R. R. Bahadur

Some limit theorems in statistics

Moment-generating Functions Chernoffs Theorem The Kullback- Leibler Information Number Some Examples of Large Deviation Probabilities Steins Lemma Asymptotic Effective Variances Exact Slopes of Test Statistics Some Examples of Exact Slopes The Existence and Consistency of Maximum Likelihood Estimates The Exact Slope of the Likelihood Ratio Statistic.

Measurable subspaces and subalgebras

a a-additive measure on S with ,u(X) = 1. Let V=.

Two Comments on "Sufficiency and Statistical Decision Functions"

(X, S, Iu) be the real Hilbert space of S-measurable functions f(x) with fxf2dM < 00o Forf and g in V, we write (f, g) ==fxf* gd, where (f. g) (x) -f(x) * g(x), and Ijjfj = (f, f) 1/2. Convergence in V is defined as usual in the norm topology, that is to say, limn_oO fn =f means limn_O llfn -fll =0. If f and g are functions in V such that IIf-ggl =0, they are regarded as identical and we writef =g. f _ g means that ,u I x:f(x) <g(x) } = O. For any real a, the function which is equal to a for every x is also denoted by a. A function f in V is said to be bounded if there exist a and ,B such that a <f <,B. A function on X which takes only the values 0 and 1 is called a characteristic function. For any set A CX, XA denotes the characteristic function which equals 1 on A and vanishes on X-A. If Si and S2 are subclasses of S, we write St C S2 if corresponding to each set A ES, there exists a BE S2 such that XB = XA; we write SI= S2 if S1CS2 and S2 Cs1. Let W be a subspace (- closed linear manifold) in V. W is algebraic

LARGE DEVIATIONS OF THE MAXIMUM LIKELIHOOD ESTIMATE IN THE MARKOV CHAIN CASE

In the following comments we employ the notation and definitions of [1]. The first comment answers a question raised in [1] by giving an example of a necessary and sufficient subfield which cannot be induced by a statistic. The second remark clarifies this example somewhat by discussing the connection between statistics and subfields in general. It was hoped that this connection would be so close as to provide the answer to another question raised in [1]: whether the existence of a necessary and sufficient subfield implies that of a necessary and sufficient statistic. However, an example given at the end of the second comment shows that such a result cannot be proved without making deeper use of sufficiency.

A Note on Quantiles in Large Samples

Publisher Summary This chapter discusses large deviations of the maximum likelihood estimate in the Markov chain case. Local optimality is a weak property. Various estimates, other than the maximum likelihood (ML) estimate, are locally optimal. However, optimality is a much stronger property; in particular, the ML estimate can be optimal only in certain special cases. The empirical transition matrix λ (n) is an ML estimate if the transition probabilities are entirely unknown, that is, if Λ is the parameter space.