Jan Swanepoel

A note on proving that the (modified) bootstrap works

Let be a sample of independent, identically distributed (i.i.d.) random variables with common distribution function F and suppose is a bootstrap sample of i.i.d. random variables from the empirical distribution function (e.d.f.) Fn of . The twofold aim of this paper consists in, firstly providing examples illustrating the fact that the custo-mary choice of m = n is frequently wrong. In fact, specifying m as some suitable function of n it can. among other things, be shown chat the bootstrap also works for the well-known counter-examples given by Bickel and Freedman (1981). Secondly, a method is suggested which can be used to show that the bootstrap method of distribution approximation is asymptotically valid. This method is based on the oscillation behavior of empirical processes rather than the equicontinuity arguments of Bickel and Freedman (1981) which are based on the Mallows metric.

Note(s): A note on the asymptotic behavior of the Bernstein estimator of the copula density

Copulas and their corresponding densities are functions of a multivariate joint distribution and the one-dimensional marginals. Bernstein estimators have been used as smooth nonparametric estimators for copulas and copula densities. The purpose of this note is to study the asymptotic distributional behavior of the Bernstein estimator of a copula density. Compared to the existing results, our general theorem does not assume known marginals. This makes our theorem applicable for real data.

A class of goodness-of-fit tests based on a new characterization of the exponential distribution

A new characterisation of the exponential distribution in the class of new better than used in expectation (NBUE) life distributions is presented. Utilising this characterisation, a new class of goodness-of-fit tests for exponentiality against NBUE alternatives is proposed. The tests are shown to be consistent, and the limiting distributions of the test statistics under the null and alternative hypotheses are derived. The newly proposed tests are compared to existing goodness-of-fit tests by means of Pitman and approximate Bahadur relative efficiencies. A limited Monte Carlo study is conducted to compare the various tests with regard to power for small and moderate sample sizes against a range of alternative distributions. Three members of the class of test statistics are identified as being at least as effective as established tests for exponentiality against NBUE alternatives.

Modifying the kernel distribution function estimator towards reduced bias

We explore the convergence rates of a kernel-based distribution function estimator with variable bandwidth. As in density estimation, a considerable bias reduction from O(h 2) to O(h 4) can be obtained by replacing the bandwidth h by h/f 1/2(X i ). We show that the necessary replacement of f 1/2 by some pilot estimator , depending on a second bandwidth g, has no penalizing effect on bias and variance, provided we undersmooth with the pilot bandwidth g, that is g/h→0 in a certain way. Owing to the considerable bias reduction, a simple plug-in normal reference bandwidth selector works effectively in practice. Distribution function estimators with good convergence properties and with simple bandwidth selectors are desirable for repetitive use in smoothed bootstrap algorithms.

Goodness-Of-Fit Tests Based on Estimated Expectations of Probability Integral Transformed Order Statistics

New goodness-of-fit tests, based on bootstrap estimated expectations of probability integral transformed order statistics, are derived for the location-scale model. The resulting test statistics are location and scale invariant, and are sensitive to discrepancies at the tails of the hypothesized distribution. The limiting null distributions of the test statistics are derived in terms of functionals of a certain Gaussian process, and the tests are shown to be consistent against a broad family of alternatives. Critical points for all sample sizes are provided for tests of normality. A simulation study shows that the proposed tests are more powerful than established tests such as Shapiro-Wilk, Cramér-von Mises and Anderson-Darling, for a wide range of alternative distributions.

Transformation Kernel Density Estimation With Applications

One of the main objectives of this article is to derive efficient nonparametric estimators for an unknown density fX. It is well known that the ordinary kernel density estimator has, despite several good properties, some serious drawbacks. For example, it suffers from boundary bias and it also exhibits spurious bumps in the tails. We propose a semiparametric transformation kernel density estimator to overcome these defects. It is based on a new semiparametric transformation function that transforms data to normality. A generalized bandwidth adaptation procedure is also developed. It is found that the newly proposed semiparametric transformation kernel density estimator performs well for unimodal, low, and high kurtosis densities. Moreover, it detects and estimates densities with excessive curvature (e.g., modes and valleys) more effectively than existing procedures. In conclusion, practical examples based on real-life data are presented.

Simple and effective number-of-bins circumference selectors for a histogram

Two very effective data-based procedures which are simple and fast to compute are proposed for selecting the number of bins in a histogram. The idea is to choose the number of bins that minimizes the circumference (or a bootstrap estimate of the expected circumference) of the frequency histogram. Contrary to most rules derived in the literature, our method is therefore not dependent on precise asymptotic analyses. It is shown by means of an extensive Monte-Carlo study that our selectors perform well in comparison with recently suggested selectors in the literature, for a wide range of density functions and sample sizes. The behaviour of one of the proposed rules is also illustrated on real data sets.

Optimal kernels when estimating non-smooth densities

The derivation of new kernel functions for the kernel estimator of an unknown density function is given. These kernels are shown to be optimal in some sense when the underlying density f is continuous but its derivative f′ is not, and consequently a solu tion is presented for an unsolved problem which was stated by van Eeden (1985). Other attractive features of these kernels are also discussed and a number of graphs are listed.

Modified bootstrap consistency rates for U-quantiles

We show that, compared to the classical bootstrap, the modified bootstrap provides faster consistency rates for the bootstrap distribution of U-quantiles. This shows that the modified bootstrap is useful, not only in cases where the classical bootstrap fails, but also in situations where it is valid.

A new powerful test for periodic pulsed emission of high-energy photons

A novel stastistical test devised for detecting periodic pulsations in a series of photon arrival times is found to perform powerfully in the detection of multiple, narrow peaked light curve shapes against a uniform background, especially if the light curve has multiple peaks. The statistical test has been applied to observations of the first 1350 COS B high-energy gamma-ray events from PSR 0833-45, the Vela pulsar. 13 refs.