Applied Multivariate Techniques by Subhash Sharma

By Subhash Sharma

This e-book makes a speciality of while to exploit many of the analytic recommendations and the way to interpret the ensuing output from the main time-honored statistical programs (e.g., SAS, SPSS).

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9. The existence and consistency of maximum likelihood estimates. In this section and the following one it is assumed that X is, or may be taken to be, a finitedimensional Euclidean space, and that $ is the field of Borel sets of X. 0* is a given set of probability measures P on ^, to be thought of as the possible distributions of x. It is assumed that & is endowed with the topology of weak convergence, s = (x : , x 2 , • • • ad inf) denotes an infinite sequence of independent replicates of x. Suppose for the moment that, as in previous sections, there is given a parametrization of ^>, say & — {Pe:9e&}, and that g(9) is a real-valued functional on 0.

1] J. KIEFER AND J. WoLFOWiiz, Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters, Ann. Math. , 27 (1956), pp. 887-906. 2] T. J. KILLEEN, T. P. HETTMANSPERGER AND G. L. , 42 (1971). 3] J. , 36 (1965), pp. 1759-1766. [K4] , Asymptotic efficiency of the Kolmogorov-Smirnov test, J. Amer. Statist. 932 938. 5] S. KULLBACK AND R. A. LEiBLER, On information and sufficiency, Ann. Math. , 22 (1951), pp. 79 86. 6] , Information Theory and Statistics, John Wiley, New York, 1959.

13) with 9 = 0°, it follows that 0 < e < EJ and n > m(e) imply that 0 < tn(e, 0°) fS oo. Since {Tn} is consistent, tn(e, 0°) ->• 0 as n -> oo. 1 that if 0 < e < e l 5 then as n -> oo. 19) holds at 9°. 1 let us say that {7^,} is an efficient estimate of g, in the sense of asymptotic effective variances if, for each fixed 0, lim n _ ao {nT^(e, 9)} exists for all sufficiently small e, say w(g, 9), and lim £ ^ 0 w(e, 9) = 0(0). At present it is an SOME LIMIT THEOREMS IN STATISTICS 25 open problem to find estimates which are efficient in this sense.

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