The Properties Of Order Statistics From Some Specific Distributions

Let X1, X2,..., Xn be non-negative independent and identically distributed with a density function f(x) and continuous cumulative distribution function F(x).The corresponding order statistics are the X1:n≤ X2:n≤...≤Xn:n, The spacing of the order statistics is Xs:n -Xr:n,1≤r≤s≤n. For the mean, variance, moment and other properties of the order statistic Xk:n, it had has complete results for specific distributions such as uniform distribution, exponential distribution and normal distributions. However, the study of order statistics spacing is less.In this paper, firstly, we study the change of spacing Xk+1:n-Xk:n when the probability density function is monotonous; secondly, we discuss when the support of the order statistics Xk:n equal to the spacing’s (Xs:n-Xr:n); Finally, when the density function is piecewise constant, we obtain the expression of the mean of the order statistic Xk:n, and got the results of EXk:n and the m th moment ofXk.n while nâ†' ∞. For our interested distribution, the expression of mean of the order statistic and simulation results are given.