Two hours MATH20802 To be supplied by the Examinations Office: Mathematical Formula Tables and Statistical Tables THE UNIVERSITY OF MANCHESTER STATISTICAL METHODS Answer any FOUR of the SIX questions. University-approved calculators may be used 1 of 7 P.T.O.
1. A random variable X is said to have the Gumbel distribution, written X Gumbel(µ, β), if its probability density function is given by f X (x) = 1 ( β exp x µ ) { ( exp exp x µ )} β β for < x <, < µ < and β > 0. (i) Show that the cumulative distribution function of X is { ( F X (x) = exp exp x µ )} β for < x <. (7 marks) (ii) Show that the moment generating function of X is M X (t) = exp(µt)γ(1 βt) for t < 1/β, where Γ( ) denotes the gamma function. (7 marks) (iii) Show that E(X) = µ βγ (1), where Γ ( ) denotes the first derivative of Γ( ). (iv) If X i Gumbel(µ, β), i = 1, 2,..., n are independent random variables then show that max(x 1, X 2,..., X n ) Gumbel(µ + β log n, β). (7 marks) 2 of 7 P.T.O.
2. (a) Suppose θ is an estimator of θ based on a random sample of size n. Define what is meant by the following: (i) θ is an unbiased estimator of θ; (ii) θ is an asymptotically unbiased estimator of θ; (iii) the bias of θ (written as bias( θ)); (iv) the mean squared error of θ (written as MSE( θ)); (v) θ is a consistent estimator of θ. (b) Suppose X 1, X 2,..., X n is a random sample from the Exp (λ) distribution. Consider the following estimators for θ = 1/λ: θ1 = (1/n) n i=1 X i and θ 2 = (1/(n + 1)) n i=1 X i. (i) Find the biases of θ 1 and θ 2. (ii) Find the variances of θ 1 and θ 2. (iii) Find the mean squared errors of θ 1 and θ 2. (iv) Which of the two estimators ( θ 1 or θ 2 ) is better and why? (3 marks) 3 of 7 P.T.O.
3. Consider the two independent random samples: X 1, X 2,..., X n from N(µ X, σ 2 ) and Y 1, Y 2,..., Y m from N(µ Y, σ 2 ), where σ 2 is assumed known. The parameters µ X and µ Y are assumed not known. (i) Write down the joint likelihood function of µ X and µ Y. (ii) Find the maximum likelihood estimators (mles) of µ X and µ Y. (10 marks) (iii) Find the mle of Pr(X < Y ), where X N(µ X, σ 2 ) and Y N(µ Y, σ 2 ) are independent random variables. (iv) Show that the mle of µ X in part (ii) is an unbiased and consistent estimator for µ X. (3 marks) (v) Show also that the mle of µ Y in part (ii) is an unbiased and consistent estimator for µ Y. 4 of 7 P.T.O.
4. Suppose X 1, X 2,..., X n is a random sample from N(µ, σ 2 ), where both µ and σ 2 are unknown. (i) Write down the joint likelihood function of µ and σ 2. (ii) Show that the maximum likelihood estimator (mle) of µ is µ = X, where X = (1/n) n i=1 X i is the sample mean. (iii) Show that the mle of σ 2 is σ 2 = (1/n) n i=1 (X i X) 2. (iv) Show that the mle, µ, is an unbiased and consistent estimator for µ. (v) Show that the mle, σ 2, is a biased and consistent estimator for σ 2. 5 of 7 P.T.O.
5. (a) Suppose we wish to test H 0 : θ = θ 0 versus H 1 : θ θ 0. Define what is meant by the following: (i) the Type I error of a test. (ii) the Type II error of a test. (iii) the significance level of a test. (iv) the power function of a test (denoted Π(θ)). (b) Suppose X 1, X 2,..., X n is a random sample from a Bernoulli distribution with parameter p. State the rejection region for each of the following tests: (i) H 0 : p = p 0 versus H 1 : p p 0. (ii) H 0 : p = p 0 versus H 1 : p < p 0. (iii) H 0 : p = p 0 versus H 1 : p > p 0. In each case, assume a significance level of α and that X = (X 1 +X 2 + +X n )/n has an approximate normal distribution. (c) Under the same assumptions as part (b), find the power function, Π(p), for each of the tests: (i) H 0 : p = p 0 versus H 1 : p p 0. (ii) H 0 : p = p 0 versus H 1 : p < p 0. (iii) H 0 : p = p 0 versus H 1 : p > p 0. (3 marks) In each case, you may express the power function, Π(p), in terms of Φ( ), the standard normal distribution function. 6 of 7 P.T.O.
6. (a) State the Neyman-Pearson test for H 0 : θ = θ 1 versus H 1 : θ = θ 2 based on a random sample X 1, X 2,..., X n from a distribution with the probability density function f(x; θ). (b) Let X 1, X 2,..., X n be a random sample from a Uniform(0, θ) distribution. (i) Find the most powerful test at significance level α for H 0 : θ = θ 1 versus H 1 : θ = θ 2, where θ 2 > θ 1 are constants. Show that the test rejects H 0 if and only if max(x 1, X 2,..., X n ) > k for some k. (ii) Determine the power function, Π(θ), of the test in part (i). (iii) Find the value of k when α = 0.05, n = 5 and θ = θ 1 = 0.5. (iv) Find β = Pr (Type II error) when n = 5, θ 1 = 0.5 and θ = θ 2 = 0.6. END OF EXAMINATION PAPER 7 of 7