Simple boundary correction for kernel density estimation

If a probability density function has bounded support, kernel density estimates often overspill the boundaries and are consequently especially biased at and near these edges. In this paper, we consider the alleviation of this boundary problem. A simple unified framework is provided which covers a number of straightforward methods and allows for their comparison: generalized jackknifing generates a variety of simple boundary kernel formulae. A well-known method of Rice (1984) is a special case. A popular linear correction method is another: it has close connections with the boundary properties of local linear fitting (Fan and Gijbels, 1992). Links with the optimal boundary kernels of Müller (1991) are investigated. Novel boundary kernels involving kernel derivatives and generalized reflection arise too. In comparisons, various generalized jackknifing methods perform rather similarly, so this, together with its existing popularity, make linear correction as good a method as any. In an as yet unsuccessful attempt to improve on generalized jackknifing, a variety of alternative approaches is considered. A further contribution is to consider generalized jackknife boundary correction for density derivative estimation. En route to all this, a natural analogue of local polynomial regression for density estimation is defined and discussed.     

Whither jackknifing in stein-rule estimation

In multi-parameter ( multivariate ) estimation, the Stein rule provides minimax and admissible estimators , compromising generally on their unbiasedness. On the other hand, the primary aim of jack-knifing is to reduce the bias of an estimator ( without necessarily compromising on its efficacy ), and, at the same time, jackknifing provides an estimator of the sampling variance of the estimator as well. In shrinkage estimation ( where minimization of a suitably defined risk function is the basic goal ), one may wonder how far the bias-reduction objective of jackknifing incorporates the dual objective of minimaxity ( or admissibility ) and estimating the risk of the estimator ? A critical appraisal of this basic role of jackknifing in shrinkage estimation is made here. Restricted, semi-restricted and the usual versions of jackknifed shrinkage estimates are considered and their performance characteristics are studied . It is shown that for Pitman-type ( local ) alternatives, usually, jackkntfing fails to provide a consistent estimator of the ( asymptotic ) risk of the shrinkage estimator, and a degenerate asymptotic situation arises for the usual fixed alternative case.     

Tests for equality of variances with paired data

The normal theory test for equality of variances with paired data is shown to be nonrobust to violation of the assumption of normality. Nonparametric tests are shown to provide a much safer alternative with little loss of efficiency.     

On the estimation of the parameters of the von mises distribution

The usual maximum likelihood estimators of the parameters of the von Mises distribution are shown to perform badly in small samples. In view of this and the fact that these estimators require a large amount of computation, alternative, simpler estimators are proposed. It is shown that these estimators are at least comparable to the traditional estimators and are, in many cases, superior to them. We also apply the procedure of jackknifing to the maximum likelihood estimator of the concentration parameter of the von Mises distribution and compare the properties of the jackknifed estimator with the other estimators considered in this paper.     

Asymptotic and finite sample behavior of the time on test estimator under random censorship when lifetimes are not Exponetial

The robustness of the time on test estimator of mean life is studied in both asymptotic and finite sample situations under random censorship. The estimator is shown t o be asymptotically normal and generally in consistent , unless the life time sare exponential . The limiting value of the estimator depends on both the life time and censorship distributions . A simulations tudy of finite sample behavior shows that biases a reslight under exponentiality and serious if exponentia lity is viol at ed . The finite sample behavior is not well described by the limiting normal distribution . Jackknifing produces a useful variance estimate, but is of little value in bias correction.     

The Berry-Esseen bound for Studentized U-statistics

Callaert and Veraverbeke (1981) recently obtained a Berry-Esseen-type bound of order n^–1/2 for Studentized nondegenerate U-statistics of degree two. The condition these authors need to obtain this order bound is the finiteness of the 4.5th absolute moment of the kernel h. In this note it is shown that this assumption can be weakened to that of a finite (4 + ?)th absolute moment of the kernel h, for some ? > 0. Our proof resembles part of Helmers and van Zwet (1982), where an analogous result is obtained for the Student t-statistic. The present note extends this to Studentized U-statistics.     

Variance reduction for quantile estimates in simulations via nonlinear controls

Linear controls are a well known simple technique for achieving variance reduction in computer simulation. Unfortunately the effectiveness of a linear control depends upon the correlation between the statistic of interest and the control, which is often low. Since statistics often have a nonlinear relation-ship with the potential control variables, nonlinear controls offer a means for improvement over linear controls. This paper focuses on the use of nonlinear controls for reducing the variance of quantile estimates in simulation. It is shown that one can substantially reduce the analytic effort required to develop a nonlinear control from a quantile estimator by using a strictly monotone transformation to create the nonlinear control. It is also shown that as one increases the sample size for the quantile estimator, the asymptotic multivariate normal distribution of the quantile of interest and the control reduces the effectiveness of the nonlinear control to that of the linear control. However, the data has to be sectioned to obtain an estimate of the variance of the controlled quantile estimate. Graphical methods are suggested for selecting the section size that maximizes the effectiveness of the nonlinear control