(a) Let X1, , Xn be a random sample with finite variance. T

1. (4 points) True/False. Please read the statements carefully, as no partial credit will be given. (a) Let X1, . .. , Xn be a random sample with finite variance. Then the sample mean is always an unbiased estimator for the population mean. (b) In hypothesis testing, the rejection region depends on the observed value of test statistic. (c) Under the regularity conditions, a maximum likelihood estimator is con- sistent and asymptotically efficient. (d) - The sample mean is a sufficient statistics for 0 in Poisson distribution, Are-0 f (ac; 0) = x = 0, 1, 2 . . . ; 0 >0. (e) A sufficient statistic is always independent of an ancillary statistic. (f ) If Xi ~ N(u, o2), for i = 1, . .., n, then E(XS2) = Mo2, where X is the sample mean and S is the sample variance. (g) The likelihood ratio test provides a most powerful rejection region of size a for testing the simple hypothesis Ho : 0 = 00 v.s. H1 : 0 = 01, (01 # 00), and it is an unbiased test. (h) A uniform most powerful (UMP) test has to be a uniform most powerful unbiased (UMPU) test