1. Introduction
Kamps [1] introduced the concept of generalized order statistics \((gos)\) as follows:
Let us note \(n\in N\), \(k\geq1\), and \(\tilde{m}=(m_1,m_2,\ldots,m_{n-1})\in\mathfrak{R}^{n-1},\) \(1\leq r\leq {n-1}\), such that
\begin{equation*}
\gamma_r=k+n-r+\sum_{j=r}^{n-1} m_{j}>0~~\, for \, ~~1\leq r \leq n-1.
\end{equation*}
The random variables \(X(1,n,\tilde{m},k),X(2,n,\tilde{m},k), \ldots,X(n,n,\tilde{m},k)\) are said to be \(gos\) from a continuous population with cumulative distribution function (\(cdf\)) \(F(x)\) and probability distribution function (\(pdf\)) \(f(x)\) if their joint \(pdf\) is of the form
\begin{equation}\label{e1.1}
k\Bigg(\prod_{j=1}^{n-1}\gamma_j\Bigg)\Bigg(\prod_{i=1}^{n-1}\big[\bar{F}(x_i)\big]^{m_i}f(x_i)\Bigg)\big[\bar{F}(x_n)\big]^{k-1}f(x_n),
\end{equation}
(1)
defined on the cone \(F^{-1}(0+)< x_1\leq x_2 \leq \ldots \leq x_n< F^{-1}(1)\) of \( \mathfrak{R}^{n}\),
where \(\bar{F}(x)=1-F(x)\).
The model of \(gos\) contains special cases such as ordinary order statistics \((\gamma_i=n-i+1; i=1,2,\dots,n ~i.e. ~ m_{1}=m_{2}=\dots=m_{n-1}=0, \,k=1)\), \(k\)-th record values \((\gamma_{i}=k ~i.e.,~ m_{1}=m_{2}=\dots=m_{n-1}=-1,\, k\in N)\), sequential order statistics \((\gamma_{i}=(n-i+1)\alpha_{i};\, \alpha_{1},\alpha_{2},\dots,\alpha_{n}> 0)\), order statistics with non-integer sample size \((\gamma_{i}=\alpha-i+1; \,\alpha > 0)\), Pfeifer’s record values \((\gamma_{i}=\beta_{i};\, \beta_{1},\beta_{2},\dots,\beta_{n} > 0)\) and progressive type II censored order statistics \((\gamma_r=n-r+1+\sum_{i=r}^{l} m_{i}, 1\leq r\leq l \leq n,\, m_{i}\in N,\, k=m_n+1)\), see [1,2,3].
Here we shall obtain the results for \(\gamma_i\ne\gamma_j\) and then deduce the results for \(\gamma_i=\gamma_j\) (\(m_{1}=m_{2}=\dots=m_{n-1}=m\ne-1\)).
Therefore, we will consider two cases:
Case I: \(\gamma_i=\gamma_j\,(m_{1}=m_{2}=\dots=m_{n-1}=m\ne-1)\) [1].
Case II: \(\gamma_i\ne\gamma_j,\,i\ne j\, i,j=1,2,\dots{,{n-1}}\) [2].
Case I: The \(pdf\) of \(r-\)th \(gos\) \(X(r,n,m,k),\) is given by
\begin{equation}\label{e1.2}
f_{X(r,n,m,k)}(x)=\frac{C_{r-1}}{(r-1)!}\,[{\bar F}(x)]^{\gamma_r-1}\,f(x)\,g_m^{r-1}(F(x)),
\end{equation}
(2)
and the joint \(pdf\) of \(X(r,n,m,k)\) and \(X(s,n,m,k)\), \(1\leq r< s\leq n\), is given by
\begin{align}
\label{e1.3}
f_{X(r,n,m,k), X(s,n,m,k)}(x,y)=&\frac{C_{s-1}}{(r-1)!\ (s-r-1)!}\, [{\bar F}(x)]^m\,g_m^{r-1}(F(x))
\end{align}
(3)
\begin{align}
&\times [h_m(F(y))-h_m(F(x))]^{s-r-1}\,[{\bar F}(y)]^{\gamma_s-1}\,f(x)\,f(y),~~x< y,
\end{align}
(4)
where
\begin{equation*}
C_{r-1}=\prod_{i=1}^r\gamma_i ,\, \,~~~\gamma_i=k+(n-i) (m+1),
\end{equation*}
\begin{equation*}
h_m(x)=\begin{cases}
-\displaystyle\frac{1}{m+1}\,(1-x)^{m+1}&,~~~ m\neq -1\\
-\ln(1-x)&,~~~m=-1
\end{cases}
\end{equation*}
and
\begin{equation*}
g_m(x)~=~h_m(x)-h_m(0)=\int_0^x(1-t)^mdt,~~x\in [0,1).
\end{equation*}
Case II: The \(pdf\) of \(r-\)th \(gos\) \(X(r,n,\tilde{m},k),\) is given by
\begin{equation}\label{e1.4}
f_{X(r,n,\tilde{m},k)}(x)~=~{C_{r-1}}\,\sum_{i=1}^{r}\,a_{i}(r)\,[{\bar F}(x)]^{\gamma_i-1}\,f(x)
\end{equation}
(5)
with the joint \(pdf\) of \(X(r,n,\tilde{m},k)\) and \(X(s,n,\tilde{m},k)\), \(1\leq r< s\leq n\),
\begin{equation}\label{e1.5}
f_{X(r,n,\tilde{m},k), X(s,n,\tilde{m},k)}(x,y)=~C_{s-1}\, \Bigg[\sum_{i=r+1}^{s}a_{i}^{(r)}(s)\bigg\{\frac{\bar{F}(y)}{\bar{F}(x)}\bigg\}^{\gamma_{i}}\Bigg]\bigg[\sum_{i=1}^{r}a_{i}(r)\big\{{\bar{F}(x)}\big\}^{\gamma_{i}}\bigg]\frac{f(x)}{\bar{F}(x)}\,\frac{f(y)}{\bar{F}(y)}
\end{equation}
(6)
where
\begin{align*}
&C_{r-1}=\prod_{i=1}^r\gamma_i,\\
&\gamma_r=k+n-r+\sum_{j=r}^{n-1} m_{j},\\
&a_{i}(r)~=~\prod_{j=1}^r\frac{1}{(\gamma_j-\gamma_i)},\ \ \ \ j\ne i,\ \ \ \ \gamma_j\ne\gamma_i, \ \ \ \ 1\leq i\leq r\leq n,\\
&a_{i}^{(r)}(s)~=~\prod_{j=r+1}^n\frac{1}{(\gamma_j-\gamma_i)},\ \ \ \ j\ne i,\ \ \ \ \gamma_j\ne\gamma_i, \ \ \ \ r+1\leq i\leq s\leq n.
\end{align*}
For \(m_{1}=m_{2}=\dots=m_{n-1}=m\ne -1\), it can be shown that [
3]:
\begin{equation}\label{e1.6}
a_{i}(r)=~\frac{(-1)^{r-i}}{(m+1)^{r-1}\,(r-1)!}\binom{r-1}{r-i}
\end{equation}
(7)
and
\begin{equation}\label{e1.8}
a_{i}^{(r)}(s)=~\frac{(-1)^{s-i}}{(m+1)^{s-r-1}\,(s-r-1)!}\binom{s-r-1}{s-i}.
\end{equation}
(8)
In this paper we are interested in a situation when a random variable \(X\) follows the Pareto-Rayleigh(P-R) distribution with \(pdf\)
\begin{equation}\label{e1.9}
f(x;\alpha,\sigma)=\frac{\alpha}{\sigma^2}\,x\,\Big(1+\frac{x^{2}}{2\sigma^2}\Big)^{-(\alpha+1)} \quad x> 0, \, {\alpha>1, \,\text{and}\, \sigma>0},
\end{equation}
(9)
and with \(df\)
\begin{equation}\label{e1.10}
{F}(x;\alpha,\sigma)=1-\Big(1+\frac{x^2}{2\sigma^2}\Big)^{-\alpha} \quad x> 0, \, {\alpha>1, \,\text{and}\, \sigma>0}.
\end{equation}
(10)
In view of (8) and (9),
\begin{equation}\label{e1.11}
\big(1+\frac{x^2}{2\sigma^2}\big)\,f(x)=\frac{\alpha}{\sigma^2}\,x\,\bar{F}(x)
\end{equation}
(11)
Pareto-Rayleigh distribution can be seen as a member of Transformed-Transformer family (or T-X family) of distributions proposed by Alzaatreh
et al., [
4]. This distribution is recognized as a good model for fitting various lifetime data, see Jebeli and Deiri [
5]. This is also confirmed in [
6] were a comparative study on the performance of Pareto-Rayleigh distribution against biased Lomax distribution was conducted. Further, for more details on Pareto-Rayleigh distribution one can see [
7,
8,
9].
Exact moments expressions of gos for
different distributions have been obtained by literature. Some examples are
exponentiated Log-logistic distribution, Burr type XII distribution, linear exponential distribution, Erlang-truncated exponential distribution, Burr distribution, power function distribution, type II exponentiated Log-logistic distribution, extended exponential distribution, generalized Pareto distribution, q-Weibull distribution; see, respectively, Athar and Nayabuddin [10], Khan et al., [11], Ahmad [12], Khan et al., [13], Khan and Khan [3],
Kumar and Khan [14], Kumar [15], Kumar and Dey [16], Malik and Kumar [17], Singh et al., [18] and Kumar et al., [19].
In this paper, we have derived explicit expression for single and product moments of Pareto-Rayleigh distribution based on gos.
2. Relations for Product Moments
In this section, we derive the exact expressions for product moments of generalized order statistics in the following theorems. Before coming to the main result, the following lemma is proved.
Lemma 1. For the Pareto-Rayleigh distribution with \(cdf\) \((1.9)\) next relations holds
\begin{equation}\label{2.1}
\Phi_{j,l}(a,b)=\frac{(2\sigma^2)^{\big(\frac{j+l}{2}+2)}}{2\,(j+2)}\,B\Big(\frac{j+l}{2}+2,\,\alpha\,b-\frac{l}{2}\Big)
_3{F}_2\Big(\frac{j}{2}+1,\,1-a\,\alpha+\frac{j}{2},\,\frac{j+l}{2}+2;\,\frac{j}{2}+2,\,\frac{j}{2}+\alpha\,b+2;1\Big)
\end{equation}
(12)
where
\begin{equation*}\label{2.2}
\Phi_{j,l}(a,b)=\int_{0}^{\infty}\int_{0}^{y}\frac{x^{j+1}}{\big(1+\frac{x^{2}}{2\sigma^2}\big)^{a\,\alpha+1}}\,\frac{y^{l+1}}{\big(1+\frac{y^{2}}{2\sigma^2}\big)^{\alpha\,b+1}}dx\,dy
\end{equation*}
and
\begin{equation*}
_{p}F_{q}[a_{1},\dots,a_{p};b_{1},\dots,b_{q};x]=\sum_{r=0}^{\infty}\Big[\prod_{j=1}^{p}\frac{\Gamma(a_{j}+r)}{\Gamma(a_{j})}\Big]\Big[\prod_{j=1}^{q}\frac{\Gamma(b_{j})}{\Gamma(b_{j}+r)}\Big]\frac{x^r}{r!},
\end{equation*}
for \(p=q+1\) and \(\sum_{j=1}^{q}b_{j}-\sum_{j=1}^{p}a_{j}\,>0\).
Proof.
We have
\begin{equation}\label{2.3}
\Phi_{j,l}(a,b)=\int_{0}^{\infty}\,\frac{y^{l+1}}{\big(1+\frac{y^{2}}{2\sigma^2}\big)^{\alpha\,b+1}}\Bigg[\int_{0}^{y}\frac{x^{j+1}}{\big(1+\frac{x^{2}}{2\sigma^2}\big)^{a\,\alpha+1}}dx\Bigg]\,dy
\end{equation}
(13)
Let
\begin{equation}\label{2.4}
B(y)=\int_{0}^{y}\frac{x^{j+1}}{\big(1+\frac{x^{2}}{2\sigma^2}\big)^{a\,\alpha+1}}dx
\end{equation}
(14)
Substituting \(1-u=\frac{1}{\big(1+\frac{x^{2}}{2\sigma^2}\big)}\) in (14), we get
\begin{align*}
B(y)&=\frac{(2\sigma^2)^{\big(1+\frac{j}{2}\big)}}{2}\,\int_{0}^{\frac{\frac{y^{2}}{2\sigma^2}}{\big(1+\frac{y^{2}}{2\sigma^2}\big)}}u^{\frac{j}{2}}\,(1-u)^{a\,\alpha-\frac{j}{2}-1}\,du\\
&=\frac{(2\sigma^2)^{\big(1+\frac{j}{2}\big)}}{2}\,B_{\frac{\frac{y^{2}}{2\sigma^2}}{\big(1+\frac{y^{2}}{2\sigma^2}\big)}}\Big(\frac{j}{2}+1,\,a\alpha-\frac{j}{2} \Big).
\end{align*}
From (13), we have
\begin{equation}\label{2.5}
\Phi_{j,l}(a,b)=\frac{(2\sigma^2)^{\big(1+\frac{j}{2}\big)}}{2}\,\int_{0}^{\infty}\,\frac{y^{l+1}}{\big(1+\frac{y^{2}}{2\sigma^2}\big)^{\alpha\,b+1}}\,B_{\frac{\frac{y^{2}}{2\sigma^2}}{\big(1+\frac{y^{2}}{2\sigma^2}\big)}}\Big(\frac{j}{2}+1,\,a\alpha-\frac{j}{2} \Big)\,dy,
\end{equation}
(15)
where
\(
B_{x}(p,q)=\,\int_{0}^{x}u^{p-1}\,(1-u)^{q-1}\,du.
\)
We know that
\begin{equation}\label{2.7}
B_{x}(p,q)=p^{-1}\,x^p\,_{2}F_{1}\,(p,1-q;\,p+1;\,x)
\end{equation}
(16)
and
\begin{equation}\label{2.8}
\int_{0}^{1}u^{a-1}\,(1-u)^{b-1}\,_{2}F_{1}\,(c,d;\,e;\,u)\,du=B(a,b)\,_{3}F_{2}\,(c,d,a;\,e,a+b;\,1)
\end{equation}
(17)
Substituting (16) and (17) in (15), we get
\begin{align}
\label{2.9}
\Phi_{j,l}(a,b)=&\frac{(2\sigma^2)^{\big(1+\frac{j}{2}\big)}}{2}\,\int_{0}^{\infty}\,\frac{y^{l+1}}{\big(1+\frac{y^{2}}{2\sigma^2}\big)^{\alpha\,b+1}}\,\Bigg({\frac{\frac{y^{2}}{2\sigma^2}}{1+\frac{y^{2}}{2\sigma^2}}}\Bigg)^{\frac{j}{2}+1}\,\Big(\frac{j}{2}+1\Big)^{-1}
\notag\\
& _2{F}_1\left[\frac{j}{2}+1,\,1-a\,\alpha+\frac{j}{2},;\,\frac{j}{2}+2;\,\left({\frac{\frac{y^{2}}{2\sigma^2}}{1+\frac{y^{2}}{2\sigma^2}}}\right)\right]\,dy.
\end{align}
(18)
Setting \(t={\frac{\frac{y^{2}}{2\sigma^2}}{1+\frac{y^{2}}{2\sigma^2}}}\) in \((18)\), we get
\begin{align*}
\Phi_{j,l}(a,b)=&\frac{(2\sigma^2)^{\left(\frac{j+l}{2}+2\right)}}{2\,(j+2)}\,\int_{0}^{1}\,t^{\frac{j+l}{2}+1}\,(1-t)^{\alpha\,b-\frac{l}{2}-1}\,_2{F}_1\left[
\frac{j}{2}+1,\,1-a\,\alpha+\frac{j}{2},;\,\frac{j}{2}+2;\,t\right]\,dt
\\&=\frac{(2\sigma^2)^{\left(\frac{j+l}{2}+2\right)}}{2\,(j+2)}\,B\left(\frac{j+l}{2}+2,\,\alpha\,b-\frac{l}{2}\right) _3{F}_2\left(\frac{j}{2}+1,\,1-a\,\alpha+\frac{j}{2},\,\frac{j+l}{2}+2;\,\frac{j}{2}+2,\,\frac{j}{2}+\alpha\,b+2;1\right).
\end{align*}
Lemma 2. Setting \(j=0\) or \(l=0\) in Lemma 1, we obtain
\begin{equation}\label{2.10}
\Phi_{0,l}(a,b)=\frac{\sigma^2}{a\,\alpha}\,[\Phi_{l}(b)-\Phi_{l}(a+b)]
\end{equation}
(19)
and
\begin{equation}\label{2.11}
\Phi_{j,0}(a,b)=\frac{\sigma^2}{b\,\alpha}\,[\Phi_{j}(a+b)]
\end{equation}
(20)
where
\[
\Phi_{j}(a)=\int_{0}^{\infty}\frac{x^{j+1}}{\left(1+\frac{x^{2}}{2\sigma}\right)^{a\,\alpha+1}}\,\ \ dx=\frac{(2\sigma^2)^{\left(1+\frac{j}{2}\right)}}{2}\,B\left(a\alpha-\frac{j}{2}, 1+\frac{j}{2}\right).
\]
Proof. Substituting \(j=0\) in (13), we get
\begin{align*}
\Phi_{0,l}(a,b)&=\int_{0}^{\infty}\,\frac{y^{l+1}}{\left(1+\frac{y^{2}}{2\sigma^2}\right)^{\alpha\,b+1}}\left[\int_{0}^{y}\frac{x}{\left(1+\frac{x^{2}}{2\sigma^2}\right)^{a\,\alpha+1}}dx\right]\,dy\\
&=\frac{\sigma^2}{a\,\alpha}\int_{0}^{\infty}\,\frac{y^{l+1}}{\left(1+\frac{y^{2}}{2\sigma^2}\right)^{\alpha\,b+1}}\left[1-\frac{1}{\left(1+\frac{y^{2}}{2\sigma^2}\right)^{a\,\alpha}}\right]\,dy\\
&=\frac{\sigma^2}{a\,\alpha}\,[\Phi_{j}(b)-\Phi_{l}(a+b)].
\end{align*}
Similarly, we get (20) by noting that
\begin{equation*}
_3{F}_2(a,\,b,\,c;\,c,\,d;1)=_2{F}_1(a,\,b;\,d;1)=\frac{\Gamma (d)\,\,\Gamma(d-a-b)}{\Gamma(d-a)\,\,\Gamma(d-b)}.
\end{equation*}
Theorem 1. Generalized product moments for Pareto-Rayleigh distribution are given as
\begin{equation}\label{2.13}
\mu^{(j,\,l)}_{r,s,n,\tilde{m},k}=E[X^j(r,n,\tilde{m},k)\,X^l(s,n,\tilde{m},k)]
=~{C_{s-1}}\left(\frac{\alpha}{\sigma^2}\right)^2\left[\sum_{t=r+1}^{s}\,a_{t}^{(r)}(s)\,\left(\sum_{i=1}^{r}\,a_{i}(r)\,\Phi_{j,\,l}(\gamma_{i}-\gamma_{t},\,\gamma_{t})\right)\right].
\end{equation}
(21)
Proof. We have
\begin{equation*}
\mu^{(j,\,l)}_{r,s,n,\tilde{m},k}=~C_{s-1}\,\int_{0}^{\infty}\int_{0}^{y}x^j\,y^{l} \left[\sum_{i=r+1}^{s}a_{i}^{(r)}(s)\left\{\frac{\bar{F}(y)}{\bar{F}(x)}\right\}^{\gamma_{i}}\right]
\left(\sum_{i=1}^{r}a_{i}(r)\left\{{\bar{F}(x)}\right\}^{\gamma_{i}}\right)\frac{f(x)}{\bar{F}(x)}\,\frac{f(y)}{\bar{F}(y)}\,dx\,dy.
\end{equation*}
which yields (21).
Corollary 2. Product moment for Pareto-Rayleigh distribution, when \(m_1=m_2=\dots=m_{n-1}=m\ne -1\) is given as
\begin{align}
\label{2.14}
\mu^{(j,\,l)}_{r,s,n,m,k}=&E\left[X^j(r,n,m,k)\,X^l(s,n,m,k)\right]\notag\\
=&\frac{C_{s-1}}{(r-1)!\,(s-r-1)!\,(m+1)^{s-2}}
\left(\frac{\alpha}{\sigma^2}\right)^2\,\sum_{i=0}^{r-1}\sum_{t=0}^{s-r-1}(-1)^{i+t}\,\binom{r-1}{i}\,\binom{s-r-1}{t}\Phi_{j,\,l}(\gamma_{r-i}-\gamma_{s-t},\,\gamma_{s-t}).
\end{align}
(22)
Remark 1. Setting \( m_{1}=m_{2}=\dots=m_{n-1}=0\) and \(k=1\) in (22), we get the result as the product moment of order statistics as
\begin{align}
\label{OSp}
\mu^{(j,\,l)}_{r,s,n,0,1}&=\mu^{j,\,l}_{r,s:n}\notag\\
&=\frac{C_{s-1}}{(r-1)!\,(s-r-1)!}\left(\frac{\alpha}{\sigma^2}\right)^2
\sum_{i=0}^{r-1}\sum_{t=0}^{s-r-1}(-1)^{i+t}\,\binom{r-1}{i}\,\binom{s-r-1}{t}\Phi_{j,\,l}(s-r-t+i,\,n-s+t+1).
\end{align}
(23)
Corollary 3. Single moments of the Pareto-Rayleigh distribution are of the form
\begin{equation}\label{2.15}
\mu^{(l)}_{s,n,\tilde{m},k}=~{C_{s-1}}\,\Big(\frac{\alpha}{\sigma^2}\Big)\sum_{i=1}^{s}\,a_{i}(s)\,\Phi_{l}(\gamma_{i}).
\end{equation}
(24)
Proof.
Putting \(j=0\) in (21) and using (19), we get
\begin{align*}
\mu^{(l)}_{r,s,n,\tilde{m},k}=
&{C_{s-1}}\left(\frac{\alpha}{\sigma^2}\right)
\left[\sum_{t=r+1}^{s}\frac{a_{t}^{(r)}(s)}{(\gamma_{i}-\gamma_{t})}\,\left(\sum_{i=1}^{r}\,a_{i}(r)\,\left\{\Phi_{l}(\gamma_{t})-\Phi_{l}(\gamma_{i})\right\}\right)\right].\\
\mu^{(l)}_{s,n,\tilde{m},k}=&{C_{s-1}}\left(\frac{\alpha}{\sigma^2}\right)
\left[\sum_{t=r+1}^{s}\,{a_{t}^{(r)}(s)}\,\Phi_{l}(\gamma_{t})\,\left(\sum_{i=1}^{r}\,\frac{a_{i}(r)}{(\gamma_{i}-\gamma_{t})}\right)\right]
\\
&+~{C_{s-1}}\left(\frac{\alpha}{\sigma^2}\right)\left[\sum_{i=1}^{r}\,a_{i}(r)\,\Phi_{l}(\gamma_{i})\,\left(\sum_{t=r+1}^{s}\,\frac{a_{t}^{(r)}(s)}{(\gamma_{i}-\gamma_{t})}\right)\right].
\end{align*}
Now using the results found in [20] we obtain
\begin{equation*}
\sum_{i=1}^{r}\frac{a_{i}(r)}{(\gamma_i-\gamma_j)}~=~\prod_{j=1}^r\frac{1}{(\gamma_i-\gamma_j)},\quad j\ne i,\quad \gamma_j\ne\gamma_i, \quad 1\leq i\leq r\leq n,
\end{equation*}
and
\begin{equation*}
\sum_{i=r+1}^{s}\frac{a_{i}^{(r)}(s)}{(\gamma_i-\gamma_j)}~=~\prod_{j=r+1}^s\frac{1}{(\gamma_i-\gamma_j)},\quad j\ne i,\quad \gamma_j\ne\gamma_i, \quad r+1\leq i\leq s\leq n.
\end{equation*}
Hence,
\begin{equation*}
\mu^{(l)}_{s,n,\tilde{m},k}=~{C_{s-1}}\Big(\frac{\alpha}{\sigma^2}\Big)\Bigg[\sum_{t=r+1}^{s}\,{a_{t}^{(r)}(s)}\,\Phi_{l}(\gamma_{t})\,\Bigg(\prod_{j=1}^r\frac{1}{(\gamma_i-\gamma_j)}\Bigg)\Bigg]
\end{equation*}
\begin{equation*}
\hspace{1.7cm}+~{C_{s-1}}\Big(\frac{\alpha}{\sigma^2}\Big)\Bigg[\sum_{i=1}^{r}\,a_{i}(r)\,\Phi_{l}(\gamma_{i})\,\Bigg(\prod_{j=r+1}^s\frac{1}{(\gamma_i-\gamma_j)}\Bigg)\Bigg],
\end{equation*}
which yields (24).
Corollary 4. Single moments of gos for Pareto-Rayleigh distribution, when \(m_1=m_2=\dots=m_{n-1}=m\ne -1\), are given as
\begin{equation}\label{2.16}
\mu^{(l)}_{s,n,m,k}=~\frac{C_{s-1}}{(s-1)!}\,\frac{1}{(m+1)^{s-1}}\Big(\frac{\alpha}{\sigma^2}\Big)\sum_{i=0}^{s-1}(-1)^i\,\binom{r-1}{i}\Phi_{j}(\gamma_{s-i}).
\end{equation}
(25)
Proof. Setting \( m_{1}=m_{2}=\dots=m_{n-1}=m\ne -1\) in (24) and using (7) we get the result as the single moment.
Remark 2. Setting \( m_{1}=m_{2}=\dots=m_{n-1}=0\) and \(k=1\) in (25), we get the result as the single moment from order statistics
\begin{equation}\label{OSs}
\mu^{(l)}_{s,n,0,1}=\mu^{(l)}_{s:n}=~\frac{C_{s-1}}{(s-1)!}\Big(\frac{\alpha}{\sigma^2}\Big)\sum_{i=0}^{s-1}(-1)^i\,\binom{s-1}{i}\Phi_{j}(n-s+i+1).
\end{equation}
(26)
Remark 3. Setting \( j=0\) and \(l=0\) in \((21)\) we get
\begin{equation}\label{2.17}
\sum_{i=1}^{r}\sum_{t=r+1}^{s}\frac{a_{i}(r)\,a_{t}^{r}(s)}{\gamma_i\,\gamma_{t}}=\frac{1}{C_{s-1}},
\end{equation}
(27)
and setting \(l=0\) in (24) we obtain
\begin{equation}\label{2.18}
\sum_{i=1}^{r}\frac{a_{i}(r)}{\gamma_i}=\frac{1}{C_{r-1}}.
\end{equation}
(28)
Combining (27) and (28), we get another identity,
\begin{equation}\label{2.19}
\sum_{t=r+1}^{s}\frac{a_{t}^{r}(s)}{\gamma_{t}}=\frac{C_{r-1}}{C_{s-1}}.
\end{equation}
(29)
When \( m_{1}=m_{2}=\dots=m_{n-1}=m\ne -1\), (29) reduces to another identity
\begin{equation}\label{2.22}
\sum_{t=0}^{s-r-1}(-1)^t \binom{s-r-1}{t}\frac{1}{\gamma_{s-t}}=\frac{C_{r-1}(s-r-1)\,!\,(m+1)^{s-r-1}}{C_{s-1}},
\end{equation}
(30)
which is obtained in [
3].
Remark 4. Setting \(\gamma_r=k+n-r+\sum_{i=r}^{l}m_{j}\), \(1\leq r\leq l\leq n\), \(m_i \in N\), in (21), then the product moments of progressive type II censored order statistics of Pareto-Rayleigh distribution can be obtained.
3. Numerical Computations
Here we have calculated means and variances for order statistics (Table 1 & 2), and generalized order statistics \((gos)\) (Table 3 & 4). All computations here we obtained using Mathematica. Mathematica like other algebraic manipulation packages allow for arbitrary precisions, so the accuracy of the given values is not an issue. In case of order statistics, the relation
\(
\sum_{r=1}^{n}\mu^{j}_{r,n,0,1}=n\,\mu^{j}_{1,1,0,1}, \qquad j=1,2,
\)
is used to evaluate the means and variancess, see [21]. It is observed that when the sample size \(n\) is fixes, increasing the value of \(r\) directly increases the means and variances, whereas, for fixed \(r\), the opposite occurs in the case when the sample size \(n\) increases.
Table 1. Means of order statistics from Pareto-Rayleigh distribution (\(\alpha\)=2, \(\sigma\)=1).
|
n |
| r |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
| 1 |
1.1107 |
0.6942 |
0.5469 |
0.4653 |
0.4120 |
0.3736 |
0.3443 |
0.3209 |
| 2 |
|
1.5272 |
0.9892 |
0.7907 |
0.6786 |
0.6040 |
0.5496 |
0.5078 |
| 3 |
|
|
1.7962 |
1.1877 |
0.9589 |
0.8279 |
0.7398 |
0.6752 |
| 4 |
|
|
|
1.9991 |
1.3403 |
1.0900 |
0.9453 |
0.8474 |
| 5 |
|
|
|
|
2.1638 |
1.4654 |
1.1984 |
1.0433 |
| 6 |
|
|
|
|
|
2.3035 |
1.5722 |
1.2916 |
| 7 |
|
|
|
|
|
|
2.4253 |
1.6658 |
| 8 |
|
|
|
|
|
|
|
2.5339 |
Table 2. Variances of order statistics from Pareto-Rayleigh distribution (\(\alpha\)=2, \(\sigma\)=1).
|
n |
| r |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
| 1 |
0.7663 |
0.1847 |
0.1011 |
0.0692 |
0.0525 |
0.0422 |
0.0353 |
0.0303 |
| 2 |
|
1.0009 |
0.2214 |
0.1176 |
0.0792 |
0.0594 |
0.0475 |
0.0395 |
| 3 |
|
|
1.1735 |
0.2464 |
0.1281 |
0.0852 |
0.0635 |
0.0504 |
| 4 |
|
|
|
1.3180 |
0.2672 |
0.1366 |
0.0900 |
0.0666 |
| 5 |
|
|
|
|
1.4450 |
0.2855 |
0.1441 |
0.0942 |
| 6 |
|
|
|
|
|
1.5599 |
0.3021 |
0.1509 |
| 7 |
|
|
|
|
|
|
1.6655 |
0.3175 |
| 8 |
|
|
|
|
|
|
|
1.7639 |
Table 3. Means of gos from Pareto-Rayleigh distribution (\(\alpha\)=2, \(\sigma\)=1, \(m\)=1, \(k\)=2).
|
n |
| r |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
| 1 |
0.3471 |
0.2327 |
0.1868 |
0.1605 |
0.1428 |
0.1300 |
0.1200 |
0.1121 |
| 2 |
|
0.2308 |
0.1622 |
0.1329 |
0.1155 |
0.1036 |
0.0947 |
0.0878 |
| 3 |
|
|
0.1325 |
0.0957 |
0.0795 |
0.0697 |
0.0628 |
0.0577 |
| 4 |
|
|
|
0.0724 |
0.0532 |
0.0447 |
0.0394 |
0.0357 |
| 5 |
|
|
|
|
0.0386 |
0.0288 |
0.0243 |
0.0216 |
| 6 |
|
|
|
|
|
0.0203 |
0.0153 |
0.0130 |
| 7 |
|
|
|
|
|
|
0.0105 |
0.0080 |
| 8 |
|
|
|
|
|
|
|
0.0054 |
Table 4. Variances of gos from Pareto-Rayleigh distribution (\(\alpha\)=2, \(\sigma\)=1, \(m\)=1, \(k\)=2).
|
n |
| r |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
| 1 |
0.2128 |
0.0887 |
0.0560 |
0.0409 |
0.0322 |
0.0266 |
0.0226 |
0.0197 |
| 2 |
|
0.2086 |
0.0971 |
0.0641 |
0.0481 |
0.0385 |
0.0321 |
0.0275 |
| 3 |
|
|
0.1480 |
0.0733 |
0.0499 |
0.0380 |
0.0308 |
0.0259 |
| 4 |
|
|
|
0.0914 |
0.0471 |
0.0327 |
0.0253 |
0.0207 |
| 5 |
|
|
|
|
0.0527 |
0.0280 |
0.0197 |
0.0154 |
| 6 |
|
|
|
|
|
0.0292 |
0.0158 |
0.0113 |
| 7 |
|
|
|
|
|
|
0.0158 |
0.0087 |
| 8 |
|
|
|
|
|
|
|
0.0084 |
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the
final manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
