Uniformly Most Powerful

Question

Let X1;X2; : : : ;X10 denote a random sample of size 10 from a population which has an exponential distribution with parameter ; > 0, i.e. with pdf f(x) =   e x if x 0; 0 otherwise. (a) Find the Uniformly Most Powerful Test (UMP) test of size = 0:01 for testing H0 : = 1 against H1 : = 2: (b) Suppose the sample selected have values 0:24; 0:34; 0:12; 0:19; 0:18; 0:68; 0:03; 0:70; 0:07; 1:31: Use this sample to test H0 against H1 at level = 0:01. (c) To test H0 : 1 against H1 : > 1 based on a random sample of size n = 10, the following critical region is prescribed: R = f(x1; x2; : : : ; x10) : Σ10 i=1 xi 1:5g: (i) Derive and sketch the power function () of this test. (ii) Give an approximate value for the size of this test. (iii) Is this test unbiased? Exp