THEORY OF ESTIMATION — DU B.Sc (hons) Statistics Previous Year Question Papers

2. Attempt six questions in all, selecting three questions from Section I and three questions from Section II. 1. (a) Let X,, 42,..,A, be a random sample of size n from a normal distribution with mean u and variance o Show that 5'=- (X, -X)' is an unbiased estimator of o Bernoulli variate X taking the value 1 with probability p and the value 0 with probability (1- p) then show that (i) I is a consistent estimator for p and (i * (1 - EX) is a consistent estimator for p(1 - p). 2. (a) Let the random variable X is distributed as P(0). Obtain an unbiased estimator for e-(k+1)e . Comment on the result. (b) State Cramer-Rao Inequality. Let X,, X2... X, be a random sample from a population having pdt. f(x, e) =/1+(x-0)j,-∞<x<,-<<. Find the Cramer-Rao Lower bound of variance of an unbiased estimator off. Also, examine whether the MVB estimator exists for 0. (7,8) 3. (a) If T, is a consistent estimator of 0, then T, is also a consistent estimator of 02. (b) Define Minimum Variance Unbiased Estimator (MVUE). Prove that an UMVUE is unique. (8,7) 4. (a) Let X,, X2...., X, be a random sample from U(0,0) population. Show that the largest order statistic (n) is a complete sufficient statistic for 0 .Using Lehman Scheffe's theorem, find the UMVUE of 0. (b) Suppose X,, X,..... X, be a random sample from the distribution having pdf f (x, 0) = de d,0 ≤ x < ∞, > O. Show that the sample mean X is sufficient for O and that .t is the only unbiased estimator of - based on F. (8,7) 5. (a) Describe the method of moments. Let X,, X,,....X, are a random sample from the distribution with pdf. f (x; a,B) = fa*-te=-Bx, x ≥ 0,0 > O,B > 0. Estimate the parameters a and B by the method of moments. (b) Find the maximum likelihood estimator of a and 2 (2 is being large) of the distribution f (x; a,2) = Fa = e a a x1-1,0 < x < ∞0,0 < 2 < ∞. You may use the large values of 6.() We ie litre er ore ened ite erig retiner obai ‹ellhood estımator of the expected itetime and reliability function in case of fallu censored sample from the following liftime distribution: f (x; o) = Le o,x > 0,0 > 0. (b) Let X be binomially distributed with pmf f (x,0) = 0*(1 - 0)1-*,x = 0,1, where -= 0 ≤ % Show that the MLE of O is (2x+1) (8,7) 7. (a) Let X,,X,,....X„ be a random sample from N(4,6'), o' is known. Obtain 100 (1 -a)% confidence interval for u. If a random sample of size n=25 from a normal population with variance o' =144 has a mean of 65.3, construct a 95% confidence interval for the population mean 4. (b) Explain the method of constructing confidence intervals for large samples. Find 100 (1 - a)% confidence interval for binomial proportion O based on a random sample of size n for large samples. (7,8) 8. Describe any three of the following: a. Method of minimum Chi-quare estimation b. Fisher-Neyman Criterion c. Properties of maximum likelihood estimator (15) d. Regularity conditions for CRLB