Approximation to the distribution of the least squares estimators in two dimensional cosine models by randomly weighted bootstrap

Recently, Kundu and Gupta （Metrika, 48：83 C 97, 1998） established the asymptotic normality of the least squares estimators in the two dimensional cosine model. In this paper, we give the approximation to the general least squares estimators by using random weights which is called the Bayesian bootstrap or the random weighting method by Rubin （Annals of Statistics, 9：130 C 134, 1981） and Zheng （Acta Math. Appl. Sinica （in Chinese）, 10（2）： 247 C 253, 1987）. A simulation study shows that this approximation works very well.     

Generalized variance estimators in the multivariate gamma models

It is already known that the uniformly minimum variance unbiased (UMVU) estimator of the generalized variance always exists for any natural exponential family. However, in practice, this estimator is often difficult to obtain. This paper provides explicit forms of the UMVU estimators for the bivariate and symmetric multivariate gamma models, which are diagonal quadratic exponential families. For the non-independent multivariate gamma models, it is shown that the UMVU and the maximum likelihood estimators are not proportional.      

Moment estimators for the two-parameter M-Wright distribution

A formal parameter estimation procedure for the two-parameter M-Wright distribution is proposed. This procedure is necessary to make the model useful for real-world applications. Note that its generalization of the Gaussian density makes the M-Wright distribution appealing to practitioners. Closed-form estimators are also derived from the moments of the log-transformed M-Wright distributed random variable, and are shown to be asymptotically normal. Tests using simulated data indicated favorable results for our estimation procedure.     

Asymptotic distribution of minimum contrast estimators

We consider the behavior of minimum contrast estimators constructed from independent not identically distributed observations. It is proved under new assumptions that the consistent estimators are asymptotically normal. For the particular case of maximum likelihood estimators we generalize the known result of A. Philippou and G. Roussas.Translated from Statisticheskie Metody, pp. 56–65, 1980.     

Method-of-moments estimators of stable distribution parameters

A method is developed for estimating the four parameters of stable Paretian distributions. Based on a procedure proposed by an earlier researcher but never developed, the method proves to be mathematically simple and easy to apply. The method is extensively tested and sampling properties of the estimators are studied, providing approximate confidence intervals for the estimators.     

A note on the performance of the gamma kernel estimators at the boundary

The gamma kernel estimator is proposed in Chen [Chen, S.X., 2000. Probability density function estimation using gamma kernels. Annals of the Institute of Statistical Mathematics 52, 471–480] to estimate densities with support $[0,\infty )$. It is shown in his paper that the gamma kernel estimator is non-negative, free of boundary bias, and achieves the optimal rate of convergence for the mean integrated squared error. Numerical results reported in Chen’s paper show that, in the boundary region, the gamma kernel estimator even outperforms some widely used boundary corrected density estimators such as the boundary kernel estimator. However, our study finds that the gamma kernel estimator at $x=0$ is actually the reflection estimator when the double exponential kernel is used and is only boundary problem free when the estimated density has a shoulder at $x=0$ (i.e., the first derivative of the density at $x=0$ is zero). For densities not satisfying the shoulder condition, we show that the gamma kernel estimator has a severe boundary problem and its performance is inferior to that of the boundary kernel estimator.     

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.     

Approximation of multivariate distribution functions

In the paper the unknown distribution function is approximated with a known distribution function by means of Taylor expansion. For this approximation a new matrix operation — matrix integral — is introduced and studied in [PIHLAK, M.: Matrix integral, Linear Algebra Appl. 388 (2004), 315–325]. The approximation is applied in the bivariate case when the unknown distribution function is approximated with normal distribution function. An example on simulated data is also given.      

Asymptotic properties of the estimators of the Weibull distribution parameters

The properties of the maximum likelihood and moment estimators are investigated for the three-dimensional Weibull distribution in the case of arbitrary values of the shape parameter.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 9–16, 1988.     

Finding maximum likelihood estimators for the three-parameter Weibull distribution

Much work has been devoted to the problem of finding maximum likelihood estimators for the three-parameter Weibull distribution. This problem has not been clearly recognized as a global optimization one and most methods from the literature occasionally fail to find a global optimum. We develop a global optimization algorithm which uses first order conditions and projection to reduce the problem to a univariate optimization one. Bounds on the resulting function and its first order derivative are obtained and used in a branch-and-bound scheme. Computational experience is reported. It is also shown that the solution method we propose can be extended to the case of right censored samples.     

Asymptotic bounds for estimators without limit distribution

Let be a general family of probability measures,κ : a functional, and the optimal limit distribution for regular estimator sequences of κ. On intervals symmetric about 0, the concentration of this optimal limit distribution can be surpassed by the asymptotic concentration of an arbitrary estimator sequence only forP in a “small” subset of . For asymptotically median unbiased estimator sequences the same is true for arbitrary intervals containing 0. The emphasis of the paper is on “pointwise” conditions for , as opposed to conditions on shrinking neighbourhoods, and on “general” rather than parametric families.     

Approximation of the variance gamma model with a finite mixture of normals

We investigate the possibility of approximating the variance gamma distribution with a finite mixture of normals. Therefore, we apply this result to derive a simple historical estimation procedure by means of the Expectation Maximization algorithm.     

Convexity of the median in the gamma distribution

We show that the median m(x) in the gamma distribution with parameter x is a strictly convex function on the positive half-line.     

Inequalities for the median of the gamma distribution

We provide several new inequalities involving λ_n, the median of the gamma distribution of order n+1 with parameter 1. Among others, we present sharp upper and lower bounds for the arithmetic mean of λ₁,λ₂,…,λ_n. For all integers n?1 we have

     

Inventory control with the gamma probability distribution

Application of inventory theory often rely on the normal and negative exponential distributions to represent the lead time demand of fast and slow moving items respectively. Yet it is now accepted that both distributions, when taken together, are incapable of adequately describing the demand characteristics of all items found in the typical inventory. Instead there has been a growing interest in the use of the gamma probability distribution because it not only encompasses both former distributions as special cases but also covers the gaps left by them. In the process a number of methods for calculating control parameters have appeared in the literature for items with gamma distributed lead time demand. As knowledge about the problem has increased there has been a general tendency towards greater simplification. This paper continues the trend by introducing an approach that depends only on concepts from basic statistics. The aim is to eliminate unnecessary complexity and make the associated theory easier to understand.     

A simulation based approach to the parameter estimation for the three-parameter gamma distribution

The gamma distribution is one of the commonly used statistical distribution in reliability. While maximum likelihood has traditionally been the main method for estimation of gamma parameters, Hirose has proposed a continuation method to parameter estimation for the three-parameter gamma distribution. In this paper, we propose to apply Markov chain Monte Carlo techniques to carry out a Bayesian estimation procedure using Hirose’s simulated data as well as two real data sets. The method is indeed flexible and inference for any quantity of interest is readily available.     

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.