Variable sample size and inference in samples from gamma populations (Q1083781)

Let G(\(\theta\),\(\alpha)\) stand for a gamma distribution with scale parameter \(\theta\) and integer shape parameter \(\alpha\). Suppose \(X_{(j_ i)}\) is the \(j_ i th\) order statistic from a random sample of size \(n_ i\) from \(G(\theta_ i,\alpha)\), \(i=1\) to k. Assume that the k samples are independent. First closed form expression for the probability density function (pdf) of \(X_{(j_ i)}\) is obtained when \(n_ i\) is fixed as well as when \(n_ i\) is having a Poisson distribution. In the latter case, an expression of the joint pdf of the ratios \(X_{(j_ i)}/X_{(j_ k)}\), \(i=1\) to k-1 is obtained. Finally, for fixed \(n_ i\) and \(\alpha =1\) (that is, exponential populations), pdf of the sum \(\sum^{k}_{i=1}X_{(j_ i)}\) is exhibited. How these distributional results are useful in inference on these populations is not made clear.