Answer Happy

3. Suppose X has the following density function: fX​(x)=θxθ−1,0≤x≤1 for some parameter θ. (a) What is the MLE for θ based on a random sample Xi​,1≤i≤n, from this distribution? (b) Show that the likelihood ratio is monotone (increasing or decreasing) in Πi=1n​Xi​ for any pair of θ0​ and θ1​. (c) What is the LRT for the case H0​:θ=θ0​ versus H1​:θ=θ1​. Consider the two cases, one where θ0​>θ1​ and the other where θ0​<θ1​. (d) Design in detail such a test for a given significance level of α in the case where n=1.