Distributional Properties of Order Statistics of Weibull Distribution

Let {Xk,1≤k≤n} be independent and identically distributed random variables,X(1)≤X(2)≤…≤X(n) be their order statistics.The joint probability density function of its order statistics and the density functions of extreme order statistics are derived,when X(k) followed Weibull distribution with parameter m and η.The mathematical expectations and variances of X(1) and X(n) are also obtained.What's more it proved the sample intervals X(1),X(2)-X(1),…,X(n)-X(n-1) are not independent and not identical distributions when the parameter m≠1,the sample intervals X(1),X(2)-X(1),…,X(n)-X(n-1) are independent but not identical distributions when the parameter m=1.