Multinomial logistic regression algorithm

The lower bound principle (introduced in Böhning and Lindsay 1988, Ann. Inst. Statist. Math., 40, 641–663), Böhning (1989, Biometrika, 76, 375–383) consists of replacing the second derivative matrix by a global lower bound in the Loewner ordering. This bound is used in the Newton-Raphson iteration instead of the Hessian matrix leading to a monotonically converging sequence of iterates. Here, we apply this principle to the multinomial logistic regression model, where it becomes specifically attractive.Supplement to Monotonicity of quadratic-approximation algorithms by Böhning and Lindsay (1988). Ann. Inst. Statist. Math., 40, 641–663.This research was supported by the German Research Foundation.     

Rank estimates in a class of semiparametric two-sample models

We consider a two-sample semiparametric model involving a real parameter and a nuisance parameter F which is a distribution function. This model includes the proportional hazard, proportional odds, linear transformation and Harrington-Fleming models (1982, Biometrika, 69, 533–546). We propose two types of estimates based on ranks. The first is a rank approximation to Huber's M-estimates (1981, Robust Statistics, Wiley) and the second is a Hodges-Lehmann type rank inversion estimate (1963, Ann. Math. Statist., 34, 598–611). We obtain asymptotic normality and efficiency results. The estimates are consistent and asymptotically normal generally but fully efficient only for special cases.Research partially supported by National Science Foundation Grant DMS-86-02083 and National Institute of General Medical Sciences Grant SSS-Y1RO1GM35416-01     

Asymptotic theorems for estimating the distribution function under random truncation

Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ.     

Multivariate symmetry via projection pursuit

Blough (1985,Ann. Inst. Statist. Math.,37, 545–555) developed a multivariate location region for a randomp-vectorX. The dimension of this region provides information on the degree of symmetry possessed by the distribution ofX. By considering all one-dimensional projections ofX, it is possible to ascertain the dimension of the location region. Projection pursuit techniques can therefore be used to study symmetry in multivariate data sets. An example from an Entomology investigation is presented illustrating these methods.     

On empirical Bayes with sequential component

Laippala (1979, Scand. J. Statist., 6, 113–118, correction note, 7, 105; 1985, Ann. Inst. Statist. Math., 37, 315–327) has defined a concept within the empirical Bayes framework that he calls floating optimal sample size. We examine this concept and show that it is one of many possibilities resulting from restricting the class of component sampling procedures in the empirical Bayes decision problem with a sequential component. All ideas are illustrated with the finite state component.     

Adaptive choice of trimming proportions

We consider Jaeckel's (1971,Ann. Math. Statist.,42, 1540–1552) proposal for choosing the trimming proportion of the trimmed mean in the more general context of choosing a trimming proportion for a trimmedL-estimator of location. We obtain higher order expansions which enable us to evaluate the effect of the estimated trimming proportion on the adaptive estimator. We find thatL-estimators with smooth weight functions are to be preferred to those with discontinuous weight functions (such as the trimmed mean) because the effect of the estimated trimming proportion on the estimator is of ordern ^–1 rather thann ^–3/4. In particular, we find that valid inferences can be based on a particular smooth trimmed mean with its asymptotic standard error and the Studentt distribution with degrees of freedom given by the Tukey and McLaughlin (1963,Sankhy Ser. A,25, 331–352) proposal.     

On sequential fixed-width confidence intervals for the mean and second-order expansions of the associated coverage probabilities

In order to construct fixed-width (2d) confidence intervals for the mean of an unknown distribution function F, a new purely sequential sampling strategy is proposed first. The approach is quite different from the more traditional methodology of Chow and Robbins (1965, Ann. Math. Statist., 36, 457–462). However, for this new procedure, the coverage probability is shown (Theorem 2.1) to be at least (1-)+Ad ²+o(d²) as d0 where (1-) is the preassigned level of confidence and A is an appropriate functional of F, under some regularity conditions on F. The rates of convergence of the coverage probability to (1-) obtained by Csenki (1980, Scand. Actuar. J., 107–111) and Mukhopadhyay (1981, Comm. Statist. Theory Methods, 10, 2231–2244) were merely O(d^1/2-q), with 0<q<1/2, under the Chow-Robbins stopping time ^*. It is to be noted that such considerable sharpening of the rate of convergence of the coverage probability is achieved even though the new stopping variable is O_p(^*). An accelerated version of the stopping rule is also provided together with the analogous second-order characteristics. In the end, an example is given for the mean estimation problem of an exponential distribution.     

Nonparametric test of restricted interchangeability

Exact and large sample distributions of the rank order test under the null hypothesis of restricted interchangeability are obtained. Under given regularity conditions and under Pitman's shift in location alternative, the asymptotic relative efficiency of this nonparametric test in comparison with Votaw's (1948, Ann. Math. Statist., 19, 447–473) likelihood ratio test is given.     

Consistent estimation of location region

Asymmetric multivariate probability distributions can be difficult to characterize in terms of their location. The works of Doksum (1975, Scand. J. Statist., 2, 11–22) and Blough (1985, Ann. Inst. Statist. Math., 37, 545–555) provide the construction of a location region for a given distribution. Any point in this closed, convex region will serve as a location parameter. It is the purpose of this paper to obtain a consistent estimator of the location region. Consistency is defined in terms of an appropriate pseudometric.     

Null distribution of the sum of squared z-transforms in testing complete independence

Brien et al. (1984, Biometrika, 71, 545–554; 1988, Biometrika, 75, 469–476) have proposed, illustrated and discussed advantages of using Fisher's z-transforms for analyzing correlation structures of multinormal data. Chen and Mudholkar (1988, Austral. J. Statist., 31, 105–110) have studied the sum of squared z-transforms of sample correlations as a test statistic for complete independence. In this paper Brown's (1987, Ann. Probab., 15, 416–422) graph-theoretic characterization of the dependence structure of sample correlations is used to evaluate moments of the test statistic. These moments are then used to approximate its null distribution accurately over a broad range of parameters, including the case where the population dimension exceeds the sample size.     

A note on the uniformly most powerful tests in the presence of nuisance parameters

When a testing problem has nuisance parameters, the uniformly most powerful (UMP) tests do not generally exist. Exceptional examples were given by Dubey (1962, Skand. Aktuarietidskr., 45, 25–38; 1963, Skand. Aktuarietidskr., 46, 1–24) and Takeuchi (1968, Ann. Math. Statist., 40, 1838–1839) for the exponential distributions. What is essential for proving the existence of UMP tests lies in a special relationship between null hypothesis and the alternative. Assuming a similar relationship between them, a similar kind of result can be shown under more general situation.     

Variable location and scale kernel density estimation

Variable (bandwidth) kernel density estimation (Abramson (1982,Ann. Statist.,10, 1217–1223)) and a kernel estimator with varying locations (Samiuddin and El-Sayyad (1990,Biometrika,77, 865–874)) are complementary ideas which essentially both afford bias of orderh ⁴ as the overall smoothing parameterh 0, sufficient differentiability of the density permitting. These ideas are put in a more general framework in this paper. This enables us to describe a variety of ways in which scale and location variation may be extended and/or combined to good theoretical effect. This particularly includes extending the basic ideas to provide new kernel estimators with bias of orderh ⁶. Technical difficulties associated with potentially overly large variations are fully accounted for in our theory.     

Shrinkage estimators of the location parameter for certain spherically symmetric distributions

We consider estimation of a location vector for particular subclasses of spherically symmetric distributions in the presence of a known or unknown scale parameter. Specifically, for these spherically symmetric distributions we obtain slightly more general conditions and larger classes of estimators than Brandwein and Strawderman (1991,Ann. Statist.,19, 1639–1650) under which estimators of the formX +ag(X) dominateX for quadratic loss, concave functions of quadratic loss and general quadratic loss.Research supported by NSF grant DMS-88-22622     

Sectional Curvature Rigidity of Asymptotically Locally Hyperbolic Manifolds

This paper presents a rigidity result of real hyperbolic quotients inanalogy to Min-Oo's result in Math. Ann. 285(4) (1989),527–539, but without the spin condition. In order to prove this,we use special Killing forms on the exterior form bundle. Moreover, wemake an assumption on the sectional curvature to obtain the necessaryeigenvalue estimates of the curvature endomorphism in theBochner–Weitzenböck formula of ^k M.     

Finite Topological Type and Volume Growth

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature Ric _M –(n – 1) and conjugateradius conj _M c > 0 has finite topological type, provided that the volume growth of geodesic balls in M is not very far from that of the balls in an n-dimensional hyperbolic space H ⁿ (–1)of sectional curvature –1. We also show that a complete open Riemannian manifold M with nonnegative intermediate Ricci curvature and quadratic curvature decay has finite topological typeif the volume of geodesic balls of M around the base point grows slowly.     

A lower bound for the volume of a four-dimensional closed manifold

A closed orientable four-dimensional manifoldM ⁴, whose sectional curvatures satisfy –1 K 1, is considered. If the product of the curvatures of orthogonal area elements is nonnegative and if at least at one point all the curvatures are different from zero, then the estimate vol(M ⁴) > (4/9)² is obtained for the volume ofM ⁴. A theorem on the local structure of a manifold with small volume whose curvatures at every point are of the same sign is established.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 3–8, 1991.     

On sequential procedures for the point estimation of the mean of a normal population

The sequential procedures developed by Starr (1966, Ann. Math. Statist., 37, 1173–1185) for estimating the mean of a normal population are further analyzed. Asymptotic properties of the regret and first two moments of the stopping rules are studied and second-order approximations are derived.     

A generalization of the results of Pillai

In a recent article Pillai (1990,Ann. Inst. Statist. Math.,42, 157–161) showed that the distribution 1–E (–x ), 0<1; 0x, whereE (x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ).     

Characterization based on conditional distributions

The field of application of a result given by Singh and Vasudeva (1984, J. Indian Statist. Assoc., 22, 93–96) which provides a way of characterizing the distribution of a random variable X, through conditional distributions of a second variable Z, given X, is extended.     

Lefschetz Complex Conditions for Complex Manifolds

For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L _1,0: H ^1,0(M) H ^{{n, n–1}(M) and L _{0, 1}: H ^{0, 1}(M) H ^{n – 1, n} (M) given by the cup product with [] ^{n – 1}, n being the complex dimension of M andH ^{*, *}(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L _{1, 0} (resp.L _{0, 1}) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L _{1, 0}characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom. 13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).