Parameters estimation for weibull distributed lifetimes under progressive censoring with random removeals

This paper considers the estimation problem when lifetimes are Weibull distributed and are collected under a Type-II progressive censoring with random removals, where the number of units removed at each failure time follows a uniform discrete distribution. The expected time of this censoring plan is discussed and compared numerically to that under a Type II censoring without removal. Maximum likelihood estimator of the parameters and their asymptotic variances are derived.     

Affiliation networks with an increasing degree sequence

Affiliation network is one kind of two-mode social network with two different sets of nodes (namely, a set of actors and a set of social events) and edges representing the affiliation of the actors with the social events. Although a number of statistical models are proposed to analyze affiliation networks, the asymptotic behaviors of the estimator are still unknown or have not been properly explored. In this article, we study an affiliation model with the degree sequence as the exclusively natural sufficient statistic in the exponential family distributions. We establish the uniform consistency and asymptotic normality of the maximum likelihood estimator when the numbers of actors and events both go to infinity. Simulation studies and a real data example demonstrate our theoretical results.     

Simulating random variables using moment-generating functions and the saddlepoint approximation

When we are given only a transform such as the moment-generating function of a distribution, it is rare that we can efficiently simulate random variables. Possible approaches such as the inverse transform using numerical inversion of the transform are computationally very expensive. However, the saddlepoint approximation is known to be exact for the Normal, Gamma, and inverse Gaussian distribution and remarkably accurate for a large number of others. We explore the efficient use of the saddlepoint approximation for simulating distributions and provide three examples of the accuracy of these simulations.     

Bayesian estimation and prediction for Weibull model with progressive censoring

This article presents the statistical inferences on Weibull parameters with the data that are progressively type II censored. The maximum likelihood estimators are derived. For incorporation of previous information with current data, the Bayesian approach is considered. We obtain the Bayes estimators under squared error loss with a bivariate prior distribution, and derive the credible intervals for the parameters of Weibull distribution. Also, the Bayes prediction intervals for future observations are obtained in the one- and two-sample cases. The method is shown to be practical, although a computer program is required for its implementation. A numerical example is presented for illustration and some simulation study are performed.     

Classical estimation of hazard rate and mean residual life functions of Pareto distribution

Abstract

In this paper we find the maximum likelihood estimates (MLEs) of hazard rate and mean residual life functions (MRLF) of Pareto distribution, their asymptotic non degenerate distribution, exact distribution and moments. We also discuss the uniformly minimum variance unbiased estimate (UMVUE) of hazard rate function and MRLF. Finally, two numerical examples with simulated data and real data set, are presented to illustrate the proposed estimates.     

Least squares estimation of varying‐coefficient hazard regression with application to breast cancer dose‐intensity data

To enhance modeling flexibility, the authors propose a nonparametric hazard regression model, for which the ordinary and weighted least squares estimation and inference procedures are studied. The proposed model does not assume any parametric specifications on the covariate effects, which is suitable for exploring the nonlinear interactions between covariates, time and some exposure variable. The authors propose the local ordinary and weighted least squares estimators for the varying‐coefficient functions and establish the corresponding asymptotic normality properties. Simulation studies are conducted to empirically examine the finite‐sample performance of the new methods, and a real data example from a recent breast cancer study is used as an illustration. The Canadian Journal of Statistics 37: 659–674; 2009 © 2009 Statistical Society of Canada     

Sums of powers of binomial coefficients

An asymptotic series for sums of powers of binomial coefficients is derived, the general term being defined and usable with a computer symbolic language. Sums of squares of coefficients in the symmetric case are shown to have a link with classical moment problems, but this property breaks down for cubes and higher powers. Problems of remainders for the asymptotic series are mentioned. Using the reflection formula for I'(.), a continuous form for a binomial function is set up, and this becomes oscillatory outstde the usual range. A new contmued fraction emerges for the logarithm of an adjusted sum of binomial squares. The note is a contribution to the problem of the interpretation of asymptotic series and processes for their convergence acceleration.     

Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t2 statlstics under normality

Let T² _i=z′_iS^?1z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ?∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n^?2. Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=I_k and (ii) k=2 by inverting the expanded characteristic function up to the orders n^?2 and n^?1, respectively. Our results can be applied to the distribution function of max (T² _i,…,T² _k) as a special case.     

A SADDLEPOINT APPROXIMATION TO THE DISTRIBUTION FUNCTION OF THE ANDERSON-DARLING TEST STATISTIC

The Anderson-Darling goodness-of-fit test has a highly skewed and non-standard limit distribution. Various attempts have been made to tabulate the associated critical points, using both theoretical approximations and simulation methods. We show that a standard saddlepoint approximation performs well in both tails of the distribution. It is markedly superior to other theoretical approximations in the lower tail of the distribution.     

Asymptotic relative efficiency of wald tests in measurement error models

In this paper, asymptotic relative efficiency (ARE) of Wald tests for the Tweedie class of models with log-linear mean, is considered when the aux¬iliary variable is measured with error. Wald test statistics based on the naive maximum likelihood estimator and on a consistent estimator which is obtained by using Nakarnura's (1990) corrected score function approach are defined. As shown analytically, the Wald statistics based on the naive and corrected score function estimators are asymptotically equivalents in terms of ARE. On the other hand, the asymptotic relative efficiency of the naive and corrected Wald statistic with respect to the Wald statistic based on the true covariate equals to the square of the correlation between the unobserved and the observed co-variate. A small scale numerical Monte Carlo study and an example illustrate the small sample size situation.