PRINCIPLES OF STATISTICS - EXAMPLES 2/4 Part II, Michaelmas 2016, Quentin Berthet (email: q.berthet@statslab.cam.ac.uk) Throughout the term 'asymptotic' always refers to a large sample limit as n -> 00 under a sampling distribution Pe = Po where, unless specified otherwise, 0 is assumed to be a fixed element of the parameter space O C RP. By a 'regular parametric model' we mean a statistical model of probability density/mass functions {f (., 0) : 0 € O} that satisfies the regularity condi- tions from lectures, ensuring asymptotic normality of the maximum likelihood estimator (which in turn may be used without proof in the solution of the respective exercise). 1. Let O C R have nonempty interior and let Sn be a sequence of random real-valued continuous functions defined on @ such that, as n-> , Sn(0) > S(0) V0 € O, where S : ( > IR is nonrandom. Suppose for some 00 in the interior of O and every & > 0 small enough we have S(00 -E) < 0 < S(00 + E), and that Sn has exactly one zero On for every n E N. Deduce that Ôn P 00 as n > co. 2. Give an example of (possibly random) functions Qn, Q defined on O C R that have unique maximisers 0n, 00, respectively, such Qn(0) -> Q(0) (almost surely) for every 0 € O as n > 00, but On > Oo (almost surely). 3. Consider an i.i.d. sample X1, ... , Xn arising from the model 0 f (x, 0) = 0x6-1 exp{x}, x > 0,0 € (0, 00) of Weibull distributions. Show that the MLE exists and is consistent. 4. Consider the maximum likelihood estimator 0 from X1, ... , Xn i.i.d. N(0, 1) where 0 € e = [0, 0). Show that Vn(0 - 0) is asymptotically normal whenever 0 > 0. What happens when 0 = 0? Comment on your findings in light of the general asypmtotic theory for maximum likelihood estimators. 5. Let X1, ... , Xn be i.i.d. random variables from a uniform U (0, 0), 0 € + = (0, 00) distribu- tion. Calculate the Fisher information for this model. Find the maximum likelihood estimator 0 of 0 and show that On = "+10 is unbiased for 0. Find the variance of On, compare it to what the Cramer-Rao inequality predicts, and discuss your findings. Finally find the asymptotic distribution of n(0 - 0). 6. Suppose one is given a regular parametric model {f(., 0) : 0 € O} with likelihood function L(0) and corresponding maximum likelihood estimator OMLE, and consider a mapping @ : 0 -> F, where O, F are subsets of R. a) Assuming that @ is one to one, show that the maximum likelihood estimator of ø in the model {f (·, ¢) : ø = @(0) for some 0 € 0} equals @(0MLE). b) Now consider a mapping § that is not necessarily one-to-one. Define the induced likelihood function L'(o) = supe:@(0)=¢ L(0)