Statistical estimators minimizing deviation

Estimators minimizing deviation are constructed. The connection between estimators minimizing deviation and the theory of function approximation is demonstrated. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 27–32, Perm, 1991.     

Asymptotic expansions for the distribution functions of Pickands-type estimators

The asymptotic expansions for the distribution functions of Pickands-type estimators in extreme statistics are obtained. In addition, several useful results on regular variation and intermediate order statistics are presented. Project supported by the National Natural Science Foundation of China (Grant No. 19601007) and Doctoral Program Foundation of Higher Education of China.     

On prior density estimators

Sufficient conditions are obtained for asymptotically unbiased and consistent estimators of prior probability density.Translated from Statisticheskie Metody, pp. 115–124, 1982.     

On shape-preserving wavelet estimators of cumulative distribution functions and densities

In a previous paper we introduced a general class of shapepreserving wavelet approximating operators (approximators) which transform cumulative distribution functions (cdf) and densities into functions of the same type. Empirical versions of these operators are used in this paper to introduce, in an unified way, shape- preserving wavelet estimators of cdf and densities, with a priori prescribed smoothness properties. We evaluate their risk for a variety of loss functions and analyze their asymptotic behavior. We study the convergence rates depending on minimal additional assumptions about the cdf/ density. These assumptions are in terms of the function belonging to certain homogeneous Besov or Triebel- Lizorkin spaces and others. As a main evaluation tool the integral p-modulus of smoothness is used     

A uniform bound for the deviation of empirical distribution functions

If X₁, …, X_n are independent R^d-valued random vectors with common distribution function F, and if F_n is the empirical distribution function for X₁, …, X_n, then, among other things, it is shown that P{sup_x F_n(x) ε} 2e²(2n)^de^−2nε2 for all nε² ≥ d². The inequality remains valid if the X_i are not identically distributed and F(x) is replaced by Σ_iP{X_i ≤ x}/n.     

On improving distribution function estimators which are not monotonic functions

In this paper we consider the problem of correcting distribution function estimators which are not nondecreasing functions (for example kernel type estimators). The method is based on the orthogonal projection in L₂ and guarantees improving of the integrated mean square error for each sample size.     

Maximal subnear-rings of functions

For a group G, let M(G) denote the near-ring of functions on G. We characterize all maximal subnear-rings of M(G) and show that for many classes of groups, E(G), the near-ring generated by the semigroup, End(G) of G, is never maximal as a subnear-ring of M ₀ (G). Received: 25 April 2008     

Admissibility of some preliminary test estimators for the mean of normal distribution

Summary Let a random variableX follow ap-variate normal distributionN _p (θ, I _p ) with an unknownp×1 vector θ andp×p identity matrixI _p . The admissibility of a preliminary test estimator using AIC (Akaike's Information Criterion) procedure will be shown ifp=1 and its inadmissibility will be shown ifp≧3 under the loss function based on Kullback-Leibler information measure. Furthermore the two sample case is also considered.     

Admissibility of linear estimators with respect to inequality constraints under some loss functions

In this paper, we study the issue of admissibility of linear estimated functions of parameters in the multivariate linear model with respect to inequality constraints under a matrix loss and a matrix balanced loss. Under the matrix loss, when the model is not constrained, the results in the class of non-homogeneous linear estimators [Xie, 1989, Chinese Sci. Bull., 1148–1149; Xie, 1993, J. Multivariate Anal., 1071–1074] showed that the admissibility under the matrix loss and the trace loss is equivalent. However, when the model is constrained by the inequality constraints, we find this equivalency is not tenable, our result shows that the admissibility of linear estimator does not depend on the constraints again under this matrix loss, but it is contrary under the trace loss [Wu, 2008, Linear Algebra Appl., 2040–2048], and it is also relative to the constraints under another matrix loss [He, 2009, Linear Algebra Appl., 241–250]. Under the matrix balanced loss, the necessary and sufficient conditions that the linear estimators are admissible in the class of homogeneous and non-homogeneous linear estimators are obtained, respectively. These results will support the theory of admissibility on the linear model with inequality constraints.     

The least prior deviation quasi-Newton update

We propose a new choice for the parameter in the Broyden class and derive and discuss properties of the resulting self-complementary quasi-Newton update. Our derivation uses a variational principle that minimizes the extent to which the quasi-Newton relation is violated on a prior step. We discuss the merits of the variational principle used here vis-a-vis the other principle in common use, which minimizes deviation from the current Hessian or Hessian inverse approximation in an appropriate Frobenius matrix norm. One notable advantage of our principle is an inherent symmetry that results in the same update being obtained regardless of whether the Hessian matrix or the inverse Hessian matrix is updated.We describe the relationship of our update to the BFGS, SR1 and DFP updates under particular assumptions on line search accuracy, type of function being minimized (quadratic or nonquadratic) and norm used in the variational principle.Some considerations concerning implementation are discussed and we also give a numerical illustration based on an experimental implementation using MATLAB.Corresponding author.     

On the value distribution theory of elliptic functions

The Nevanlinna characteristic of a nonconstant elliptic function φ (z) satisfiesT(r, φ)=Kr ² (1+o(1)) asr→∞ whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , having coefficientsa(z) satisfyingT(r, a)=o(r ²) asr→∞, will the meromorphic functionh _Q (z)=Q(z, ?(z),...,?⁽ⁿ⁾(z)) either be identically zero or satisfyN(r, 1/h _Q )=o(r ²) asr→∞? In fact, we answer this question for rational functionsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , and also obtain analogous results for the Weierstrass functions ζ(z) and σ(z).     

A duality theory for some non-convex functions of matrices

Abstract We study a special class of non-convex functions which appear in nonlinear elasticity, and we prove that they have a well-defined Legendre transform. Several examples are given, and an application to a nonlinear eigenvalue problem. Keywords: Duality, Legendre transform, Nonlinear elasticity Mathematics Subject Classification (2000): 05C38, 15A15, 05A15, 15A18     

A note on the distribution of some additive functions

Let f denote an additive arithmetical function with continuous limiting distribution F on the integers. Then f also has a limiting distribution G on shifted primes. Under some growth conditions on the values of f at primes, we provide optimal lower bounds for the modulus of continuity of F and G, at all points from a specified infinite set.     

Asymptotic distribution theory of statistical functionals: The compact derivative approach for robust estimators

Summary Derivatives of statistical functionals have been used to derive the asymptotic distributions ofL-,M- andR-estimators. This approach is often heuristic because the types of derivatives chosen have serious limitations. The Gateaux derivative is too weak and the Fréchet derivative is too strong. In between lies the compact derivative. This paper obtains strong results in a rigorous manner using the compact derivative onC ₀(R). This choice of space allows results for a broader class of functionals than previous choices, and the fact that is often tight provides the compact set required. A major result is the derivation of the compact derivative of the inverse c.d.f. when the range space is endowed with the uniform norm. It has applications to the asymptotic theory ofL-,M- andR-estimators. We illustrate the power of this result by applications toL-estimators in settings including the one sample problem, data grouped by quantiles, and censored survival time data.