Moment generating functions of record values from a class of doubly truncated distributions

Abstract

In this paper, we consider a class of doubly truncated distributions. Based on this class, recurrence relation for moment generating functions of record values is derived. Recurrence relations for single and product moments of record values are deduced from the main results. Several of the known distributions can be seen as special cases of the used class, which includes the left, right and non-truncated distributions. So, we discuss in detail some examples of these distributions based on the main results. In general, for any distribution in the considered class, we reach to a general form to find recurrence relations of moment generating function and moments.

KEYWORDS:

• Record values
• moment generating function
• moments
• doubly truncated distributions

AMS SUBJECT CLASSIFICATIONS:

• 62E10
• 62G30
• 62G99

1. Introduction

There is a large volume of works based on the study of recurrence relations between moment generating functions of record values and characterizations based on these relations. The moments have considerable importance in the statistical literature. Many authors have studied and derived several identities and recurrence relations for the single and product moments. For example, from Burr Type XII distribution, Ahmad [1] derived recurrence relations for single and product moments of record values and their characterizations. Ahmad and Fawzy [2] studied the recurrence relations for moments of generalized order statistics (gos). Recurrence relation for moments of record values is derived by Kamps [3]. Recurrence relations for moment generating functions of record values and order statistics are considered by AL-Hussaini et al. [4]. For more details, the reader is referred to Saran and Singh [5], Khan and Zia [6], Kumar [7] and Bashir and Ahmad [8]. Record values have great importance since they use in several real-life problems involving economics, weather, industry and sports. It can be demonstrated as order statistics from a sample where its size is determined by the values and order of occurrence of the observations. Numerous papers have appeared discussing characterizations based on record values, such as, Balakrishnan and Balasubramanian [9], Franco and Ruiz [10,11], Wu [12,13], Bairamov [14]. For more details for survey of record values, see Ahsanullah et al. [15] and Hassan et al. [16]. In this paper, our focus is on recurrence relations for moment generating functions of record values from a class of continuous distributions in the doubly truncated case. Section 2 gives a recurrence relation for moment generating functions of record values, which can be applied to obtain relations for single and product moments of record values from this class. In Section 3, we apply such recurrence relation for moment generating functions of record values to characterize the doubly truncated class of continuous distributions.

Suppose that a random variable (rv) X has an absolutely continuous distribution function (df), as considered by AL-Hussaini [17] and it is given by $F\left(x\right)\equiv F_x(x;\theta )=1-e^{-\lambda \left(x\right)},x>0,$so that the probability density function (pdf) is written as $f\left(x\right)={\lambda }^{\mathrm{\prime }}\left(x\right)e^{-\lambda \left(x\right)},$where $\lambda \left(x\right)\equiv \lambda (x;\theta )$ is a nonnegative, strictly increasing and differentiable function of x such that $\lambda \left(x\right)\to 0$ as $x\to 0^{+}$ and $\lambda \left(x\right)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$. The function ${\lambda }^{\mathrm{\prime }}\left(x\right)$ is the derivative of $\lambda \left(x\right)$ with respect to x and represents the failure rate whereas $e^{-\lambda \left(x\right)}$represents the corresponding survival function. We shall write the class of distributions as (1) $\mathrm{\Im }=\left\{F:F\left(x\right)=1-e^{-\lambda \left(x\right)},x>0\right\}.$(1)

A doubly truncated pdf on $[P_1,Q_1]$, denoted by $f_d\left(x\right)$, is given by (2) $f_d\left(x\right)=A_d{\lambda }^{\mathrm{\prime }}\left(x\right)e^{-\lambda \left(x\right)},P_1\le x\le Q_1,(P_1\ge 0,Q_1<\mathrm{\infty }),$(2) where (3) $A_d=1/[e^{-\lambda \left(P_1\right)}-e^{-\lambda \left(Q_1\right)}].$(3)

The doubly truncated df and survival function (sf) are given, respectively, for $0\le P_1\le x\le Q_1\le \mathrm{\infty }$, by (4) $F_d\left(x\right)=A_d\left[e^{-\lambda \left(P_1\right)}-e^{-\lambda \left(x\right)}\right],$(4) and (5) ${\overline{F}}_d\left(x\right)=Q_2+\frac{f_d\left(x\right)}{{\lambda }^{\mathrm{\prime }}\left(x\right)},$(5) where ${\overline{F}}_d(.)=1-F_d(.)$,$F_d(.)$ is given by (4) and (6) $Q_2=-A_d{e}^{-\lambda \left(Q_1\right)}=e^{-\lambda \left(Q_1\right)}/\left[e^{-\lambda \left(Q_1\right)}-e^{-\lambda \left(P_1\right)}\right].$(6)

Note that ${\overline{F}}_d\left(P_1\right)=1-F_d\left(P_1\right)=1$ and ${\overline{F}}_d\left(Q_1\right)=0$.

The doubly truncated class denoted by ${\mathrm{\Im }}_d$. Then, for $P_1\le x\le Q_1,P_1\ge 0,Q_1<\mathrm{\infty }),$ (7) ${\mathrm{\Im }}_d=\left\{F_d:F_d\left(x\right)=\left[e^{-\lambda P_1}-e^{-\lambda \left(x\right)}\right]/\left[e^{-\lambda P_1}-e^{-\lambda \left(Q_1\right)}\right]\right\}.$(7)

The non-truncated, right- and left-truncated classes are denoted by $\mathrm{\Im }$, ${\mathrm{\Im }}_R$ and ${\mathrm{\Im }}_L$, respectively. The right truncated class ${\mathrm{\Im }}_R$ is given by (8) $\begin{array}{rl}{\mathrm{\Im }}_R=\{F_R:F_R(x)& =[1-e^{-\lambda \left(x\right)}]/[1-e^{-\lambda \left(Q_1\right)}]\\ & 0\le x\le Q_1,Q_1<\mathrm{\infty }\},\end{array}$(8) where $\lambda \left(x\right)\to 0$ as $x\to 0^{+}$.

The left-truncated class is given by (9) ${\mathrm{\Im }}_L=\left\{F_L:F_L\left(x\right)=1-e^{-\left[\lambda \right(x)-\lambda (P_1\left)\right]},x\ge P_1,P_1>0\right\},$(9) where $\lambda \left(x\right)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$.

2. Recurrence relation for moment generating functions

Let $X_{U\left(n_1\right)}\le X_{U\left(n_2\right)}\le \cdot \cdot \cdot \le X_{U\left(n_k\right)}$, be k upper record values based on a sequence $\left\{X_j,j\ge 1\right\}$ of identically and independently distributed (i.i.d) rvs having pdf (2). The joint pdf of $X_{U\left(n_1\right)},X_{U\left(n_2\right)},\dots ,X_{U\left(n_k\right)}$,$n_1<n_2<\cdot \cdot \cdot <n_k,n_0=0$, for $0\le x_1<x_2<\cdot \cdot \cdot <x_k<\mathrm{\infty },$is given by (10) $\begin{array}{rl}& f_{n_1,n_2,\dots ,n_k}(x_1,x_2,\dots ,x_k)\\ & =\prod _{i=0}^{k-1}\frac{{\left[R\right(x_{i+1})-R(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}r\left(x_{i+1}\right),\end{array}$(10) where $R\left(z\right)=-\ln \left[\overline{F}\right(z\left)\right],n_0=0,R\left(x_0\right)=0,r\left(x_k\right)=f\left(x_k\right),$ (11) $r\left(x_i\right)=\frac{f\left(x_i\right)}{\overline{F}\left(x_i\right)},i=1,\dots ,k-1.$(11)

For more details, see Arnold et al. [18]. Note that (10) is based on k upper records, whereas the corresponding expression in the previous reference is based on k + 1 upper records (including the zeroth record). We shall use $M_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)$ and ${\mu }_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}$, to denote the joint moment generating function and joint moments of $X_{U\left(n_1\right)}^{a_1},X_{U\left(n_2\right)}^{a_2},\dots ,X_{U\left(n_k\right)}^{a_k}$, respectively, where $a_1,\dots ,a_k$are nonnegative real numbers. For simplicity, we usually drop $U\left(n_i\right)$ in the computations and be satisfied with i, so, for example we write $x_i$ instead of $x_{U\left(n_i\right)}$. In other words, (12) $\begin{array}{rl}& M_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =E\left[\exp \left\{\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\right]\\ & ={\int }_0^{\mathrm{\infty }}{\int }_{x_1}^{\mathrm{\infty }}\cdot \cdot \cdot {\int }_{x_{k-1}}^{\mathrm{\infty }}\\ & \times \exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times f_{n_1,n_2,\dots ,n_k}(x_1,x_2,\dots ,x_k)d{x}_k\dots d{x}_{2}d{x}_1,\end{array}$(12) and (13) $\begin{array}{rl}& {\mu }_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}=E\left[\prod _{i=1}^k{X}_{U\left(n_i\right)}^{a_i}\right]\\ & ={\int }_0^{\mathrm{\infty }}{\int }_{x_1}^{\mathrm{\infty }}\cdot \cdot \cdot {\int }_{x_{k-1}}^{\mathrm{\infty }}\left[\prod _{i=1}^k{x}_i^{a_i}\right]\\ & \times f_{n_1,n_2,\dots ,n_k}(x_1,x_2,\dots ,x_k)d{x}_k\dots d{x}_{2}d{x}_1,\end{array}$(13) where $f_{n_1,n_2,\dots ,n_k}(x_1,x_2,\dots ,x_k)$is the joint pdf given by (10).

It may be observed that $X_{U\left(n_i\right)}<X_{U(n_k+1)}$, then we can write a joint moment generating function as; $\begin{array}{rl}& E\left[\exp \left\{\sum _{i=1}^{k-1}{t}_i{X}_{U\left(n_i\right)}^{a_i}+t_k{X}_{U(n_k+1)}^{a_k}\right\}\right]\\ & =M_{n_1,n_2,\dots ,n_{k-1},n_k+1}^{(a_1,a_2,\dots ,a_{k-1},a_k)}(t_1,\dots ,t_{k-1},t_k).\end{array}$

In the following theorem, a recurrence relation for moment generating functions from the doubly truncated df (4), is obtained for record values.

Theorem 2.1

Let $X_{U\left(n_1\right)},X_{U\left(n_2\right)},\dots ,X_{U\left(n_k\right)}$be k upper records based on a sequence $\left\{X_j,j\ge 1\right\}$of i.i.d rvs having the doubly truncated pdf (2). For nonnegative integers $a_1,a_2,\dots ,a_k\ge 1,1\le n_1<n_2<\cdot \cdot \cdot <n_k,$the following recurrence relation for the moment generating function of $X_{U\left(n_1\right)}^{a_1},X_{U\left(n_2\right)}^{a_2},\dots ,X_{U\left(n_k\right)}^{a_k}$, is satisfied.

(14) $\begin{array}{rl}& M_{n_1,n_2,\dots ,n_{k-1},n_k+1}^{(a_1,a_2,\dots ,a_{k-1},a_k)}(t_1,\dots ,t_{k-1},t_k)\\ & -M_{n_1,n_2,\dots ,n_{k-1},n_k}^{(a_1,a_2,\dots ,a_{k-1},a_k)}(t_1,\dots ,t_{k-1},t_k)\\ & =G_{n_1,\dots ,n_{k-2},n_{k-1},n_k+1}^{(a_1,\dots a_{k-2},a_{k-1},a_k)}(t_1,\dots ,t_{k-2},t_{k-1},t_k),\end{array}$(14) where (15) $\begin{array}{rl}& G_{n_1,\dots ,n_{k-2},n_{k-1},n_k+1}^{(a_1,\dots a_{k-2},a_{k-1},a_k)}(t_1,\dots ,t_{k-2},t_{k-1},t_k)\\ & =t_k{a}_{k}E\left[\frac{X_{U(n_k+1)}^{a_k-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(n_k+1)}\right)}\right.\\ & \times \left.\exp \left\{\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\left\{1-e^{\lambda \left(X_{U(n_k+1)}\right)-\lambda \left(Q_1\right)}\right\}\right].\end{array}$(15)

On the other hand, if condition (14) is satisfied, then the survival function corresponding to $F_d\left(x\right)$ is given by (5). All of the expectations and moment generating functions involved are assumed to be exist.

Proof:

By replacing f and $\overline{F}$in (10) and (11) by $f_d$ and ${\overline{F}}_d$, respectively, the resulting joint $f_{n_1,n_2,\dots ,n_k}(x_1,x_2,\dots ,x_k)$ is used in definition (12) to obtain the joint moment generating function

(16) $\begin{array}{rl}& M_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)\\ & ={\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times \left\{\prod _{i=0}^{k-1}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times d{x}_k\dots d{x}_{2}d{x}_1\\ & ={\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-2}}^{Q_1}\exp \left\{\sum _{i=1}^{k-1}{t}_i{x}_i^{a_i}\right\}I\left(x_{k-1}\right)\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times d{x}_{k-1}\dots d{x}_{2}d{x}_1,\end{array}$(16) where $\begin{array}{rl}I\left(x_{k-1}\right)& ={\int }_{x_{k-1}}^{Q_1}\exp \left(t_k{x}_k^{a_k}\right)\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}-1}}{(n_k-n_{k-1}-1)!}{f}_d\left(x_k\right)d{x}_k\\ & =\frac{1}{(n_k-n_{k-1}-1)!}{\int }_{x_{k-1}}^{Q_1}\exp \left(t_k{x}_k^{a_k}\right){\overline{F}}_d\left(x_k\right)\\ & \times d{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}},\end{array}$and $R_d\left(x\right)=-\ln \left[{\overline{F}}_d\right(x\left)\right]$. Using integrating by parts, we obtain $\begin{array}{rl}I\left(x_{k-1}\right)& ={\int }_{x_{k-1}}^{Q_1}\exp \left(t_k{x}_k^{a_k}\right)\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}{f}_d\left(x_k\right)d{x}_k\\ & -t_k{a}_k{\int }_{x_{k-1}}^{Q_1}{x}_k^{a_k-1}\exp \left(t_k{x}_k^{a_k}\right)\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}{\overline{F}}_d\left(x_k\right)d{x}_k.\end{array}$

By substituting $I\left(x_{k-1}\right)$in (16), we obtain (17) $\begin{array}{rl}& M_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)\\ & ={\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}{f}_d\left(x_k\right)\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}d{x}_k\dots d{x}_{2}d{x}_1\\ & -t_k{a}_k{\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}{x}_k^{a_k-1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}{\overline{F}}_d\left(x_k\right)\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times d{x}_k\dots d{x}_{2}d{x}_1.\end{array}$(17)

The first term in (17) is then $M_{n_1,n_2,\dots ,n_k+1}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)$, and Equation (17) becomes $\begin{array}{rl}& M_{n_1,n_2,\dots ,n_k+1}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)-M_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =t_k{a}_k{\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}{x}_k^{a_k-1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}{\overline{F}}_d\left(x_k\right)\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times d{x}_k\dots d{x}_{2}d{x}_1.\end{array}$

From (2), (5) and (6), we have $\frac{{\overline{F}}_d\left(x\right)}{f_d\left(x\right)}=\frac{1-e^{\lambda \left(x\right)-\lambda \left(Q_1\right)}}{{\lambda }^{\mathrm{\prime }}\left(x\right)}.$

Using this relation in the second term, we obtain (18) $\begin{array}{rl}& M_{n_1,n_2,\dots ,n_k+1}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)-M_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =t_k{a}_k{\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}{x}_k^{a_k-1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}\frac{{\overline{F}}_d\left(x_k\right)}{f_d\left(x_k\right)}\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times f_d\left(x_k\right)d{x}_k\dots d{x}_{2}d{x}_1\\ & =t_k{a}_k{\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}{x}_k^{a_k-1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}{\overline{F}}_d\left(x_k\right)\\ & \times \frac{1-e^{-\left[\lambda \right(x_k)-\lambda (Q_1\left)\right]}}{{\lambda }^{\mathrm{\prime }}\left(x_k\right)}\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times f_d\left(x_k\right)d{x}_k\dots d{x}_{2}d{x}_1\\ & =t_k{a}_{k}E\left[\frac{X_{U(n_k+1)}^{a_k-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(n_k+1)}\right)}\exp \left\{\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\right.\\ & \times \left.\left\{1-e^{\lambda \left(X_{U(n_k+1)}\right)-\lambda \left(Q_1\right)}\right\}\phantom{\frac{X_{U(n_k+1)}^{a_k-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(n_k+1)}\right)}}\right].\end{array}$(18)

On the other hand, if condition (14) is satisfied, then using (18), (12) and (10), such condition may be rewritten as $\begin{array}{rl}& {\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}{x}_k^{a_k-1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}\\ & \times \frac{1-e^{-\left[\lambda \right(x_k)-\lambda (Q_1\left)\right]}}{{\lambda }^{\mathrm{\prime }}\left(x_k\right)}\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times f_d\left(x_k\right)d{x}_k\dots d{x}_{2}d{x}_1\\ & ={\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}{x}_k^{a_k-1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}{\overline{F}}_d\left(x_k\right)\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times d{x}_k\dots d{x}_{2}d{x}_1,\end{array}$or, equivalently, $\begin{array}{rl}& {\int }_{P_1}^{Q_1}{\int }_{x_1}^{Q_1}\cdot \cdot \cdot {\int }_{x_{k-1}}^{Q_1}{x}_k^{a_k-1}\exp \left\{\sum _{i=1}^k{t}_i{x}_i^{a_i}\right\}\\ & \times \frac{{\left[R_d\right(x_k)-R_d(x_{k-1}\left)\right]}^{n_k-n_{k-1}}}{(n_k-n_{k-1})!}\\ & \times \left[{\overline{F}}_d\left(x_k\right)-\frac{1-e^{\lambda \left(x_k\right)-\lambda \left(Q_1\right)}}{{\lambda }^{\mathrm{\prime }}\left(x_k\right)}{f}_d\left(x_k\right)\right]\\ & \times \left\{\prod _{i=0}^{k-2}\frac{{\left[R_d\right(x_{i+1})-R_d(x_i\left)\right]}^{n_{i+1}-n_i-1}}{(n_{i+1}-n_i-1)!}{r}_d\left(x_{i+1}\right)\right\}\\ & \times d{x}_k\dots d{x}_{2}d{x}_1=0.\end{array}$

Applying the extension of Müntz-Szász theorem [see, Hwang and Lin [19]], we obtain $\begin{array}{rl}0& ={\overline{F}}_d\left(x_k\right)-\frac{1-e^{\lambda \left(x_k\right)-\lambda \left(Q_1\right)}}{{\lambda }^{\mathrm{\prime }}\left(x_k\right)}{f}_d\left(x_k\right)\\ & ={\overline{F}}_d\left(x_k\right)-\left[\frac{f_d\left(x_k\right)}{{\lambda }^{\mathrm{\prime }}\left(x_k\right)}+\left\{-A_d{e}^{-\lambda \left(Q_1\right)}\right\}\right].\end{array}$

Thus $f_d\left(x\right)$ has the pdf as given in (2).

Special cases

(1) By differentiating (14) with respect to $t_1,\dots ,t_k$and putting $t_1=\cdot \cdot \cdot =t_k=0$, we obtain the following recurrence relation for product moments of record values (19) ${\mu }_{n_1,n_2,\dots ,n_k+1}^{(a_1,a_2,\dots ,a_k)}-{\mu }_{n_1,n_2,\dots ,n_k}^{(a_1,a_2,\dots ,a_k)}=G_{n_1,n_2,\dots ,n_k+1}^{(a_1,a_2,\dots ,a_k)},$(19) where (20) $\begin{array}{rl}& G_{n_1,n_2,\dots ,n_k+1}^{(a_1,a_2,\dots ,a_k)}\\ & =a_{k}E\left[\frac{\prod _{i=1}^{k-1}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(n_k+1)}\right)}\right.\\ & \times \left.\left\{1-e^{\lambda \left(X_{U(n_k+1)}\right)-\lambda \left(Q_1\right)}\right\}\phantom{\frac{\prod _{i=1}^{k-1}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(n_k+1)}\right)}}\right],\end{array}$(20) and ${\mu }_{n_1,n_2,\dots ,n_k+1}^{(a_1,a_2,\dots ,a_k)}=E\left[\prod _{i=1}^{k-1}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k}\right].$

(2) If we put $a_1=\cdot \cdot \cdot =a_{k-1}=0$, $a_k=a,n_k=m$, in (20) and, in addition, $t_k=t$, in (15), we obtain recurrence relations for the moment generating functions and moments for the single record value $X_{U\left(m\right)}^a$. The moment generating function is given by (21) $M_{m+1}^{\left(a\right)}\left(t\right)-M_m^{\left(a\right)}\left(t\right)=G_{m+1}^{\left(a\right)}\left(t\right),$(21) where (22) $\begin{array}{rl}& G_{m+1}^{\left(a\right)}\left(t\right)\\ & =taE\left.\left[\frac{X_{U(m+1)}^{a-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(m+1)}\right)}{e}^{t{X}_{U\left(m\right)}^a}\right.\left\{1-e^{\lambda \left(X_{U(m+1)}\right)-\lambda \left(Q_1\right)}\right\}\right].\end{array}$(22)

This leads to the moment generating function of the $X_{U\left(m\right)}^a$ record as (23) $M_m^{\left(a\right)}\left(t\right)=M_1^{\left(a\right)}\left(t\right)+\sum _{j=2}^m{G}_j^{\left(a\right)}\left(t\right),m=2,3,\dots ,$(23) where $M_1^{\left(a\right)}\left(t\right)=E\left[e^{t{X}_{U\left(1\right)}^a}\right]$ and $G_j^{\left(a\right)}\left(t\right)$is given by (22) after replacing $(m+1)$ by j.

A recurrence relation for the moments of single upper records is given by (24) ${\mu }_{m+1}^{\left(a\right)}-{\mu }_m^{\left(a\right)}=G_{m+1}^{\left(a\right)},$(24) where (25) $G_{m+1}^{\left(a\right)}=aE\left.\left[\frac{X_{U(m+1)}^{a-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(m+1)}\right)}\right.\left\{1-e^{\lambda \left(X_{U(m+1)}\right)-\lambda \left(Q_1\right)}\right\}\right].$(25)

This leads to the expectation of the $X_{U\left(m\right)}^a$, which can be written as follows (26) ${\mu }_m^{\left(a\right)}={\mu }_1^{\left(a\right)}+\sum _{j=2}^m{G}_j^{\left(a\right)},m=2,3,\dots ,$(26) where ${\mu }_1^{\left(a\right)}=E\left[X_{U\left(1\right)}^a\right]$ and $G_j^{\left(a\right)}$ is given by (25) after replacing $(m+1)$ by j.

(3) If we put $a_1=\cdot \cdot \cdot =a_{k-2}=0$, $a_{k-1}=a,a_k=b,n_{k-1}=m,n_k=n$, and$t_{k-1}=t_1,t_k=t_2$, in (14) and (20), we obtain recurrence relations for the moment generating functions and moments of $X_{U\left(m\right)}^a$ and $X_{U\left(m\right)}^b$. The moment generating function is given by (27) $M_{m,n+1}^{(a,b)}(t_1,t_2)-M_{m,n}^{(a,b)}(t_1,t_2)=G_{m,n+1}^{(a,b)}(t_1,t_2),$(27) where (28) $\begin{array}{rl}{G}_{m,n+1}^{(a,b)}(t_1,t_2)& =b{t}_{2}E\left[\frac{X_{U(n+1)}^{b-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(n+1)}\right)}{e}^{[t_1{X}_{U\left(m\right)}^a+t_2{X}_{U(n+1)}^b]}\right.\\ & \times \left.\left\{1-e^{\lambda \left(X_{U(n+1)}\right)-\lambda \left(Q_1\right)}\right\}\phantom{\frac{X_{U(n+1)}^{b-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(n+1)}\right)}}\right].\end{array}$(28)

This leads to (29) $\begin{array}{rl}& M_{m,n}^{(a,b)}(t_1,t_2)-M_{m,1}^{(a,b)}(t_1,t_2)\\ & =\sum _{j=2}^n{G}_{m,j}^{(a,b)}(t_1,t_2),n=2,3,\dots ,\end{array}$(29) where $M_{m,1}^{(a,b)}(t_1,t_2)=e^{[t_1{X}_{U\left(m\right)}^a+t_2{X}_{U\left(1\right)}^b]}$and $G_{m,j}^{(a,b)}(t_1,t_2)$ is given by (28) after replacing $(n+1)$ by j.

A recurrence relation for the mixed moments is given by (30) ${\mu }_{m,n}^{(a,b)}-{\mu }_{m,1}^{(a,b)}=G_{m,n+1}^{(a,b)},$(30) where (31) $G_{m,n+1}^{(a,b)}=bE\left[\left.\frac{X_{U\left(m\right)}^a{X}_{U(n+1)}^{b-1}}{{\lambda }^{\mathrm{\prime }}\left(X_{U(n+1)}\right)}\left\{1-e^{\lambda \left(X_{U(n+1)}\right)-\lambda \left(Q_1\right)}\right\}\right]\right..$(31)

This leads to (32) ${\mu }_{m,n}^{(a,b)}-{\mu }_{m,1}^{(a,b)}=\sum _{j=2}^n{G}_{m,j}^{(a,b)},$(32) where ${\mu }_{m,1}^{(a,b)}=E[X_{U\left(m\right)}^a+X_{U\left(1\right)}^b]$and $G_{m,j}^{(a,b)}$ is given by (31) after replacing $(n+1)$ by j.

Remark 2.1:

In all special cases of the doubly truncated distributions, recurrence relations remain the same, yet the following should be observed when computing the expectations.

In the non-truncated or left-truncated cases, where $Q_2=0$, (or $Q_1=\mathrm{\infty }$), we use $\overline{F}(.)$ such that $\overline{F}(.)=\frac{f(.)}{{\lambda }^{\mathrm{\prime }}(.)}$ for the non-truncated case and ${\overline{F}}_L(.)=\frac{f_L(.)}{{\lambda }^{\mathrm{\prime }}(.)}$ for the left-truncated case.

In the right truncated case, we use the following equation in the computation of the expectations, ${\overline{F}}_R(.)=Q_2+\frac{f_R(.)}{{\lambda }^{\mathrm{\prime }}(.)},\text{ where }{Q}_2=-e^{-\lambda \left(Q_1\right)}/[1-e^{-\lambda \left(Q_1\right)}].$

3. Examples

In this section, we shall apply previous results to write recurrence relations (RR) based on some members of class (4).

3.1. Doubly truncated Weibull distribution

Let $\lambda \left(x\right)=\text{β}{x}^{\gamma }$and ${\lambda }^{\mathrm{\prime }}\left(x\right)=\text{β}\gamma x^{\gamma -1}$.

Then the CDF of doubly truncated Weibull distribution is given by $F_d\left(x\right)=\frac{e^{-\text{β}{P_1}^{\gamma }}-e^{-\text{β}{x}^{\gamma }}}{e^{-\text{β}{P_1}^{\gamma }}-e^{-\text{β}{Q_1}^{\gamma }}}.$

(i) For $a_k\ge \gamma ,k=1,2,\dots ,$which satisfy the conditions of the theorem, it follows from (15), so that we can write $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =\frac{t_k{a}_k}{\text{β}\gamma }E\left[X_{U(n_k+1)}^{a_k-\gamma }\exp \left\{\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\right.\\ & \times \left.\left\{1-e^{\text{β}(X_{U(n_k+1)}^{\gamma }-Q_1^{\gamma })}\right\}\phantom{\sum _{i=1}^k}\right].\end{array}$

From (20), the rhs of RR (19) is given, by $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}\\ & =\frac{a_k}{\text{β}\gamma }E\left[\prod _{i=1}^{k-1}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k-\gamma }\left\{1-e^{\text{β}(X_{U(n_k+1)}^{\gamma }-Q_1^{\gamma })}\right\}\right].\end{array}$

(ii) For $a\ge \gamma ,m=1,2,\dots ,$it follows from (22) and it is easy to get that $G_{m+1}^{\left(a\right)}\left(t\right)=\frac{at}{\text{β}\gamma }E\left[X_{U(m+1)}^{a-\gamma }{e}^{t{X}_{U\left(m\right)}^a}\left\{1-e^{\text{β}(X_{U(m+1)}^{\gamma }-Q_1^{\gamma })}\right\}\right].$

From (25), the rhs of RR (24) is given by $G_{m+1}^{\left(a\right)}=\frac{a}{\text{β}\gamma }E\left[X_{U(m+1)}^{a-\gamma }\left\{1-e^{\text{β}(X_{U(m+1)}^{\gamma }-Q_1^{\gamma })}\right\}\right].$

(iii) For $b\ge \gamma ,m,n=1,2,\dots ,$and using (28), we obtain $\begin{array}{rl}& G_{m,n+1}^{(a,b)}(t_1,t_2)\\ & =\frac{b{t}_2}{\text{β}\gamma }E[X_{U(n+1)}^{b-\gamma }{e}^{[t_1{X}_{U\left(m\right)}^a+t_2{X}_{U(n+1)}^b]}\\ & \times \left\{1-e^{\text{β}(X_{U(n+1)}^{\gamma }-Q_1^{\gamma })}\right\}].\end{array}$

Also, using (31), we get $G_{m,n+1}^{(a,b)}=\frac{b}{\text{β}\gamma }E\left[X_{U(n+1)}^{b-\gamma }{X}_{U\left(m\right)}^a\left\{1-e^{\text{β}(X_{U(n+1)}^{\gamma }-Q_1^{\gamma })}\right\}\right].$

Remark 3.1:

(i) In the non-truncated case, $Q_1=\mathrm{\infty }$. So that, the terms $\left\{1-e^{\text{β}(X_{U(n+1)}^{\gamma }-Q_1^{\gamma })}\right\}$ reduce to 1. In this case, it follows from (22), so that we can write $G_{j+1}^{\left(a\right)}\left(t\right)=\frac{at}{\text{β}\gamma }E\left[X_{U(j+1)}^{a-\gamma }{e}^{t{X}_{U\left(j\right)}^a}\right].$

It can be shown that $\begin{array}{rl}{M}_1^{\left(a\right)}\left(t\right)& =E\left[e^{t{X}_{U\left(1\right)}^a}\right]\\ & ={\text{β}}^2\gamma {\int }_0^{\mathrm{\infty }}{x}^{2\gamma -1}\exp [-(\text{β}{x}^{\gamma }-t{x}^a\left)\right]dx,\end{array}$and (33) $\begin{array}{rl}{G}_{j+1}^{\left(a\right)}\left(t\right)& =\frac{at{\text{β}}^{j+1}}{j!}{\int }_0^{\mathrm{\infty }}{I}_{n+1}\left(y\right)y^{a-1}{e}^{-\text{β}{y}^{\gamma }}dy,I_{j+1}\left(y\right)\\ & ={\int }_0^y{x}^{\gamma (j+1)-1}{e}^{t{x}^a}dx.\end{array}$(33)

This is true since in the non-truncated Weibull $(\text{β},\gamma )$ case, we obtain (34) $\begin{array}{rl}{G}_{j+1}^{\left(a\right)}\left(t\right)& =\frac{at}{\text{β}\gamma }E\left[X_{U(j+1)}^{a-\gamma }{e}^{t{X}_{U\left(j\right)}^a}\right]\\ & =\frac{at}{\text{β}\gamma }{\int }_0^{\mathrm{\infty }}{\int }_0^y{y}^{a-\gamma }{e}^{t{x}^a}{x}^{\gamma (j+1)-1}{f}_{j,j+1}(x,y)dxdy,\end{array}$(34) where $f_{j,j+1}(x,y)$ is given by Arnold et al. [19] as $f_{j,j+1}(x,y)=\frac{{[-\ln \overline{F}(x\left)\right]}^j}{j!}\frac{f\left(x\right)f\left(y\right)}{\overline{F}\left(x\right)},x<y.$

In the non-truncated Weibull $(\text{β},\gamma )$ case, we can see that $\overline{F}\left(x\right)=e^{-\text{β}{x}^{\gamma }}\text{and}f\left(x\right)=\text{β}\gamma x^{\gamma -1}{e}^{-\text{β}{x}^{\gamma }},x>0.$

By substituting in (34), we obtain $\begin{array}{rl}{G}_{j+1}^{\left(a\right)}\left(t\right)& =\frac{at}{\text{β}\gamma }{\int }_0^{\mathrm{\infty }}{\int }_0^y{y}^{a-\gamma }{e}^{t{x}^a}\\ & \times \frac{{\left(\text{β}{x}^{\gamma }\right)}^j}{j!}\text{β}\gamma x^{\gamma -1}\text{β}\gamma y^{\gamma -1}{e}^{-\text{β}{y}^{\gamma }}dxdy\\ & =\frac{at\gamma {\text{β}}^{j+1}}{j!}{\int }_0^{\mathrm{\infty }}{I}_{n+1}\left(y\right)y^{a-1}{e}^{-\text{β}{y}^{\gamma }}dy,\end{array}$where $I_{j+1}\left(y\right)={\int }_0^y{x}^{\gamma (j+1)-1}{e}^{t{x}^a}dx.$

By substituting the values of $M_1^a\left(t\right)$ and $G_{j+1}^a\left(t\right)$ in (23), we obtain a RR for the mgf of powers of single upper record values.

In the non-truncated Weibull $(\text{β},\gamma )$ case, the same RR (26) is obtained except that in this case, $G_j^{\left(a\right)}=\frac{a}{\text{β}\gamma }E\left[X_{U\left(j\right)}^{a-\gamma }\right],j=2,\dots ,m.$ It then follows, from (26), so we can write that $\begin{array}{rl}{\mu }_1^{\left(a\right)}& =\frac{1}{{\text{β}}^{a/\gamma }}\mathrm{\Gamma }(2+\frac{a}{\gamma }),\\ {\mu }_2^{\left(a\right)}& =\mathrm{\Gamma }(1+\frac{a}{2\gamma }){\mu }_1^{\left(a\right)},\dots ,\\ {\mu }_{m+1}^{\left(a\right)}& =\mathrm{\Gamma }(1+\frac{a}{(m+1)\gamma }){\mu }_m^{\left(a\right)}.\end{array}$

In this case, it can be shown that ${\mu }_n^{\left(a\right)}=\frac{\mathrm{\Gamma }(n+1+\frac{a}{\gamma })}{\left[{\text{β}}^{a/\gamma }\mathrm{\Gamma }\right(n+1\left)\right]},n=1,2,\dots .$

This RR coincides with the RR obtained by Balakrishnan and Chan [20] where they used $\text{β}=1,\gamma =c\text{and}a=k.$

(ii) The exponential distribution is obtained by setting $\gamma =1$ in the Weibull $(\text{β},\gamma )$ distribution. That is, in the exponential $\left(\text{β}\right)$ distribution, $\lambda \left(x\right)=\text{β}x$.

(iii) The Rayleigh distribution is obtained by setting $\gamma =2$ in the Weibull $(\text{β},\gamma )$ distribution. That is, in the Rayleigh $\left(\text{β}\right)$ distribution, $\lambda \left(x\right)=\text{β}{x}^2$.

3.2. Doubly truncated compound Weibull (three-parameter Burr Type XII) distribution

Let $\lambda \left(x\right)=\gamma \ln \left(1+x^{\theta }/\text{β}\right)\text{and}{\lambda }^{\mathrm{\prime }}\left(x\right)=\gamma \theta x^{\theta -1}/\left(\text{β}+x^{\theta }\right).$

Then the CDF of doubly truncated compound Weibull distribution is given by $F_d\left(x\right)=\frac{{(1+{P_1}^{\text{β}}/\text{β})}^{-\gamma }-{(1+x^{\text{β}}/\text{β})}^{-\gamma }}{{(1+{P_1}^{\text{β}}/\text{β})}^{-\gamma }-{(1+{Q_1}^{\text{β}}/\text{β})}^{-\gamma }}.$

(i) For $a_k\ge \theta ,k=1,2,\dots ,$ it follows from (15), we can obtain that $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =\frac{t_k{a}_k}{\theta \gamma }E\left[X_{U(n_k+1)}^{a_k-\theta }(\text{β}+x_{U(n_k+1)}^{\theta })\exp \left\{\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\right.\\ & \times \left.\left\{1-{\left(\frac{\text{β}+x_{U(n_k+1)}^{\theta }}{\text{β}+Q_1^{\theta }}\right)}^{\gamma }\right\}\phantom{\sum _{i=1}^k}\right].\end{array}$

From (20), we have $\begin{array}{rl}{G}_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}& =\frac{a_k}{\theta \gamma }E\left[\prod _{i=1}^{k-1}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k-\theta }(\text{β}+X_{U(n_k+1)}^{\theta })\right.\\ & \times \left.\left\{1-{\left(\frac{\text{β}+X_{U(n_k+1)}^{\theta }}{\text{β}+Q_1^{\theta }}\right)}^{\gamma }\right\}\phantom{\prod _{i=1}^{k-1}}\right].\end{array}$

(ii) For $a\ge \theta ,m=1,2,\dots ,$it follows from (22) and we can write $\begin{array}{rl}{G}_{m+1}^{\left(a\right)}\left(t\right)& =\frac{at}{\theta \gamma }E\left[\phantom{\left\{1-{\left(\frac{\text{β}+X_{U(m+1)}^{\theta }}{\text{β}+Q_1^{\theta }}\right)}^{\gamma }\right\}}{X}_{U(m+1)}^{a-\theta }(\text{β}+X_{U(m+1)}^{\theta })e^{t{X}_{U\left(m\right)}^a}\right.\\ & \times \left.\left\{1-{\left(\frac{\text{β}+X_{U(m+1)}^{\theta }}{\text{β}+Q_1^{\theta }}\right)}^{\gamma }\right\}\right].\end{array}$

From (25), we obtain $\begin{array}{rl}{G}_{m+1}^{\left(a\right)}& =\frac{a}{\theta \gamma }E\left[\phantom{\frac{\text{β}+X_{U(m+1)}^{\theta }}{\text{β}+Q_1^{\theta }}}{X}_{U(m+1)}^{a-\theta }(\text{β}+X_{U(m+1)}^{\theta })\right.\\ & \times \left.\left\{1-{\left(\frac{\text{β}+X_{U(m+1)}^{\theta }}{\text{β}+Q_1^{\theta }}\right)}^{\gamma }\right\}\right].\end{array}$

(iii) For $b\ge \theta ,m,n=1,2,\dots ,$it follows from (28) and it is easy to see that $\begin{array}{rl}& G_{m,n+1}^{(a,b)}(t_1,t_2)\\ & =\frac{b{t}_2}{\theta \gamma }E\left[\phantom{\frac{\text{β}+X_{U(n+1)}^{\theta }}{\text{β}+Q_1^{\theta }}}{X}_{U(n+1)}^{b-\theta }(\text{β}+X_{U(m+1)}^{\theta })e^{[t_1{X}_{U\left(m\right)}^a+t_2{X}_{U(n+1)}^b]}\right.\\ & \times \left.\left\{1-{\left(\frac{\text{β}+X_{U(n+1)}^{\theta }}{\text{β}+Q_1^{\theta }}\right)}^{\gamma }\right\}\right].\end{array}$

From (31), we have $\begin{array}{rl}& G_{m,n+1}^{(a,b)}\\ & =\frac{b}{\theta \gamma }E\left[X_{U(n+1)}^{b-\theta }{X}_{U\left(m\right)}^a\left\{1-{\left(\frac{\text{β}+X_{U(n+1)}^{\theta }}{\text{β}+Q_1^{\theta }}\right)}^{\gamma }\right\}\right].\end{array}$

Recurrence relations of the doubly truncated (Lomax and Rayleigh) distributions can be obtained by setting $\theta =1$ and $\theta =2$ respectively, in the previous RR’s.

For example, to check the results, when $\theta =1$,$\lambda \left(x\right)=\gamma \ln (1+\frac{x}{\text{β}})\text{and}{\lambda }^{\mathrm{\prime }}\left(x\right)=\gamma /(\text{β}+x)$, we have the compound exponential (Lomax $(\text{β},\gamma )$) distribution. From RR (26), it is not difficult to get that ${\mu }_m^{\left(a\right)}={\mu }_1^{\left(a\right)}+\sum _{j=2}^m{G}_j^{(a+1)},m=2,3,\dots ,$where ${\mu }_1^{\left(a\right)}=E\left[X_{U\left(1\right)}^{a+1}\right]$, for $j=2,\dots ,m$. The non-truncated Lomax$(\text{β},\gamma )$ distribution yields $G_j^{\left(a\right)}=\frac{a+1}{\gamma }E\left[X_{U\left(j\right)}^a(1+X_{U\left(j\right)})\right]=\frac{a+1}{\gamma }[{\mu }_j^{\left(a\right)}+{\mu }_j^{(a+1)}].$

Therefore, we get $\begin{array}{rl}{\mu }_m^{(a+1)}& ={\mu }_1^{(a+1)}+\frac{a+1}{\gamma }\sum _{j=2}^m[{\mu }_j^{\left(a\right)}+{\mu }_j^{(a+1)}]\\ & ={\mu }_1^{(a+1)}+\frac{a+1}{\gamma }\sum _{j=2}^{m-1}[{\mu }_j^{\left(a\right)}+{\mu }_j^{(a+1)}]\\ & +\frac{a+1}{\gamma }[{\mu }_m^{\left(a\right)}+{\mu }_m^{(a+1)}]\\ & ={\mu }_{m-1}^{(a+1)}+\frac{a+1}{\gamma }[{\mu }_m^{\left(a\right)}+{\mu }_m^{(a+1)}].\end{array}$$\Rightarrow$ $\left(\frac{\gamma -a-1}{\gamma }\right){\mu }_{m-1}^{(a+1)}=\frac{a+1}{\gamma }{\mu }_m^{\left(a\right)}+{\mu }_m^{(a+1)}.$$\Rightarrow$ $\begin{array}{rl}{\mu }_m^{(a+1)}& =\frac{a+1}{\gamma -a-1}{\mu }_m^{\left(a\right)}+\frac{\gamma }{\gamma -a-1}{\mu }_{m-1}^{(a+1)},\\ & m=2,3,\dots ,a=1,2,\dots .\end{array}$

This recurrence relation coincides with the recurrence relation obtained, differently, by Balakrishnan and Ahsanullah [21] in which they used $k,n,\vartheta$instead of $a,m$and $\gamma$, respectively.

For the non-truncated Lomax $(\text{β},\gamma )$ distribution, it can be shown that $\begin{array}{rl}{\mu }_n^{\left(a\right)}& ={\text{β}}^a\sum _{i=0}^a\left[{(-1)}^{a-i}\left(\begin{array}{c}a\\ i\end{array}\right)/{(1-\frac{i}{\gamma })}^{n+1}\right],\\ & a<\gamma ,n=1,2,\dots ,a=1,2,\dots .\end{array}$

3.3. Doubly truncated Pareto (Type I) distribution (belongs to class =${\mathrm{\Im }}_L$)

Let $\lambda \left(x\right)=-\gamma \ln \left(\text{β}/x\right)\text{and}{\lambda }^{\mathrm{\prime }}\left(x\right)=\gamma /x.$

Then the CDF of doubly truncated Pareto distribution is given by $F_d\left(x\right)=\frac{{\left(\text{β}/P_1\right)}^{-\gamma }-{\left(\text{β}/x\right)}^{-\gamma }}{{\left(\text{β}/P_1\right)}^{-\gamma }-{\left(\text{β}/Q_1\right)}^{-\gamma }}.$

(i) For $a_k>0,k=1,2,\dots ,$ it follows from (15), so we can write that $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =\frac{t_k{a}_k}{\gamma }E\left[X_{U(n_k+1)}^{a_k}\exp \left\{\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\right.\\ & \times \left.\left\{1-{\left(\frac{X_{U(n_k+1)}}{Q_1}\right)}^{\gamma }\right\}\phantom{\sum _{i=1}^k}\right].\end{array}$

From (20), we have $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}\\ & =\frac{a_k}{\gamma }E\left[\prod _{i=1}^{k-1}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k}\right.\left.\left\{1-{\left(\frac{X_{U(n_k+1)}}{Q_1}\right)}^{\gamma }\right\}\phantom{\prod _{i=1}^{k-1}}\right].\end{array}$

(ii) For $a>0,m=1,2,\dots ,$ and using (22), the rhs of RR (21) is given by $G_{m+1}^{\left(a\right)}\left(t\right)=\frac{ta}{\gamma }E\left[\phantom{\frac{X_{U(m+1)}}{Q_1}}{X}_{U(m+1)}^a{e}^{t{X}_{U\left(m\right)}^a}\left\{1-{\left(\frac{X_{U(m+1)}}{Q_1}\right)}^{\gamma }\right\}\right].$

From (25), we obtain $G_{m+1}^{\left(a\right)}=\frac{a}{\gamma }E\left[X_{U(m+1)}^a\left\{1-{\left(\frac{\left(X_{U(m+1)}\right)}{Q_1}\right)}^{\gamma }\right\}\right].$

(iii) For $a,b>0,m,n=1,2,\dots ,$and using (28), we get $\begin{array}{rl}& G_{m,n+1}^{(a,b)}(t_1,t_2)\\ & =\frac{b{t}_2}{\gamma }E\left[\phantom{\frac{X_{U(n+1)}}{Q_1}}{X}_{U(n+1)}^b{e}^{[t_1{X}_{U\left(m\right)}^a+t_2{X}_{U(n+1)}^b]}\right.\\ & \times \left.\left\{1-{\left(\frac{X_{U(n+1)}}{Q_1}\right)}^{\gamma }\right\}\right].\end{array}$

From (31), we have $G_{m,n+1}^{(a,b)}=\frac{b}{\gamma }E\left[X_{U(n+1)}^b{X}_{U\left(m\right)}^a\left\{1-{\left(\frac{x_{U(n+1)}}{Q_1}\right)}^{\gamma }\right\}\right].$

It can be shown that RR (26) reduces to the following RR in the case of non-truncated Pareto I $(\text{β},\gamma )$ where $x>\text{β}$: $\begin{array}{rl}{\mu }_m^{(a+1)}& =\frac{\gamma }{\gamma -a-1}{\mu }_{m-1}^{(a+1)},\\ & m=2,3,\dots ,a>0,\gamma >a.\end{array}$

Therefore, we get ${\mu }_n^{\left(a\right)}=\frac{{\text{β}}^a}{{(1-\frac{1}{\gamma })}^{n+1}},\gamma >1,n=1,2,\dots .$

3.4. Doubly truncated beta distribution (belongs to class =${\mathrm{\Im }}_R$)

Let $\lambda \left(x\right)=-\text{β}\ln (1-x)\text{and}{\lambda }^{\mathrm{\prime }}\left(x\right)=\frac{\text{β}}{(1-x)}.$

Then the CDF of doubly truncated beta distribution is given by $F_d\left(x\right)=\frac{{(1-P_1)}^{-\text{β}}-{(1-x)}^{-\text{β}}}{{(1-P_1)}^{-\text{β}}-{(1-Q_1)}^{-\text{β}}}.$

(i) For $a_k=1,2,\dots ,k=1,2,\dots ,$ it follows from (15), we get $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =\frac{t_k{a}_k}{\text{β}}E\left[X_{U(n_k+1)}^{a_k-1}(1-X_{U(n_k+1)})\exp \left\{\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\right.\\ & \times \left.\left\{1-{\left(\frac{1-Q_1}{1-X_{U(n_k+1)}}\right)}^{\text{β}}\right\}\phantom{\sum _{i=1}^k}\right].\end{array}$

We note that in the non-truncated case, since $0<x<1$, then $Q_1=1$.

From (20), we have $\begin{array}{rl}{G}_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}& =\frac{a_k}{\text{β}}E\left[\phantom{\frac{1-Q_1}{1-X_{U(n_k+1)}}}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k}(1-X_{U(n_k+1)})\right.\\ & \times \left.\left\{1-{\left(\frac{1-Q_1}{1-X_{U(n_k+1)}}\right)}^{\text{β}}\right\}\right].\end{array}$

(ii) For $a=1,2,\dots ,$ and using (22), we get that $\begin{array}{rl}{G}_{m+1}^{\left(a\right)}\left(t\right)& =\frac{ta}{\text{β}}E\left[\phantom{\frac{1-Q_1}{1-X_{U(n_k+1)}}}{X}_{U(m+1)}^{a-1}(1-X_{U(m+1)})e^{t{X}_{U\left(m\right)}^a}\right.\\ & \times \left.\left\{1-{\left(\frac{1-Q_1}{1-X_{U(m+1)}}\right)}^{\text{β}}\right\}\right].\end{array}$

From (25), we have $\begin{array}{rl}{G}_{m+1}^{\left(a\right)}& =\frac{a}{\text{β}}E\left[\phantom{\frac{1-Q_1}{1-X_{U(n_k+1)}}}{X}_{U(m+1)}^{a-1}(1-X_{U(m+1)})\right.\\ & \times \left.\left\{1-{\left(\frac{1-Q_1}{1-X_{U(m+1)}}\right)}^{\text{β}}\right\}\right].\end{array}$

(iii) For $a,b=1,2,\dots ,m,n=1,2,\dots ,$ it follows from (28) and it is easy to get that $\begin{array}{rl}& G_{m,n+1}^{(a,b)}(t_1,t_2)\\ & =\frac{b{t}_2}{\text{β}}E\left[X_{U(n+1)}^{b-1}(1-X_{U(n+1)})e^{[t_1{X}_{U\left(m\right)}^a+t_2{X}_{U(n+1)}^b]}\right.\\ & \times \left.\left\{1-{\left(\frac{1-Q_1}{1-X_{U(n+1)}}\right)}^{\text{β}}\right\}\right].\end{array}$

From (31), we have $\begin{array}{rl}{G}_{m,n+1}^{(a,b)}& =\frac{b}{\text{β}}E\left[\phantom{\frac{1-Q_1}{1-X_{U(n_k+1)}}}{X}_{U(n+1)}^{b-1}{X}_{U\left(m\right)}^a(1-X_{U(n+1)})\right.\\ & \times \left.\left\{1-{\left(\frac{1-Q_1}{1-X_{U(n+1)}}\right)}^{\text{β}}\right\}\right].\end{array}$

In the non-truncated beta $(1,\text{β})$case (note that, in this case, $Q_1=1$), RR (26) reduces to the RR $\begin{array}{rl}{\mu }_m^{(a+1)}& =\frac{a}{a+\text{β}}{\mu }_{m-1}^{(a+1)}+\frac{\text{β}}{a+\text{β}}{\mu }_m^{\left(a\right)},\\ & m=2,3,\dots ,a=1,2,\dots .\end{array}$and it can be shown that $\begin{array}{rl}{\mu }_n^{\left(a\right)}& =\sum _{i=0}^a\left[{(-1)}^i\left(\begin{array}{c}a\\ i\end{array}\right)/{(1-\frac{i}{\text{β}})}^{n+1}\right],\\ & n=1,2,\dots ,a=1,2,\dots .\end{array}$

3.5. Doubly truncated Gompertz distribution

Let $\lambda \left(x\right)=(e^{\gamma x}-1)/\text{β}\gamma$and ${\lambda }^{\mathrm{\prime }}\left(x\right)=e^{\gamma x}/\text{β}$.

Then the CDF of doubly truncated Gompertz distribution is given by $F_d\left(x\right)=\frac{e^{(1-e^{\gamma P_1})/\text{β}\gamma }-e^{(1-e^{\gamma x})/\text{β}\gamma }}{e^{(1-e^{\gamma P_1})/\text{β}\gamma }-e^{(1-e^{\gamma Q_1})/\text{β}\gamma }}.$

(i) For $a_k=1,2,\dots ,k=1,2,\dots ,$ it follows from (15), so we can get that $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =\text{β}{a}_k{t}_{k}E\left[X_{U(n_k+1)}^{a_k-1}\exp \left\{-\gamma X_{U(n_k+1)}+\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\right.\\ & \times \left.\left\{1-\exp \left(\frac{e^{\gamma X_{U(n_k+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }\right)\right\}\phantom{\sum _{i=1}^k}\right].\end{array}$

From (20), we have $\begin{array}{rl}{G}_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}& =\text{β}{a}_{k}E\left[\prod _{i=1}^{k-1}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k-1}{e}^{-\gamma X_{U(n_k+1)}}\right.\\ & \times \left.\left\{1-\exp \left(\frac{e^{\gamma X_{U(n_k+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }\right)\right\}\right].\end{array}$

(ii) For $a=1,2,\dots ,$and using (22), we obtain that $\begin{array}{rl}{G}_{m+1}^{\left(a\right)}\left(t\right)& =\text{β}atE\left[X_{U(m+1)}^{a-1}\exp \left\{-\gamma X_{U(m+1)}+t_i{X}_{U\left(m\right)}^a\right\}\right.\\ & \times \left.\left\{1-\exp \left(\frac{e^{\gamma X_{U(m+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }\right)\right\}\right].\end{array}$

From (25), we obtain $\begin{array}{rl}{G}_{m+1}^{\left(a\right)}& =\text{β}aE\left[\phantom{\frac{e^{\gamma X_{U(m+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }}{X}_{U(m+1)}^{a-1}\exp \left\{-\gamma X_{U(m+1)}\right\}\right.\\ & \times \left.\left\{1-\exp \left(\frac{e^{\gamma X_{U(m+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }\right)\right\}\right].\end{array}$

(iii) For $a,b=1,2,\dots ,$it follows from (28), so we can get that $\begin{array}{rl}& G_{m,n+1}^{(a,b)}(t_1,t_2)\\ & =\text{β}b{t}_{2}E\left[\phantom{\frac{e^{\gamma X_{U(n+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }}{X}_{U(n+1)}^{b-1}\right.\\ & \times \exp \{-\gamma X_{U(n+1)}+t_1{X}_{U\left(m\right)}^a+t_2{X}_{U(n+1)}^b\}\\ & \times \left.\left\{1-\exp \left(\frac{e^{\gamma X_{U(n+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }\right)\right\}\right].\end{array}$

Using (31), we have $\begin{array}{rl}{G}_{m,n+1}^{(a,b)}& =\text{β}bE\left[\phantom{\frac{e^{\gamma X_{U(n+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }}{X}_{U\left(m\right)}^a{X}_{U(n+1)}^{b-1}\exp \{-\gamma X_{U(n+1)}\}\right.\\ & \times \left.\left\{1-\exp \left(\frac{e^{\gamma X_{U(n+1)}}-e^{\gamma Q_1}}{\text{β}\gamma }\right)\right\}\right].\end{array}$

3.6. Doubly truncated compound Gompertz distribution

$\lambda \left(x\right)=\delta \ln [1+(e^{\gamma x}-1\left)/\text{β}\gamma \right]$and ${\lambda }^{\mathrm{\prime }}\left(x\right)=\gamma \delta /[1+(\text{β}\gamma -1\left)e^{-\gamma x}\right]$.

Then the CDF of doubly truncated Gompertz distribution is given by $F_d\left(x\right)=\frac{\left[1+(e^{\gamma P_1}-1)/\text{β}\gamma \right]-\left[1+(e^{\gamma x}-1)/\text{β}\gamma \right]}{\left[1+(e^{\gamma P_1}-1)/\text{β}\gamma \right]-\left[1+(e^{\gamma Q_1}-1)/\text{β}\gamma \right]}.$

(i) For $a_k=1,2,\dots ,k=1,2,\dots ,$ and from (15), the rhs of RR (14) is given by $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}(t_1,\dots ,t_k)\\ & =\frac{t_k{a}_k}{\gamma \delta }E\left[\phantom{{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }}{X}_{U(n_k+1)}^{a_k-1}\exp \{1+(\text{β}\gamma -1\left)e^{-\gamma X_{U(n_k+1)}}\right\}\right.\\ & \times \exp \left\{\sum _{i=1}^k{t}_i{X}_{U\left(n_i\right)}^{a_i}\right\}\\ & \times \left.\left\{1-{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n_k+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }\right\}\right].\end{array}$

Using (20), the rhs of RR (19) is given by $\begin{array}{rl}& G_{n_1,\dots ,n_k+1}^{(a_1,\dots ,a_k)}\\ & =\frac{a_k}{\gamma \delta }E\left[\prod _{i=1}^{k-1}{X}_{U\left(n_i\right)}^{a_i}{X}_{U(n_k+1)}^{a_k-1}\right.\\ & \times \exp \left\{1+(\text{β}\gamma -1)e^{-\gamma X_{U(n_k+1)}}\right\}\\ & \times \left.\left\{1-{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n_k+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }\right\}\right].\end{array}$

(ii) Using (22) and for $a=1,2,\dots ,$the rhs of RR (21) is given by $\begin{array}{rl}& G_{m+1}^{\left(a\right)}\left(t\right)\\ & =\frac{ta}{\gamma \delta }E\left[\phantom{{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }}{X}_{U(m+1)}^{a-1}\right.\\ & \times \exp \{1+(\text{β}\gamma -1)e^{-\gamma X_{U(m+1)}}+t{X}_{U\left(m\right)}^a\}\\ & \times \left.\left\{1-{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(m+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }\right\}\right].\end{array}$

From (25), the rhs of RR (24) is obtained by $\begin{array}{rl}{G}_{m+1}^{\left(a\right)}& =\frac{a}{\gamma \delta }E\left[\phantom{{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }}{X}_{U(m+1)}^{a-1}\exp \{1+(\text{β}\gamma -1\left)e^{-\gamma X_{U(m+1)}}\right\}\right.\\ & \times \left.\left\{1-{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(m+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }\right\}\right].\end{array}$

(iii) For $a,b=1,2,\dots ,$and from (28), the rhs of RR (27) is given by $\begin{array}{rl}& G_{m,n+1}^{(a,b)}(t_1,t_2)\\ & =\frac{b{t}_2}{\gamma \delta }E\left[\phantom{{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }}{X}_{U(n+1)}^{b-1}\exp \{1+(\text{β}\gamma -1)e^{-\gamma X_{U(n+1)}}\right.\\ & +t_1{X}_{U\left(m\right)}^a+t_2{X}_{U(n+1)}^b\}\\ & \times \left.\left\{1-{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }\right\}\right].\end{array}$

Using (31), the rhs of RR (30) can be written as $\begin{array}{rl}& G_{m,n+1}^{(a,b)}\\ & =\frac{b}{\gamma \delta }E\left[\phantom{{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }}{X}_{U\left(m\right)}^a{X}_{U(n+1)}^{b-1}\exp \{1+(\text{β}\gamma -1\left)e^{-\gamma X_{U(n+1)}}\right\}\right.\\ & \times \left.\left\{1-{\left(\frac{\text{β}\gamma -1+e^{\gamma X_{U(n+1)}}}{\text{β}\gamma -1+e^{\gamma Q_1}}\right)}^{\delta }\right\}\right].\end{array}$

4. Conclusion

Based on a general class of doubly truncated distributions, recurrence relations for the moment generating functions and product moments of powers of upper record values are considered for a sequence of iid random variables. So, a general form in (14) can be used to find recurrence relations of moment generating function and moments of powers for any distribution includes in the class in (7) for any sequence of iid random variables. The considered class includes the right, left and non-truncated distributions as special cases. In addition, it includes the most important distributions than can be used in the life testing and other applied area of statistics. Some special cases are presented using (14) such as recurrence relations for moment generating function and moments of powers for univariate and bivariate upper record values. Also, the results of some of these special cases have been checked to ensure the agreement with already existing results in the literature.

Disclosure statement

No potential conflict of interest was reported by the author(s).