Bl. (a) (b) In a test about a real parameter 0, the null and alternative hypotheses are: H:00, and H:0>0. Explain what is meant by a Uniformly Most Powerful (UMP) test at significance level a. (3 Marks) A random sample X, X.....X, is drawn from a population with probability density function (pdf) f(x,0)= (i) P(-1) Aexp 20 Aexp . *so x>0 where a>0 is an unknown parameter and A= all x values with x > 0. 2 (1+0) What is the well-known distribution when a = 1? (2 marks) A test is required of the null hypothesis Ho=1 against the alternative hypothesis H:=o', where o'>1 is a fixed value. Show that the Neyman-Pearson's approach leads to the rejection of the null hypothesis when >k for some suitably chosen k, where x is the sum of for 0 1. (4 marks) (iv) Perform the Generalised Likelihood Ratio Test (GLRT) of H:0=1 against the alternative H:01 when Here is the unrestricted Maximum Likelihood Estimate (MLE) of a, and you should use an approximate 5% significance level. (8 marks) Bl. (a) (b) In a test about a real parameter 0, the null and alternative hypotheses are: H:00, and H:0>0. Explain what is meant by a Uniformly Most Powerful (UMP) test at significance level a. (3 Marks) A random sample X, X.....X, is drawn from a population with probability density function (pdf) f(x,0)= (i) P(-1) Aexp 20 Aexp . *so x>0 where a>0 is an unknown parameter and A= all x values with x > 0. 2 (1+0) What is the well-known distribution when a = 1? (2 marks) A test is required of the null hypothesis Ho=1 against the alternative hypothesis H:=o', where o'>1 is a fixed value. Show that the Neyman-Pearson's approach leads to the rejection of the null hypothesis when >k for some suitably chosen k, where x is the sum of for 0 1. (4 marks) (iv) Perform the Generalised Likelihood Ratio Test (GLRT) of H:0=1 against the alternative H:01 when Here is the unrestricted Maximum Likelihood Estimate (MLE) of a, and you should use an approximate 5% significance level. (8 marks)