Elements of Statistical Inference — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

(iii) Invariance property of a consistent estimator (iv) Point and Interval estimation (v) Critical region and Level of Significance (vi) Difference between MVUE and UMVUE. Nameofthecourse : B.Sc. (P)/B.A. (P) with 1.` (a) Define an unbiased and consistent estimator of a parameter 0 in a distribution. Show that 71(71-1) is an unbiased estimator of G2, for the sample values 1 or a with respective probabilities G and (1 - ®. (b) Define Neyman Factorization theorem. Let JF], .X2, .........,. X„ be a random sample from uniform population [0, ©]. Find a sufficient estimator for 2. (a) Prove that minimum variance unbiased (MVU) estimator is unique in the sense that if T] and T2 are MVU estimators for Y(a), then T] = T.2, almost surely. (b) A random sample (JY], .X2, .Y3) of size 3 is drawn from a normal population with mean H and variance o2. Let r„ r2 and r3 are the estimators of Lt, where (i) If W be a most powerful critical region of size c¥ for testing fro: © = ©o against (ii) If W be uniformly most powerful test of size c¥ for testing fro: © = © against (b) Let JF], JY2' .......- Yn be a random sample of size n from Normal distribution N(0, o2) where a2 is known. Obtain UMP test for testing fro: © = Oo against fro: 0 > Oo. Also find the power function of the test. sample of size n from Normal distribution (8,7) 8. Explain any five from the followings (5x3=15) (i) Errors of first and second kinds. used to determine the most powerful critical region for testing a simple null hypothesis against a simple alternative hypothesis. (b) Let X have a pdf : Find the power of the test for testing Ho: G = 2 against Hi: G = 1„ based upon a random sample xl, x2 and critical region: W={((Xi,#2) : 7. (a) Prove that in the critical region W is necessarily unbiased if (i) Find ^ such that r3 is an unbiased estimator for LL. (ii) Are r] and r2 unbiased estimators of pr? (iii) Find the best estimator among T], r2 and r3. 3. (a) Let X be distributed as Poisson (0). Show that the only unbiased estimator of e{-(k+1)G),k > o is I = (-K)#, where TCx) < O,if x is odd. (b) State Cramer-Rao inequality with mentioning its underlying regularity conditions. A random sample X,, X2 ,........., Xn is taken from a normal population with mean zero and variance o2. 5. (a) Obtain 100(1 -c¥)% confidence limits for parameter (cz)G ¢7?d (a) o2 of the distribution r(„a)=±exp{-:(#)2},_cO<#<cO, Examine if Zr=Lx[Z/7iisaMVB is a MVB estimator of o2. (b) Let p be the probability tliat a coin will fall head 4. (a) Describe the method of maximum likelihood estimation. Find MLE for the parameter ^ of a Poisson (^) distribution on the basis of a random sample of size n. Also find its variance. in a single toss in order to test Ho:P = : against fr .j q', (b) State optimum properties of maximum likelihood estimators. Let x2, I;2 ,.......... x,, be a random sample from the uniform distribution with pdf/(x,