arXiv subjects

Ismihan Bairamov

Publications and source records attributed to Ismihan Bairamov.

4 recordsLinked to original sources

arXiv2019

Joint distribution of a random sample and an order statistic: A new approach with an application in reliability analysis

This paper considers the joint distribution of elements of a random sample and an order statistic of the same sample. \ The motivation for this work stems from the important problem in reliability analysis, to estimate the number of inspections we need in order to detect failed components in a coherent system. We consider an $(n-r+1)$-out-of-$n$ system, which is intact until at least $n-r+1$ of the components are alive, and it fails if the number of failed components exceeds $r$. The life time of the system is the $% r$th order statistic. Assuming that some of the components failed but the system \ is still functioning, \ using the results presented in this paper it is possible to find an expected value of the number of inspections we need to do for detecting certain number of failed components.

math.ST↗

arXiv2011

Concomitants and majorization bounds for bivariate distribution function

Let ($X,Y)$ be a random vector with distribution function $F(x,y),$ and $(X_{1},Y_{1}),(X_{2},Y_{2}),...,(X_{n},Y_{n})$ are independent copies of ($X,Y).$ Let $X_{i:n}$ be the $i$th order statistics constructed from the sample $X_{1},X_{2},...,X_{n}$ of the first coordinate of the bivariate sample and $Y_{[i:n]}$ be the concomitant of $X_{i:n}.$ Denote $F_{i:n}% (x,y)=P\{X_{i:n}\leq x,Y_{[i:n]}\leq y\}.$ Using majorization theory we write upper and lower bounds for $F$ expressed in terms of mixtures of joint distributions of order statistics and their concomitants, i.e. ${\dsum \limits_{i=1}^{n}}% {\sum\limits_{i=1}^{n}} p_{i}F_{i:n}(x,y)$ and ${\dsum \limits_{i=1}^{n}}% {\sum\limits_{i=1}^{n}} p_{i}F_{n-i+1:n}(x,y).$ It is shown that these bounds converge to $F$ for a particular sequence $(p_{1}(m),p_{2}(m),...,p_{n}(m)),m=1,2,..$ as $m\rightarrow\infty.$

math.ST↗

arXiv2011

Majorization bounds for distribution function

Let $X$ be a random variable with distribution function $F,$ and $X_{1},X_{2},...,X_{n}$ are independent copies of $X.$ Consider the order statistics $X_{i:n},$ $i=1,2,...,n$ and denote $F_{i:n}(x)=P\{X_{i:n}\leq x\}.$ Using majorization theory we write upper and lower bounds for $F$ expressed in terms of mixtures of distribution functions of order statistics, i.e. $\sum \limits_{i=1}^{n}p_{i}F_{i:n}$ and $\sum \limits_{i=1}^{n}p_{i}F_{n-i+1:n}.$ It is shown that these bounds converge to $F$ \ for a particular sequence $(p_{1}(m),p_{2}(m),...,p_{n}(m)),m=1,2,..$ as $m\rightarrow\infty.$

math.ST↗

arXiv2011

Copula representations and order statistics for conditionally independent random variables

The copula representations for conditionally independent random variables and the distribution properties of order statistics of these random variables are studied.

math.ST↗