The IPW estimator for the joint distribution function of the gap times from recurrent event data

Lee et al. in 2016 proposed a nonparametric estimator of the joint distribution of the gap time between transplant and the first infection and the following gap times between consecutive infections. In this article, we propose an alternative estimator based on the inverse-probability weighted (IPW) approach. Asymptotic properties of the proposed estimator are established . Simulation results indicate that the IPW estimator performs as well as the estimator proposed by Lee et al. We also propose an IPW estimator for estimating the joint distribution function of the gap times between consecutive recurrent events beyond the first episode.     

Principal varying coefficient estimator for high-dimensional models

ABSTRACT

We consider principal varying coefficient models in the high-dimensional setting, combined with variable selection, to reduce the effective number of parameters in semiparametric modelling. The estimation is based on B-splines approach. For the unpenalized estimator, we establish non-asymptotic bounds of the estimator and then establish the (asymptotic) local oracle property of the penalized estimator, as well as non-asymptotic error bounds. Monte Carlo studies reveal the favourable performance of the estimator and an application on a real dataset is presented.     

Test for the mean matrix in a Growth Curve model for high dimensions

We consider the problem of estimating and testing a general linear hypothesis in a general multivariate linear model, the so-called Growth Curve model, when the p × N observation matrix is normally distributed.

The maximum likelihood estimator (MLE) for the mean is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. We modify the MLE to an unweighted estimator and propose new tests which we compare with the previous likelihood ratio test (LRT) based on the weighted estimator, i.e., the MLE. We show that the performance of these new tests based on the unweighted estimator is better than the LRT based on the MLE.     

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted oscillating Brownian motion. For this continuously observed diffusion, the maximum likelihood estimator coincides with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results of the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient, or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations.     

Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring

In this paper, we consider the maximum likelihood estimator (MLE) of the scale parameter of the generalized exponential (GE) distribution based on a random censoring model. We assume the censoring distribution also follows a GE distribution. Since the estimator does not provide an explicit solution, we propose a simple method of deriving an explicit estimator by approximating the likelihood function. In order to compare the performance of the estimators, Monte Carlo simulation is conducted. The results show that the MLE and the approximate MLE are almost identical in terms of bias and variance.     

基于梯度统计量的逆抽样下风险差的置信区间构建

通过逆抽样过程获得的分布又称为负二项分布,在流行病学研究和二分类变量分布的研究中应用极为广泛。因此,提出两种基于梯度统计量的逆抽样下风险差的置信区间的构建方法,分别依据风险差的极大似然估计(MLE)和方差最小无偏一致估计量(UMVUE)。与现有的WALD方法和得分方法相比,该方法所构建置信区间的优点在于:置信区间构建方法既不需要计算Fisher信息阵也不需要计算其逆矩阵,可使计算得以大大简化;对所提出的基于梯度统计量的置信区间构建方法进行蒙特卡洛模拟研究,模拟结果表明提出的构建方法可以得到很好的覆盖概率和较短的区间宽度。     

The estimation of treatment effect for the censored bivariate data under the univariate censoring

This paper establishes a nonparametric estimator for the treatment effect on censored bivariate data under unvariate censoring. This proposed estimator is based on the one from Lin and Ying(1993)'s nonparametric bivariate survival function estimator, which is itself a generalized version of Park and Park(1995)' quantile estimator. A Bahadur type representation of quantile functions were obtained from the marginal survival distribution estimator of Lin and Ying' model. The asymptotic property of this estimator is shown below and the simulation studies are also given     

Estimation for the Generalized Pareto Distribution Using Maximum Likelihood and Goodness of Fit

In this article, we introduce a new estimator for the generalized Pareto distribution, which is based on the maximum likelihood estimation and the goodness of fit. The asymptotic normality of the new estimator is shown and a small simulation. From the simulation, the performance of the new estimator is roughly comparable with maximum likelihood for positive values of the shape parameter and often much better than maximum likelihood for negative values.     

Bayesian inference using product of spacings function for Progressive hybrid Type-I censoring scheme

This article is devoted to the development of product of spacings estimator for a Progressive hybrid Type-I censoring scheme with binomial removals. The experimental units are assumed to follow inverse Lindley distribution. We propose a Bayes estimator of associated scale parameter based on the product of spacings function and simultaneously compare it with that obtained under a usual Bayesian estimation procedure. The estimators are obtained under the squared error loss function along with corresponding HP intervals evaluated by using the Markov chain Monte-Carlo technique. The classical product of spacings estimator has also been derived and compared with the maximum likelihood estimator in addition to 95% average asymptotic confidence intervals. The applicability of the proposed methods is demonstrated by analysing a real data of guinea pigs affected with tuberculosis for the considered censoring scheme.     

Parametric Estimation of Menarcheal Age Distribution Based on Recall Data

Menarche, the onset of menstruation, is an important maturational event of female childhood. Most of the studies of age at menarche make use of dichotomous (status quo) data. More information can be harnessed from recall data, but such data are often censored in a informative way. We show that the usual maximum likelihood estimator based on interval censored data, which ignores the informative nature of censoring, can be biased and inconsistent. We propose a parametric estimator of the menarcheal age distribution on the basis of a realistic model of the recall phenomenon. We identify the additional information contained in the recall data and demonstrate theoretically as well as through simulations the advantage of the maximum likelihood estimator based on recall data over that based on status quo data.     

α-stable laws for noncoding regions in DNA sequences

In this work, we analyze the long-range dependence parameter for a nucleotide sequence in several different transformations. The long-range dependence parameter is estimated by the approximated maximum likelihood method, by a novel estimator based on the spectral envelope theory, by a regression method based on the periodogram function, and also by the detrended fluctuation analysis method. We study the length distribution of coding and noncoding regions for all Homo sapiens chromosomes available from the European Bioinformatics Institute. The parameter of the tail rate decay is estimated by the Hill estimator ?α. We show that the tail rate decay is greater than 2 for coding regions, while for almost all noncoding regions it is less than 2.     

Hodges-Lehmann Scale Estimator for Cauchy Distribution

We draw here on the relation between the Cauchy and hyperbolic secant distributions to prove that the MLE of the scale parameter of the Cauchy distribution is log-normally distributed and to study the properties of a Hodges-Lehmann type estimator for the scale parameter. This scale estimator is slightly biased but performs well even on small samples regardless of the location parameter. The asymptotic efficiency of the estimator is 98%.     

Exact linear inference for scaled exponential distribution based on doubly type-ii censored samples

In this paper, we make use of an algorithm of Huffer and Lin (2000) in order to develop exact interval estimation for the scale parameter to of an exponential distribution based on doubly Type-II censored samples. We also evaluate the accuracy of a chi-square approximation proposed by Balakrishnan and Gupta (1998). We present the MAPLE program for the determination of the exact percentage points of the pivotal quantity based on the best linear unbiased estimator. Finally, we present a couple of examples to illustrate the method of inference developed here.     

Bayes Estimation of Intraclass Correlation Coefficients Under Unequal Family Sizes

Bayesian inference for the intraclass correlation ρ is considered under unequal family sizes. We obtain the posterior distribution of ρ and then compare the performance of the Bayes estimator (posterior mean of ρ) with that of Srivastava's (1984) estimator through simulation. Simulation study shows that the Bayes estimator performs better than the Srivastava's estimator in terms of lower mean square error. We also obtain large sample posteriors of ρ based on the asymptotic posterior distribution and based on the Laplace approximation.     

A hypothesis testing procedure for the evaluation on the lifetime performance index of products with Burr XII distribution under progressive type I interval censoring

ABSTRACT

It is a very important topic these days to assessing the lifetime performance of products in manufacturing or service industries. Lifetime performance indices C_L is used to measure the larger-the-better type quality characteristics to evaluate the process performance for the improvement of quality and productivity. The lifetimes of products are assumed to have Burr XII distribution. The maximum likelihood estimator is used to estimate the lifetime performance index based on the progressive type I interval censored sample. The asymptotic distribution of this estimator is also developed. We use this estimator to build the new hypothesis testing algorithmic procedure with respect to a lower specification limit. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.     

Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.     

Exploring the distribution for the estimator of Rosenthal's ‘fail-safe’ number of unpublished studies in meta-analysis

The present article discusses the statistical distribution for the estimator of Rosenthal's ‘file-drawer’ number N_R, which is an estimator of unpublished studies in meta-analysis. We calculate the probability distribution function of N_R. This is achieved based on the central limit theorem and the proposition that certain components of the estimator N_R follow a half-normal distribution, derived from the standard normal distribution. Our proposed distributions are supported by simulations and investigation of convergence.     

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

When estimating loss distributions in insurance, large and small losses are usually split because it is difficult to find a simple parametric model that fits all claim sizes. This approach involves determining the threshold level between large and small losses. In this article, a unified approach to the estimation of loss distributions is presented. We propose an estimator obtained by transforming the data set with a modification of the Champernowne cdf and then estimating the density of the transformed data by use of the classical kernel density estimator. We investigate the asymptotic bias and variance of the proposed estimator. In a simulation study, the proposed method shows a good performance. We also present two applications dealing with claims costs in insurance.     

On estimating the common mean in two normal distributions after a preliminary test for equality of variances

Given two random samples of equal size from two normal distributions with common mean but possibly different variances, we examine the sampling performance of the pre-test estimator for the common mean after a preliminary test for equality of variances. It is shown that when the alternative in the pretest is one-sided, the Graybill-Deal estimator is dominated by the pre-test estimator if the critical value is chosen appropriately. It is also shown that all estimators, the grand mean, the Graybill-Deal estimator and the pre-test estimator, are admissible when the alternative in the pre-test is two-sided. The optimal critical values in the two-sided pre-test are sought based on the minimax regret and the minimum average risk criteria, and it is shown that the Graybill-Deal estimator is most preferable under the minimum average risk criterion when the alternative in the pre-test is two-sided.     

On a Family of Trimodal Distributions

In this article, a family of trimodal distributions is presented. The distributional properties and some of the inferential aspects of this family of trimodal distributions are discussed. We propose a moment based estimator as well as a maximum likelihood estimator of the parameters. A numerical simulation is conducted to evaluate the finite sample performances of the proposed estimators. A real data example is analyzed for illustration.