Optimizing the smoothed bootstrap

In this paper we develop the technique of a generalized rescaling in the smoothed bootstrap, extending Silverman and Young's idea of shrinking. Unlike most existing methods of smoothing, with a proper choice of the rescaling parameter the rescaled smoothed bootstrap method produces estimators that have the asymptotic minimum mean (integrated) squared error, asymptotically improving existing bootstrap methods, both smoothed and unsmoothed. In fact, the new method includes existing smoothed bootstrap methods as special cases. This unified approach is investigated in the problems of estimation of global and local functionals and kernel density estimation. The emphasis of this investigation is on theoretical improvements which in some cases offer practical potential.     

Empirical likelihood confidence intervals for hazard and density functions under right censorship

In this paper, we use smoothed empirical likelihood methods to construct confidence intervals for hazard and density functions under right censorship. Some empirical log-likelihood ratios for the hazard and density functions are obtained and their asymptotic limits are derived. Approximate confidence intervals based on these methods are constructed. Simulation studies are used to compare the empirical likelihood methods and the normal approximation methods in terms of coverage accuracy. It is found that the empirical likelihood methods provide better inference.     

A note on polyharmonic functions

In this note we prove the following theorem:Suppose 0<p<∞ and α>−1. Then there is a constant C=C(p,m,n,α) such that

     

Optimum calibration points estimating distribution functions

The calibration method has been widely discussed in the recent literature on survey sampling, and calibration estimators are routinely computed by many survey organizations. The calibration technique was introduced in [12] to estimate linear parameters as mean or total. Recently, some authors have applied the calibration technique to estimate the finite distribution function and the quantiles. The computationally simpler method in [14] is built by means of constraints that require the use of a fixed value t₀. The precision of the resulting calibration estimator changes with the selected point t₀. In the present paper, we study the problem of determining the optimal value t₀ that gives the best estimation under simple random sampling without replacement. A limited simulation study shows that the improvement of this optimal calibrated estimator over possible alternatives can be substantial.     

A note on W-I-continuous functions

In [1], Aç?kgöz et al. introduced and investigated the notions of w-I-continuous and w*-I-continuous functions in ideal topological spaces. In this paper, we investigate their relationships with continuous and θ-continuous functions.     

A note on pair-formation functions

In this paper, we take a closer look at two harmonic mean functions [1, 2] and two minimum functions (moving dominance function [3] and group-specific minimum function), in two sex multigroup populations. Comparisons between these functions are focused on proportionate mixing. We show that under some special conditions, the two harmonic mean functions are identical; and under the mixing framework of Castillo-Chavez and Busenberg [4], the two minimum functions are also identical (for both proportionate and nonproportionate mixing). Simulations of a simple demographic model with the four functions are also performed to confirm the above mentioned identity and to illustrate the behavior of these functions.     

A note on canonical functions

We construct a generic extension in which the ℵ₂nd canonical function on ℵ₁ exists. Supported by NSF and by a Fulbright grant. Publ. 378. Partially supported by the B.S.F.     

A note on maximum-type functions

The paper studies the differential properties of functions of the form

$g(x) = \mathop {\max }\limits_{y \in Y} f(x,y),$

where x ∈ X (X is an open convex set from ? ^m ) and y ∈ Y (Y is a compact from ? ⁿ ). Apart from the conventional smoothness conditions imposed on f(x, y), the condition of the concavity of g(x) on X is also imposed.

The differentiability of function g(x) on X is proved.The results of the study facilitate the derivation of the conditions ensuring the sufficiency of Pontryagin’s maximum principle.     

A note on schur functions

We extend the Vandermonde determinant and then extend the classical presentation of Schur functions as quotients of such determinants. Partially supported by NSA Grant MDA904-91-H-0050 and by NSF Grant DMS-9101488.     

A stochastic framework for recursive computation of spline functions: Part II,smoothing splines

Many of the optimal curve-fitting problems arising in approximation theory have the same structure as certain estimation problems involving random processes. We develop this structural correspondence for the problem of smoothing inaccurate data with splines and show that the smoothing spline is a sample function of a certain linear least-squares estimate. Estimation techniques are then used to derive a recursive algorithm for spline smoothing.     

A nonlinear time series model and estimation of missing observations

This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.Authors were supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

A note on functional a posteriori estimates for elliptic optimal control problems

In this work, new results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed, and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two‐sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 403–424, 2017     

A note on feebly continuous functions

A function f from to is said to be feebly continuous at a point (x,y) if there exist sequences x_nx and y_ny with lim_n→∞lim_m→∞f(x_n,y_m)=f(x,y). Dales asked if every function has a point of feeble continuity. Our aim in this short note is to show that (assuming the Continuum Hypothesis) the answer is ‘no’. Dales also asked what happens for functions taking only two values: we show that in this case the answer is ‘yes’.     

A note on locally Lipschitzian functions

An oversight in a paper of Correa and Lemaréchal (this journal, 1993) is noted; a counterexample is given.     

A note onK-convex functions

In this article we concern ourselves with a generalized version of convex functions widely used in the analysis of optimization problems, especially those involving inventory control and/or production-planning. We give some characterizations of these socalledK-convex functions and derive a few of their properties concerning measurability and continuity.

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Zusammenfassung In der vorliegenden Arbeit untersuchen wir eine Klasse von verallgemeinert konvexen Funktionen, die in der Modellierung von Lagerhaltungs- und/oder Produktionsproblemen verbreitete Anwendung gefunden haben. Wir leiten einige Charakterisierungen dieser sogenanntenK-konvexen Funktionen her und beweisen einige ihrer Eigenschaften, die Meßbarkeits- und Stetigkeitsfragen betreffen.

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Dedicated To K.F. Hinderer On His Fiftieth Anniversary