Question 3 (30 marks; 30 minutes) A researcher plans on using a random sample X1, ..., Xn of size n from a population with probability density function e'mZ/Z'B a: 2 0 mm) ={ 0 elsewhere, where parameter L? > 0. (a) Derive an expression for the test statistic LRS = 2 lnA for undertaking a likeli- hood ratio test of the null hypothesis H0 : B = 1 against the alternative hypothesis H0 : B a 1 when the sample size n is large (say, n = 500). (b) Explain the test procedure to the researcher, paying attention to the assumptions, distribution and interpretation. (c) Explain how you would use simulation methods to determine the small sample distribution of the test statistic LRS and how it could be used to undertake the above hypothesis test in small samples