Expressions for moments of order statistics and records from the skew-normal distribution in terms of multivariate normal orthant probabilities

Abstract

In designs of experiments and reliability analyze, order statistics (OS) are used for various purposes including model checking, estimations of parameters and prediction. Most of these procedures are defined on the basis of expectations of OS problems. In this paper, explicit expressions for central moments of OS coming from the skew-normal (SN) distribution are derived. The SN model enjoys interesting properties from the normal distribution while captures asymmetric behavior in the parent population. Another important topic which is related to OS is record statistics. These data are arising in some practical situations including shock models, sports and epoch times of a non-homogeneous Poisson processes. Here, we derive moments of the upper and the lower record values arising from the skew normal distribution. The obtained results may be used for prediction purposes such as predictive maintenance in a repairable system and prediction of performance of a transmission oil pipeline. Some real data sets are analyzed using the results obtained for illustration purposes.

Appendix

Proof of Lemma 3

Assuming $g(t)=\varPhi _{SN}^{k}(t)$, a simple integration yields [see also Chiogna (1998)]

$$\begin{aligned} kE\left[ \phi _{SN}(Z_{\lambda };\lambda )\varPhi _{SN}^{k-1}(Z_{\lambda };\lambda )\right] \le kE\left[ \phi _{SN}(Z_{\lambda };\lambda )\right] =\frac{2k}{\pi ^{1.5}}\tan ^{-1}\sqrt{1+\lambda ^2}<\infty .\nonumber \\ \end{aligned}$$

(43)

So, one can use Lemma 2 for computing the expectation $E\left[ Z_{\lambda }\varPhi _{SN}^k(Z_{\lambda };\lambda )\right]$. Substituting $\lambda _1=\lambda$ and $\lambda _2=\rho =0$ into (11) and (12), we have (for $k\ge 2$)

$$\begin{aligned} E\left[ Z_{\lambda }\varPhi _{SN}^k(Z_{\lambda };\lambda )\right]= & {} kE \left[ \phi _{SN}(Z_{\lambda };\lambda )\varPhi _{SN}^{k-1}(Z_{\lambda };\lambda )\right] \nonumber \\&\quad +\,\frac{2\alpha }{\sqrt{2\pi }} E\left[ \varPhi _{SN}^{k}\left( \frac{Z_{0}}{\sqrt{1+\lambda ^2}};\lambda \right) \right] , \end{aligned}$$

(44)

where $\alpha =\frac{\lambda }{\sqrt{1+\lambda ^2}}$ and $Z_0\sim N(0,1)$. The first term in right hand side (RHS) of (44) can be expressed as

$$\begin{aligned} E\left[ \phi _{SN}(Z_{\lambda };\lambda )\varPhi _{SN}^{k-1}(Z_{\lambda };\lambda )\right]= & {} \int _{-\infty }^{+\infty }\phi ^2_{SN}(z;\lambda )\varPhi _{SN}^{k-1}(z;\lambda )dz\nonumber \\= & {} \frac{2}{\sqrt{\pi }}\int _{-\infty }^{+\infty }\phi (z)\varPhi ^2\left( \frac{\lambda }{\sqrt{2}}z\right) \varPhi _{SN}^{k-1}\left( \frac{z}{\sqrt{2}};\lambda \right) dz\nonumber \\= & {} \frac{\cos ^{-1}\left( \frac{-\lambda ^2}{2+\lambda ^2}\right) }{\pi ^ {1.5}}\nonumber \\&\quad \!\times \!\, \int _{-\infty }^{+\infty }\phi _{GSN}\left( z;\frac{\lambda }{\sqrt{2}},\frac{\lambda }{\sqrt{2}},0\right) \varPhi _{SN}^{k-1}\left( \frac{z}{\sqrt{2}};\lambda \right) dz\nonumber \\= & {} \frac{\cos ^{-1}\left( \frac{-\lambda ^2}{2+\lambda ^2}\right) }{\pi ^ {1.5}} E\left[ \varPhi _{SN}^{k-1}\left( \frac{Z_{\frac{\lambda }{\sqrt{2}},\frac{\lambda }{\sqrt{2}}}}{\sqrt{2}};\lambda \right) \right] , \end{aligned}$$

(45)

where $Z_{\frac{\lambda }{\sqrt{2}},\frac{\lambda }{\sqrt{2}}}\sim GSN\left( \frac{\lambda }{\sqrt{2}},\frac{\lambda }{\sqrt{2}},0\right)$. Let $Z_{\lambda }^{(n)}$’s, $1\le n \le k-1$ be iid random variables coming from the $SN(\lambda )$-distribution. Thus

$$\begin{aligned} E\left[ \varPhi _{SN}^{k-1}\left( \frac{Z_{\frac{\lambda }{\sqrt{2}},\frac{\lambda }{\sqrt{2}}}}{\sqrt{2}};\lambda \right) \right]= & {} \Pr \left\{ Z_{\lambda }^{(1)}\le Z_{\frac{\lambda }{\sqrt{2}},\frac{\lambda }{\sqrt{2}}}, {\ldots }, Z_{\lambda }^{(k-1)}\le Z_{\frac{\lambda }{\sqrt{2}},\frac{\lambda }{\sqrt{2}}} \right\} . \end{aligned}$$

From (2) and (5) with $\gamma =\rho =0$, we have

$$\begin{aligned}= & {} \Pr \left\{ X_1\le \frac{X}{\sqrt{2}}, {\ldots }, X_{k-1}\le \frac{X}{\sqrt{2}} \bigg |~ \lambda X_1>Y_1, {\ldots }, \lambda X_{k-1}>Y_{k-1},\frac{\lambda }{\sqrt{2}} X>Y^{\star }_1,\right. \nonumber \\&\quad \left. \frac{\lambda }{\sqrt{2}} X>Y^{\star }_2\right\} ,\nonumber \\= & {} \Pr \left\{ \mathbf {V}_1\le \mathbf {0}|~\mathbf {W}_1>\mathbf {0}\right\} , \end{aligned}$$

(46)

where $X,Y_1^{\star },Y_2^{\star },X_{n},Y_{n},1\le n \le k-1$, are iid random variables arising from $N(0,1)$ and

$$\begin{aligned} \left( \begin{array}{l} \mathbf {W}_1\\ \mathbf {V}_1 \end{array}\right) \sim N_{2k}\left( \left( \begin{array}{l} \mathbf {0}\\ \mathbf {0} \end{array}\right) , \left( \begin{array}{ll} \mathbf {I}_{k+1}&{}\quad {\varvec{\Delta }}_1^T\\ {\varvec{\Delta }}_1&{}\quad {\varvec{\Sigma }}_{k-1,1} \end{array}\right) \right) , \end{aligned}$$

(47)

with partitioned covariance matrix ${\varvec{\Delta }}_1$ as

$$\begin{aligned} {\varvec{\Delta }}^{(1)}_{{k,\lambda }}=\left( \begin{array}{lllll} \frac{\lambda }{\sqrt{1.5(1+\lambda ^2)}}\mathbf {I}_{k-1}&\vdots&-\frac{\lambda }{\sqrt{6+3\lambda ^2}}\mathbf {1}_{k-1}&\vdots&-\frac{\lambda }{\sqrt{6+3\lambda ^2}}\mathbf {1}_{k-1} \end{array}\right) _{(k-1)\times (k+1)} \end{aligned}$$

(48)

and

$$\begin{aligned} {\varvec{\Sigma }}_{k-1,1}=\left( \begin{array}{llll} 1&{}\quad &{}\quad &{}\quad \\ \frac{1}{3}&{}\quad 1&{}\quad &{}\quad \\ \vdots &{}\quad &{}\quad \ddots &{}\quad \\ \frac{1}{3}&{}\quad \cdots &{}\quad \frac{1}{3}&{}\quad 1 \end{array}\right) _{(k-1)\times (k-1)}. \end{aligned}$$

(49)

From (6), (7) and (46), we conclude that

$$\begin{aligned} E\left[ \varPhi _{SN}^{k-1}\left( \frac{1}{\sqrt{2}}Z_{\frac{\lambda }{\sqrt{2}},\frac{\lambda }{\sqrt{2}}};\lambda \right) \right] =\varPhi _{ SUN _{k-1,k+1}}\left( \mathbf {0}_{k-1};\mathbf {0}_{k-1},\mathbf {0}_{k+1}, {\varvec{\Sigma }}_{k-1,1},\mathbf {I}_{k+1},{\varvec{\Delta }}_1 \right) .\nonumber \\ \end{aligned}$$

(50)

Similarly, for the second expression in RHS of (44), we have

$$\begin{aligned} E\left[ \varPhi _{SN}^{k}\left( \frac{Z_{0}}{\sqrt{1+\lambda ^2}};\lambda \right) \right] = \Pr \left\{ \mathbf {V}_2\le \mathbf {0}_{k}~|~\mathbf {W}_2>\mathbf {0}_{k}\right\} \end{aligned}$$

where

$$\begin{aligned} \left( \begin{array}{l} \mathbf {W}_2\\ \mathbf {V}_2 \end{array}\right) \sim N_{2k}\left( \mathbf {0}_{2k},\left( \begin{array}{ll} \mathbf {I}_{k}&{}\quad {\varvec{\Delta }}_2^T\\ {\varvec{\Delta }}_2&{}\quad {\varvec{\Sigma }}_{k,\lambda } \end{array}\right) \right) ,\quad {\varvec{\Delta }}^{(2)}_{{k,\lambda }}=\frac{\lambda }{\sqrt{2+\lambda ^2}}\mathbf {I}_{k} \end{aligned}$$

(51)

and

$$\begin{aligned} {\varvec{\Sigma }}_{k,\lambda }=\left( \begin{array}{llll} 1&{}\quad &{}\quad &{}\quad \\ \frac{1}{2+\lambda ^2}&{}\quad 1&{}\quad &{}\\ \vdots &{}\quad &{}\quad \ddots &{}\quad \\ \frac{1}{2+\lambda ^2}&{}\quad \cdots &{}\quad \frac{1}{2+\lambda ^2}&{}\quad 1 \end{array}\right) _{k\times k}. \end{aligned}$$

(52)

Thus

$$\begin{aligned} E\left[ \varPhi _{SN}^{k}\left( \frac{Z_{0}}{\sqrt{1+\lambda ^2}};\lambda \right) \right] = \varPhi _{ SUN _{k,k}}\left( \mathbf {0}_{k};\mathbf {0}_{k},\mathbf {0}_{k}, {\varvec{\Sigma }}_{k,\lambda },\mathbf {I}_{k},{\varvec{\Delta }}^{(2)}_{{k,\lambda }} \right) .\quad \end{aligned}$$

(53)

By substituting (45)–(53) into (44) and using Lemma 1, the desired result follows provided that $k\ge 2$. For $k=1$, with a similar manner we have

$$\begin{aligned} E\left[ Z_{\lambda }\varPhi _{SN}(Z_{\lambda };\lambda )\right]= & {} \frac{2}{\pi ^{1.5}}\tan ^{-1}\sqrt{1+\lambda ^2}\nonumber \\&\quad +\,\frac{2\lambda }{\sqrt{2\pi (1+\lambda ^2)}}\varPhi _{ SUN _{1,1}}\left( 0;0,0,1,1,\frac{\lambda }{\sqrt{2+\lambda ^2}}\right) . \end{aligned}$$

(54)

Finally, using Lemma 1 and the fact that $\varPhi _2\left( \mathbf {0};\left( \begin{array}{ll} 1&{}\quad \rho \\ \rho &{}\quad 1\\ \end{array}\right) \right) =\frac{1}{4}+\frac{1}{2\pi }\sin ^{-1}\rho$ [see for example Kotz et al. (2000)], the proof is completed. $\square$

Proof of Proposition 1

Set $d+i-1=j$, and from the density (13), we have

$$\begin{aligned} M_{Z_{(i{:}n),\lambda }}(s;\lambda )= & {} i\left( \begin{array}{l} n\\ i \end{array}\right) \sum \limits _{d=0}^{n-i}\left( \begin{array}{l} n-i\\ d \end{array}\right) (-1)^dI(s,j;\lambda ), \end{aligned}$$

(55)

where

$$\begin{aligned} I(s,j;\lambda )=\int _{-\infty }^{\infty }e^{sz}\phi _{SN}(z;\lambda )\varPhi _{SN}^{j}(z;\lambda )dz. \end{aligned}$$

(56)

From (1) and a change of variable, the integration $I(s,j;\lambda )$ in (56) simplifies to

$$\begin{aligned} I(s,j;\lambda )= & {} 2e^{\frac{s^2}{2}}\varPhi \left( \frac{\lambda s}{\sqrt{1+\lambda ^2}}\right) \int _{-\infty }^{\infty }\phi _{ ESN }(u;\lambda ,\lambda s)\varPhi _{SN}^{j}(u+s;\lambda )du\nonumber \\= & {} 2e^{\frac{s^2}{2}}\varPhi \left( \frac{\lambda s}{\sqrt{1+\lambda ^2}}\right) E\left[ \varPhi _{SN}^{j}\left( Z^{\star }_{\lambda ,\lambda s}+s;\lambda \right) \right] , \end{aligned}$$

(57)

where $Z^{\star }_{\lambda ,\lambda s}\sim ESN (\lambda ,\lambda s)$. Let $Z_{\lambda }^{(1)}, {\ldots },Z_{\lambda }^{(j)}\mathop {\sim }\limits ^{iid}SN(\lambda )$ and be independent of the random variable $Z^{\star }_{\lambda ,\lambda s}$ in (57). Then

$$\begin{aligned} E\left[ \varPhi _{SN}^{j}\left( Z^{\star }_{\lambda ,\lambda s}+s;\lambda \right) \right] = \Pr \left\{ Z_{\lambda }^{(1)}-Z^{\star }_{\lambda ,\lambda s}\le s, {\ldots },Z_{\lambda }^{(j)}-Z^{\star }_{\lambda ,\lambda s}\le s\right\} , \end{aligned}$$

and the stochastic representation (2) imply

$$\begin{aligned} E\left[ \varPhi _{SN}^{j}\left( Z^{\star }_{\lambda ,\lambda s}+s;\lambda \right) \right] = \Pr \left\{ \mathbf {V}\le \frac{s}{\sqrt{2}}\mathbf {1}_{j}~|~\mathbf {W}\ge \mathbf {0}_{j+1}\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathbf {V}= & {} \frac{1}{\sqrt{2}}\left( X_1-X^{\star }, {\ldots },X_j-X^{\star }\right) ^T,\\ \mathbf {W}= & {} \frac{1}{\sqrt{1+\lambda ^{2}}} \left( \lambda X_1-Y_1, {\ldots },\lambda X_j-Y_j, \lambda X^{\star }-Y^{\star }+\lambda s \right) ^T, \end{aligned}$$

and $X^{\star },Y^{\star },X_{n},Y_{n},1\le n \le j$, are copies from the standard normal distribution. Finally, by (6) and (7) we get

$$\begin{aligned} M_{Z_{(i{:}n),\lambda }}(s;\lambda )= & {} 2i\left( \begin{array}{l} n\\ i \end{array}\right) e^{\frac{s^2}{2}}\varPhi \left( \frac{\lambda s}{\sqrt{1+\lambda ^2}}\right) \sum \limits _{d=0}^{n-i}(-1)^{d}\left( \begin{array}{l}n-i \\ d \end{array}\right) \nonumber \\&\quad \times \,\varPhi _{ SUN _{j,j+1}}\left( \frac{s}{\sqrt{2}}\mathbf {1}_{j}; \mathbf {0}_j,{\varvec{\gamma }},\frac{1}{2}[\mathbf {1}_{j}\mathbf {1}_{j}^{T}+ \mathbf {I}_{j}],\mathbf {I}_{j+1},{\varvec{\Delta }}\right) . \end{aligned}$$

(58)

By applying Lemma 1, the required result follows. $\square$

Proof of Eq. (36)

From (30) and (35), the expectation of the $m$th lower record from the $SN(\lambda )$-distribution is

$$\begin{aligned} E[L_{(m),\lambda }]= & {} \frac{1}{(m-1)!}\int _{-\infty }^{+\infty }u\left[ \sum \limits _{i=1}^{+\infty }\frac{\bar{\varPhi }_{SN}^{i}(u;\lambda )}{i}\right] ^{m-1}\phi _{SN}(u;\lambda )du\nonumber \\= & {} \frac{1}{(m-1)!}\sum \limits _{i_1=1}^{+\infty }\ldots \sum \limits _{i_{m-1}=1}^{+\infty }\sum \limits _{l_1=0}^{i_1}\ldots \sum \limits _{l_{m-1}=0}^{i_{m-1}}\frac{1}{i_1\ldots i_{m-1}}\prod \limits _{r=1}^{m-1}\left[ \left( \begin{array}{l} {i_r}\\ {l_r} \end{array}\right) (-1)^{l_r}\right] \nonumber \\&\quad \times \,\int _{-\infty }^{+\infty }u\phi _{SN}(u;\lambda )\left[ \varPhi _{SN}(u;\lambda )\right] ^{l_1+\cdots +l_{m-1}}du\nonumber \\= & {} \frac{1}{(m-1)!}\sum \limits _{i_1=1}^{+\infty }\ldots \sum \limits _{i_{m-1}=1}^{+\infty }\sum \limits _{l_1=0}^{i_1}\ldots \sum \limits _{l_{m-1}=0}^{i_{m-1}}\frac{1}{i_1\ldots i_{m-1}}\prod \limits _{r=1}^{m-1}\left[ \left( \begin{array}{l} {i_r}\\ {l_r} \end{array}\right) (-1)^{l_r}\right] \nonumber \\&\quad \times \,E\left( Z_{\lambda }[\varPhi _{SN}(Z_{\lambda };\lambda )]^{l_1+\cdots +l_{m-1}}\right) , \end{aligned}$$

where $Z_{\lambda }\sim SN(\lambda )$. So, the proof is completed. $\square$