Mei Ling Huang

A level crossing quantile estimation method

We introduce a nonparametric quantile estimation method by applying a level crossing empirical function which will be defined in this paper, and also introduce a computational method for the new estimator. A comparison of the new quantile estimation method with the usual kernel quantile estimation method based on the classical empirical distribution function is included. Computational results show that the new method is more efficient than the usual method in many cases.

R-distribution and its applications

An R-distribution (RD) is introduced in this paper as the distribution of the sum of n independent but not identically distributed right truncated Poisson variables. Its p.d.f. can be expressed in terms of an R-number and an incomplete exponential function, both of which are defined and tabulated here. Properties of this R-distribution are investigated. A MVU estimate of the p.d.f. of RD is obtained. Some examples of its applications are given together with tables to facilitate the calculations.

An Algorithm for Fitting Heavy-Tailed Distributions via Generalized Hyperexponentials

In this paper, we propose an algorithm to fit heavy-tailed (HT) distribution functions by generalized hyperexponential (GH) distribution functions. A discussion of the steps, usage, and accuracy of the GH algorithm is given. Several examples in this paper show that the proposed method can be applied to fit HT distributions with a completely monotone probability density function (pdf) very well, like the Pareto distribution and the Weibull distribution with the shape parameter less than one, as well as HT distributions whose pdf is not completely monotone, like the lognormal distribution. In addition, we provide an example that shows that the proposed method can be applied to density estimation of real data presenting a heavy tail.

On a distribution-free quantile estimator

Abstract A distribution free non-kernel quantile estimator HD p of the p th population quantile was introduced by Harrell and Davis (Biometrika 69(3) (1982) 635). In this paper, we use a level crossing empirical distribution function to propose a new estimator HD p (lc) which is a weighted version of HD p . The exact efficiency and simulation efficiency of HD p (lc) relative to the HD p quantile estimator are studied. From both the theoretical and computational points of view, the new estimator is more efficient in many cases, especially for the tails of the distributions and small sample sizes.

A note on moments of the maximum of Cesàro summation

Let {Xn; n [greater-or-equal, slanted] 1} be a sequence of independent real-valued random variables and {an,k; k [greater-or-equal, slanted] 1, n [greater-or-equal, slanted] 1} an infinite matrix of real numbers with supn an, k 0. This result is used to establish some results on moments of the maximum of normed weighted averages, in particular, the maximum of Cesaro summation.

A weighted linear quantile regression

In this article, we introduce a new weighted quantile regression method. Traditionally, the estimation of the parameters involved in quantile regression is obtained by minimizing a loss function based on absolute distances with weights independent of explanatory variables. Specifically, we study a new estimation method using a weighted loss function with the weights associated with explanatory variables so that the performance of the resulting estimation can be improved. In full generality, we derive the asymptotic distribution of the weighted quantile regression estimators for any uniformly bounded positive weight function independent of the response. Two practical weighting schemes are proposed, each for a certain type of data. Monte Carlo simulations are carried out for comparing our proposed methods with the classical approaches. We also demonstrate the proposed methods using two real-life data sets from the literature. Both our simulation study and the results from these examples show that our proposed method outperforms the classical approaches when the relative efficiency is measured by the mean-squared errors of the estimators.

ERROR ESTIMATES FOR DOMINICI’S HERMITE FUNCTION ASYMPTOTIC FORMULA AND SOME APPLICATIONS

The aim of this paper is to find a concrete bound for the error involved when approximating the nth Hermite function (in the oscillating range) by an asymptotic formula due to D. Dominici. This bound is then used to study the accuracy of certain approximations to Hermite expansions and to Fourier transforms. A way of estimating an unknown probability density is proposed.

The D compound Poisson distribution

A new extension of the Neyman Type A distribution is presented in this paper. It is called the D Compound Poisson distribution (D-CPD) and is based on the D distribution, D numbers and an incomplete exponential function. The properties of D-CPD are studied. The maximum likelihood estimation of the parameters, and a minimum variance unbiased estimator (MUVE) of the probability function of the D-CPD are given. It is interesting to observe that this MVUE depends on only three D numbers. An example of the applications of D-CPD is provided at the end.

Recurrence relations for the r-distribution

Recurrence relations for R-numbers are given and used to derive similar relations for the associated probability distribution. The probability recurrence relations do not depend on R-numbers and are used to tabulate some numerical values.

A more generalized stirling distribution of the second kind

A more generalized stirling distribution of the second kind (MGSDSK) is introduced in this paper as the distribution of the sum of the independent but not identically distributed left truncated Poisson variables. Properties of MGSDSK are studied. The recursion relation and decomposition of MGSDSK are obtained. The rth moment is also found and a new recurrence relationship for them are given. A new incomplete exponential function is utilized in the derivations. A MVU estimate of the p, d, f. of MGSDSK is obtained.