Computation of Power Loss in Likelihood Ratio Tests for Probability Densities Extended by Lehmann Alternatives

We compute the loss of power in likelihood ratio tests when we test the original parameter of a probability density extended by the first Lehmann alternative. 1 Distributions Generated by Lehmann Alternatives In the context of parametric models for lifetime data, [Gupta et alii 1998] disseminated the study of distributions generated by Lehmann alternatives, cumulative distributions that take one of the following forms: G1 (x, λ) = [F (x)] λ or G2 (x, λ) = 1− [1− F (x)] (1) where F (x) is any cumulative distribution and λ > 0. In the present note, we are going to call both G distributions generated distributions or extended distributions. It is easy to see that for integer values of λ, G1 and G2 are, respectively, the distribution of the maximum and the minimum of a sample of size λ, the support of the two distribution is the same of F , and that the associated density functions are g1 (x, λ) = λf(x) [F (x)] λ−1 and g2 (x, λ) = λf(x) [1− F (x)] (2) where f(x) is the density function associated with F . Suppose that we generate a distribution G(x|λ) based on the distribution F (x), and want to generate another distribution G′(x|λ, λ′) repeating the process; It is easy to see that the distribution G′ will be the same as G, for the new parameter of the distribution, λλ′ may be summarized as a single one. This has 1 ar X iv :0 70 4. 07 39 v2 [ m at h. ST ] 1 1 A pr 2 00 7 the interesting side effect that the standard uniparametric exponential distribution may be seen as a distribution generated by the second Lehmann alternative from the distribution F (x) = 1− e−x. To compute the moments of distribution generated by Lehmann alternatives, we use the change of variables u = F (x) in the expression