GIVE DETAILED EXPLANATIONS

11.2.1. Consider iid variables x1 ,... , xn distributed as uniform over [a, b], b > a. (1) Show that the largest order statistic is a consistent estimator for b; (2) Is the smallest order statistic consistent for a? Prove your assertion.

11.2.2. Let f(x) = cx2 , 0 x and zero elsewhere. (1) Compute c if f(x) is a density; (2) compute Pr{y2 0.8} and Pr{y1 0.1} where y1 and y2 are the smallest and largest order statistics for a sample of size 5 coming from the population in (1). 11.2.3. Let f(x) = 3 3 x 2 , 0 x and zero elsewhere. For a sample of size 6 from this population, compute (1) Pr{y1 4 }; (2) Pr{y2 3 4 }, where y1 and y2 are the smallest and largest order statistics, respectively. 11.2.4. Consider the same population as in Exercise

11.2.3. Is the largest order statistic for a sample of size n from this population (1) unbiased for ; (2) consistent for ? 11.2.5. Consider the function f(x) = c, elsewhere. If f(x) is a density function then (1) compute c; (2) the density of (a) the smallest order statistic, (b) the largest order statistic, for a sample of size 3 from this population. 11.2.6. For the population in Exercise

11.2.5 compute the probabilities (1) Pr{y1 2 }; (2) Pr{y2 3 2 }, where y1 and y2 are the smallest and largest order statistics, respectively. 11.2.7. For the order statistics in Exercise 11.2.6 is y1 or y2 unbiased for ; (2) consistent for ?

11.2.8. Show that (11.7) is a density function for max{x1 ,... , xn } = m = a given number.

11.2.9. Show that the conditional statement in (11.6) is in fact a density function for x= m = a given number.

11.2.10. Let x1 ,... , xn be a sample from some population with all parameters denoted by . Let u1 = x1 ,... , un = xn be n statistics. Then show that u1 ,... , un are jointly sufficient for all the parameters .

11.3.1. By using the method of moments estimate the parameters and in (1) type-1 beta population with parameters (, ); (2) type-2 beta population with the parameters (, ). 11.3.2. Prove Lemmas 11.1 and 11.2. 11.3.3. Prove Lemmas 11.3 and 11.4. 11.3.4. Consider the populations (1) f 1 (x) = x1 , 0 x 1, > 0, and zero elsewhere; (2) f 2 (x) = (1 x) 1 , 0 x 1, > 0 and zero elsewhere. Construct two unbiased estimators each for the parameters and in (1) and (2).

11.3.5. Show that the sample mean is unbiased for (1) in a Poisson population f 1 (x) = x x! e , x = 0, 1, 2,... , > 0 and zero elsewhere; (2) in the exponential population f 2 (x) = 1 e x/ , x 0, > 0 and zero elsewhere.11.3.6. For in a uniform population over [0, ] construct an estimate T = c1 x1 + c2 x2 + c3 x3 , where x1 , x2 , x3 are iid, as uniform over [0, ], such that E(T) = . Find c1 , c2 , c3 such that two unbiased estimators T1 and T2 are obtained where T1 = 2x, x= the sample mean. Compute E[T1 ] 2 and E[T2 ] 2 . Which is relatively more efficient? 11.3.7. Consider a simple random sample of size n from a Laplace density or double exponential density

11.3.6. For in a uniform population over [0, ] construct an estimate T = c1 x1 + c2 x2 + c3 x3 , where x1 , x2 , x3 are iid, as uniform over [0, ], such that E(T) = . Find c1 , c2 , c3 such that two unbiased estimators T1 and T2 are obtained where T1 = 2x, x= the sample mean. Compute E[T1 ] 2 and E[T2 ] 2 . Which is relatively more efficient? 11.3.7. Consider a simple random sample of size n from a Laplace density or double exponential density

An experiment started with n = 20 rabbits. But rabbits die out one by one before the experiment is completed. Let x be the number that survived and let p be the true probability of survival for reach rabbit. Then x is a binomial random variable with parameters (p, n = 20). Suppose that in this particular experiment 15 rabbits survived at the completion of the experiment. This p need not be the same for all experimental rabbits. Suppose that p has a type-1 beta density with parameters ( = 3, = 5). Compute the Bayes' estimate of p in the light of the observation x = 15.

11.4.1. Derive the Bayes' estimator of the parameter in a Poisson population if has a prior (1) exponential distribution with known scale parameter, (2) gamma distribution with known scale and shape parameters. 11.4.2. If the conditional density of x, given , is given by f(x|) = c1 x 1 e x for > 0, > 0, > 0, > 0, x 0 and zero elsewhere and has a prior density of the form g() = c2 1 e for > 0, > 0, > 0, > 0 and zero elsewhere, (1) evaluate the normalizing constants c1 and c2 , (2) evaluate the Bayes' estimate of if , and are known. 11.4.3. Write down your answer in Exercise 11.4.2 for the following special cases: (1) = 1; (2) = 1, = 1; (3) = 1, = 1, = 1; (4) = 1, = 1. 11.4.4. Derive the best estimator, best in the minimum mean square sense, of y at preassigned values of x, if the joint density of x and y is given by the following

11.4.10. Verify (a) Cramer-Rao inequality, (b) Rao-Blackwell theorem with reference to the MLE of the parameter (1) p in a Bernoulli population; (2) in a Poisson population; (3) in an exponential population, by taking suitable sufficient statistics whenever necessary. Assume that a simple random sample of size n is available.