Václav Fabian

On Estimation and Adaptive Estimation for Locally Asymptotically Normal Families

SummaryLocally asymptotically minimax (LAM) estimates are constructed for locally asymptotically normal (LAN) families under very mild additional assumptions. Adaptive estimation is also considered and a sufficient condition is given for an estimate to be locally asymptotically minimax adaptive. Incidently, it is shown that a well known lower bound due to Hájek (1972) for the local asymptotic minimax risk is not sharp.

On the Problem of Interactions in the Analysis of Variance

Abstract For the two-way analysis of variance without the assumption of zero interactions, consider the following recommendation: Test the zero-interactions hypothesis and, if the hypothesis is not rejected, proceed as if the interactions were zero. This recommendation is unwarranted and may give incorrect decisions with a large probability, as has been pointed out in several articles beginning in the year 1935. Despite such isolated criticism, it is a recommendation suggested in most applied books in the field and used in countless applications. An apology for the recommendation is that it can be easily improved by considering the power of the test and thus obtaining information on the interactions neglected. We consider this improved recommendation for the goals of (i) an interval estimate of one of the cell expectations, (ii) a simultaneous interval estimate of the cell expectations, and (iii) an estimate of the cell with the largest expectation. We show that the improved recommendation leads to method...

Stochastic approximation of global minimum points

Abstract A method is proposed for approximating a point of a global minimum of a function f, defined on a subset D of R k, when values of f can be estimated at points determined by the method. The method combines a stochastic approximation method with a nonparametric regression estimate. If f has only one global minimum point, then the new estimate has the same asymptotic behavior as the component stochastic approximation method, except that the new method does not require f to have only one stationary point. Similar properties are obtained if there are multiple global minimum points. The best behavior is obtained if f attains its global minimum on a set with a nonempty interior.

On uniform convergence of measures

SummaryA new and simpler proof is given in Section 3 for the sufficiency part of Theorem 3.1 in Ranga Rao [6] and its generalization by Billingsley and TopsØe [1]. Essential for the proof, which does not require the topological space X to be metric, is Lemma 2.1. As examples of possible wider application of this lemma, simple proofs are given for a well known result on uniformity in convergence of distribution functions (Example 2.3) and of Theorem 4.2 in Ranga Rao [6]. The derivation of the latter from the lemma is substantially simpler than the derivation from Theorem 3.1 in Ranga Rao [6]. Another result on uniformity, given in Rubin [7], is closely related to the Ascoli theorem, but outside the scope of applicability of our lemma.

On the Effect of Jury Size

Abstract In Williams vs. Florida, the Supreme Court compares twelve-man and six-man juries and claims that “there is no discernible difference between the results reached by the two different sized juries.” Walbert (1971) uses a simple probabilistic argument to refute the Courts opinion. Subsequently, Gelfand and Solomon (1974, p. 36) seem to contradict Walbert and side with the Court. This note reaffirms Walberts result.

Complete cubic spline estimation of non-parametric regression functions

SummaryFor regression functions on [0, 1] with bounded fourth derivatives, a complete cubic spline estimate is proposed and shown to have an asymptotically optimal error rate among all estimates. The error is measured by the supremum norm.

The local asymptotic minimax adaptive property of a recursive estimate

A locally asymptotically normal estiamtion problem generated by independent and identically distributed generalized random variables is considered. A recursive estimate based on a stochastic approximation method, a modification of an estimate proposed by Sakrison, is shown to be locally asymptotically minimax. For a nonparametric generalization of the estimation problem, a modification of the estimate is shown locally asymptotically minimax adaptive.

A Local Asymptotic Minimax Optimality of an Adaptive Robbins Monro Stochastic Approximation Procedure

Results, obtained earlier, on asymptotic properties of an adaptive RM stochastic approximation method, are strengthened. The method is shown optimal in the sense of local asymptotic minimax risk and robust with respect to small changes of the unknown density and small changes of the estimated parameter.

New modifications of the Bechhofer method

Abstract The method proposed by Bechhofer (Ann. Math. Statist. 25 (1954) 16–39) for the selection of the largest of k expectations μ1,…,μk of k independent normal random variables X1,…,Xk is considered. The strongest assertion the method gives is that the expectation μI, corresponding to the largest observation, is the largest among these expectations. This is a somewhat disappointing statement if the difference between the largest and second largest Xi is large. The Tukey method and the method proposed by Edwards and Hsu (J. Amer. Statist. Assoc. 78 (1983) 965–971) do not have this disadvantage, but have weaker assertions than the Bechhofer method concerning the difference μ 1 − Max i μ i for some observations. Two new modifications are proposed here that give stronger assertions that the Tukey and the Edwards and Hsu methods concerning both μ I − Max i μ i and μ I − Max i≠I μ i .