x\right)>0\}$ (the left endpoint of the DF $F$), $\overline{r}\left(F\right)=sup\{x:F\left(x\right)<1\}$ (the right endpoint of the DF $F$) and ${\gamma }_j^{\left(n\right)}=k+(n-j)(m+1),j=1,2,\dots ,n$. On the other hand, the RVs $X_{1,n;m,k}^{\star }\ge X_{2,n;m,k}^{\star }\ge \cdots \ge X_{n,n;m,k}^{\star }$ are said to be $m$-DGOS, if their JPDF is given by (cf. Burkschat et al., 2003)

$$f_{1,2,\dots ,n:n}^{\star (m,k)}(x_1,x_2,\dots ,x_n)=\left(\prod _{j=1}^n{\gamma }_j^{\left(n\right)}\right)\left(\prod _{j=1}^{n-1}{F}^m\left(x_j\right)f\left(x_j\right)\right)F^{k-1}\left(x_n\right)f\left(x_n\right),$$

where $\underline{r}\left(F\right)\le x_n\le \cdots \le x_1\le \overline{r}\left(F\right)$.

From now onward, we omit $m$ of the notations $m$-GOS and $m$-DGOS to make them less cumbersome. In this paper, we consider only the case when $m\ne -1$ (i.e., we exclude the case of record values). When the $m>-1$, the marginal DFs of the $r$th GOS $X(r,n;m,k)$ (cf. Kamps, 1995) and DGOS $X^{\ast }(r,n;m,k)$ (cf. Burkschat et al., 2003) are given, respectively, by

$$f_{X(r,n,m,k)}=\frac{C_{r-1}^{\left(n\right)}}{(r-1)!{(m+1)}^{r-1}}{\overline{F}}^{{\gamma }_r^{\left(n\right)}-1}\left(x\right){\left[1-{\overline{F}}^{m+1}\left(x\right)\right]}^{r-1}f\left(x\right),\underline{r}\left(F\right)\le x\le \overline{r}\left(F\right)$$

and

$$f_{X^{\ast }(r,n,m,k)}=\frac{C_{r-1}^{\left(n\right)}}{(r-1)!{(m+1)}^{r-1}}{F}^{{\gamma }_r^{\left(n\right)}-1}\left(x\right){\left[1-F^{m+1}\left(x\right)\right]}^{r-1}f\left(x\right),\underline{r}\left(F\right)\le x\le \overline{r}\left(F\right),$$

where $C_{r-1}^{\left(n\right)}={\prod }_{i=1}^r{\gamma }_i^{\left(n\right)},1\le r\le n$. Clearly, (1.2) is obtained from (1.1) just by replacing $\overline{F}$ by $F$. Moreover, in (1.1), (1.2) by convention, we will write $X(0,n;m,k)=X{(n+1,n;m,k)}^{\star }=\underline{r}\ge -\infty$ and $X(n+1,n;m,k)=X{(0,n;m,k)}^{\star }=\overline{r}\le \infty$.