Some Properties About Order Statistics And Their Concomitants Of Random Samples

Based on stochastic comparison of some common stochastic orders can reflect the application of order statistics and its concomitants. Aiming at random and non-random samples size, we investigate the stochastic comparison of their order statistics and concomitants. We also study the dependence of the minimus and maximum order statistics.In this dissertation, we study three aspects. Firstly, the stochastic comparisons of their minimum and maximum order statistics corresponding random and non-random sample size respectively were studied. If the random variables were independent and identically distribution, we show that the minimum order statistics of random N sample is better than non-random n sample by using of the Jensen inequality when the random variable N satisfies E(N) ≤n. There is an inverse conclusion for the maximum order statistics. If the random variables are identically distribution but not independent, implementing an exchangeable copula function to measure the dependent among the random variables when we assume that the dependent relationships between random variables are exchangeable. At the same time, by using of the convexity of discrete functions we can obtain the same results. In system reliability analysis, the minimum and maximum order statistics corresponds to the lifetime of series and parallel systems respectively. Naturally, the conclusion which reveals the lifetime of series system composed of N components is better than that of the system which is composed of non-random n components under the usual stochastic order; meanwhile, the lifetime of parallel stytem which is composed of non-random n components is better than that of the system which is composed of N components under the usual stochastics order. We will reflect the application of these results through six examples in this paper. In addition, if the lifetime of all components obey exponential distribution, we research the MTTF of k/ n(G) and n-k+1/ n(G) system when E(N) ≤n.Secondly, for random and non-random sample size, the stochastic comparisons of their concomitants were considered when( X,Y) satisfy Farlie-Gumbel-Morgenstern type bivarivate uniform distributions. We obtained the relationship of random N sample and the non-random n sample under the usual stochastic order when E(N) ≤n. The stochastic comparison of concomitants are concerned further when random variables1 N and2N satisfy different relationship. This work extends the available results on the stochastic comparisons of concomitants in the literature, which usually not consider the stochastic comparisons of concomitants from random and non-random sample.Finally, samples 1 2,,,nX X X and1 2,,,nY Y Y from( X,Y), we got the copula function of(1:n)X and(n:n)Y which must keep same quadrant dependence if the copula of( X,Y) had quadrant dependence. In addition,(1:n)X and(n:n)Y is asymptotic independence when n â†'∞.