Automatic choice of driving values in Monte Carlo likelihood approximation via posterior simulations

For models with random effects or missing data, the likelihood function is sometimes intractable analytically but amenable to Monte Carlo approximation. To get a good approximation, the parameter value that drives the simulations should be sufficiently close to the maximum likelihood estimate (MLE) which unfortunately is unknown. Introducing a working prior distribution, we express the likelihood function as a posterior expectation and approximate it using posterior simulations. If the sample size is large, the sample information is likely to outweigh the prior specification and the posterior simulations will be concentrated around the MLE automatically, leading to good approximation of the likelihood near the MLE. For smaller samples, we propose to use the current posterior as the next prior distribution to make the posterior simulations closer to the MLE and hence improve the likelihood approximation. By using the technique of data duplication, we can simulate from the sharpened posterior distribution without actually updating the prior distribution. The suggested method works well in several test cases. A more complex example involving censored spatial data is also discussed.     

Tests and confidence intervals for the shape parameter of a gamma distribution

Inference based on the Central Limit Theorem has only first order accuracy. We give tests and confidence intervals (CIs) of second orderaccuracy for the shape parameter ρ of a gamma distribution for both the unscaled and scaled cases.

Tests and CIs based on moment and cumulant estimates are considered as well as those based on the maximum likelihood estimate (MLE).

For the unscaled case the MLE is the moment estimate of order zero; the most efficient moment estimate of integral order is the sample mean, having asymptotic relative efficiency (ARE) .61 when ρ= 1.

For the scaled case the most efficient moment estimate is a functionof the mean and variance. Its ARE is .39 when ρ = 1.

Our motivation for constructing these tests of ρ = 1 and CIs forρ is to provide a simple and convenient method for testing whether a distribution is exponential in situations such as rainfall models where such an assumption is commonly made.     

一种简单有效的二项分布比例参数似然比置信区间的求法

对二项分布比例参数p的似然比置信区间,提出一种简便求解方法。在平均覆盖率、平均区间长度及区间长度的95%置信区间准则下与WScore、Plus4、Jeffreys置信区间进行模拟比较。试验表明,在二项分布b(n,p)的参数n≥20且p∈(0.1,0.9)时,该方法获取的似然比置信区间性能优良。当点估计p值不是接近于0或1且n≥20时,推荐使用本方法获取p的置信区间。     

Firth adjustment for Weibull current-status survival analysis

Abstract

Analysis of right-censored data is problematic due to infinite maximum likelihood estimates (MLE) and potentially biased estimates, especially for small numbers of events. Analyzing current-status data is especially troublesome because of the extreme loss of precision due to large failure intervals. We extend Firth’s method for regular parametric problems to current-status modeling with the Weibull distribution. Firth advocated a bias reduction method for MLE by systematically correcting the score equation. An advantage is that it is still applicable when the MLE does not exist. We present simulation studies and two illustrative analyses involving RFM mice lung tumor data.     

Likelihood methods in randomized trials with noncompliance and subsequent nonresponse

This article considers likelihood methods for estimating the causal effect of treatment assignment for a two-armed randomized trial assuming all-or-none treatment noncompliance and allowing for subsequent nonresponse. We first derive the observed data likelihood function as a closed form expression of the parameter given the observed data where both response and compliance state are treated as variables with missing values. Then we describe an iterative procedure which maximizes the observed data likelihood function directly to compute a maximum likelihood estimator (MLE) of the causal effect of treatment assignment. Closed form expressions at each iterative step are provided. Finally we compare the MLE with an alternative estimator where the probability distribution of the compliance state is estimated independent of the response and its missingness mechanism. Our work indicates that direct maximum likelihood inference is straightforward for this problem. Extensive simulation studies are provided to examine the finite sample performance of the proposed methods.     

Inconsistency of the MLE for the Joint Distribution of Interval-Censored Survival Times and Continuous Marks

Abstract.  This paper considers the non-parametric maximum likelihood estimator (MLE) for the joint distribution function of an interval-censored survival time and a continuous mark variable. We provide a new explicit formula for the MLE in this problem. We use this formula and the mark-specific cumulative hazard function of Huang & Louis (1998) to obtain the almost sure limit of the MLE. This result leads to necessary and sufficient conditions for consistency of the MLE, which imply that the MLE is inconsistent in general. We show that the inconsistency can be repaired by discretizing the marks. Our theoretical results are supported by simulations.     

Estimation of the scale parameter of the Rayleigh distribution with multiply type–II censored sample

Based on a multiply type-II censored sample, the maximum likelihood estimator (MLE) and Bayes estimator for the scale parameter and the reliability function of the Rayleigh distribution are derived. However, since the MLE does not exist an explicit form, an approximate MLE which is the maximizer of an approximate likelihood function will be given. The comparisons among estimators are investigated through Monte Carlo simulations. An illustrative example with the real data concerning the 23 ball bearing in the life test is presented.     

Statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data

In this article, the statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data is discussed. The estimation of the parameters for Gompertz distribution is obtained using maximum likelihood method (MLE) and Bayesian method under three different loss functions. We also proved the existence and uniqueness of the MLE. The one-sample Bayesian prediction intervals are obtained. The work is done for different values of the parameters. We apply the Monto Carlo simulation to compare the proposed methods, also an example is discussed to construct the Prediction intervals.     

A new three-parameter extension of the log-logistic distribution with applications to survival data

In this article, a new three-parameter extension of the two-parameter log-logistic distribution is introduced. Several distributional properties such as moment-generating function, quantile function, mean residual lifetime, the Renyi and Shanon entropies, and order statistics are considered. The estimation of the model parameters for complete and right-censored cases is investigated competently by maximum likelihood estimation (MLE). A simulation study is conducted to show that these MLEs are consistent in moderate samples. Two real datasets are considered; one is a right-censored data to show that the proposed model has a superior performance over several existing popular models.     

Estimation of the scale parameter of the Rayleigh distribution under general progressive censoring

Based on a general progressively type II censored sample, the maximum likelihood estimator (MLE), Bayes estimator under squared error loss and credible intervals for the scale parameter and the reliability function of the Rayleigh distribution are derived. Also, the Bayes predictive estimator and highest posterior density (HPD) prediction interval for future observation are considered. Comparisons among estimators are investigated through Monte Carlo simulations. An illustrative example with real data concerning 23 ball bearings in a life test is presented.     

Constrained estimation and likelihood intervals for censored data

The authors propose a reduction technique and versions of the EM algorithm and the vertex exchange method to perform constrained nonparametric maximum likelihood estimation of the cumulative distribution function given interval censored data. The constrained vertex exchange method can be used in practice to produce likelihood intervals for the cumulative distribution function. In particular, the authors show how to produce a confidence interval with known asymptotic coverage for the survival function given current status data.     

Exact inference for competing risks model with generalized type-I hybrid censored exponential data

Recently, exact inference under hybrid censoring scheme has attracted extensive attention in the field of reliability analysis. However, most of the authors neglect the possibility of competing risks model. This paper mainly discusses the exact likelihood inference for the analysis of generalized type-I hybrid censoring data with exponential competing failure model. Based on the maximum likelihood estimates for unknown parameters, we establish the exact conditional distribution of parameters by conditional moment generating function, and then obtain moment properties as well as exact confidence intervals (CIs) for parameters. Furthermore, approximate CIs are constructed by asymptotic distribution and bootstrap method as well. We also compare their performances with exact method through the use of Monte Carlo simulations. And finally, a real data set is analysed to illustrate the validity of all the methods developed here.     

Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data

In this paper, inference for the scale parameter of lifetime distribution of a k-unit parallel system is provided. Lifetime distribution of each unit of the system is assumed to be a member of a scale family of distributions. Maximum likelihood estimator (MLE) and confidence intervals for the scale parameter based on progressively Type-II censored sample are obtained. A β-expectation tolerance interval for the lifetime of the system is obtained. As a member of the scale family, half-logistic distribution is considered and the performance of the MLE, confidence intervals and tolerance intervals are studied using simulation.     

Gompertz-power series distributions

ABSTRACT

In this article, we introduce the Gompertz power series (GPS) class of distributions which is obtained by compounding Gompertz and power series distributions. This distribution contains several lifetime models such as Gompertz-geometric (GG), Gompertz-Poisson (GP), Gompertz-binomial (GB), and Gompertz-logarithmic (GL) distributions as special cases. Sub-models of the GPS distribution are studied in details. The hazard rate function of the GPS distribution can be increasing, decreasing, and bathtub-shaped. We obtain several properties of the GPS distribution such as its probability density function, and failure rate function, Shannon entropy, mean residual life function, quantiles, and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented, and simulation studies are performed for evaluation of this estimation for complete data, and the MLE of parameters for censored data. At the end, a real example is given.     

Inference on constant stress accelerated life tests for log-location-scale lifetime distributions with type-I hybrid censoring

In this paper, we consider a constant stress accelerated life test terminated by a hybrid Type-I censoring at the first stress level. The model is based on a general log-location-scale lifetime distribution with mean life being a linear function of stress and with constant scale. We obtain the maximum likelihood estimators (MLE) and the approximate maximum likelihood estimators (AMLE) of the model parameters. Approximate confidence intervals, likelihood ratio tests and two bootstrap methods are used to construct confidence intervals for the unknown parameters of the Weibull and lognormal distributions using the MLEs. Finally, a simulation study and two illustrative examples are provided to demonstrate the performance of the developed inferential methods.     

AN ALTERNATIVE PARAMETRIC APPROACH FOR DISCRETE MISSING DATA PROBLEMS

We propose an iterative method of estimation for discrete missing data problems that is conceptually different from the Expectation–Maximization (EM) algorithm and that does not in general yield the observed data maximum likelihood estimate (MLE). The proposed approach is based conceptually upon weighting the set of possible complete-data MLEs. Its implementation avoids the expectation step of EM, which can sometimes be problematic. In the simple case of Bernoulli trials missing completely at random, the iterations of the proposed algorithm are equivalent to the EM iterations. For a familiar genetics-oriented multinomial problem with missing count data and for the motivating example with epidemiologic applications that involves a mixture of a left censored normal distribution with a point mass at zero, we investigate the finite sample performance of the proposed estimator and find it to be competitive with that of the MLE. We give some intuitive justification for the method, and we explore an interesting connection between our algorithm and multiple imputation in order to suggest an approach for estimating standard errors.     

Maximum likelihood estimation for the proportional hazards model with partly interval-censored data

Summary. The maximum likelihood estimator (MLE) for the proportional hazards model with partly interval-censored data is studied. Under appropriate regularity conditions, the MLEs of the regression parameter and the cumulative hazard function are shown to be consistent and asymptotically normal. Two methods to estimate the variance–covariance matrix of the MLE of the regression parameter are considered, based on a generalized missing information principle and on a generalized profile information procedure. Simulation studies show that both methods work well in terms of the bias and variance for samples of moderate size. An example illustrates the methods.     

Exact confidence intervals for the shape parameter of the gamma distribution

In this paper exact confidence intervals (CIs) for the shape parameter of the gamma distribution are constructed using the method of Bølviken and Skovlund [Confidence intervals from Monte Carlo tests. J Amer Statist Assoc. 1996;91:1071–1078]. The CIs which are based on the maximum likelihood estimator or the moment estimator are compared to bootstrap CIs via a simulation study.     

Retrospective change detection in categorical time series

The aim of this paper is to propose methods of detecting change in the coefficients of a multinomial logistic regression model for categorical time series offline. The alternatives to the null hypothesis of stationarity can be either the hypothesis that it is not true, or that there is a temporary change in the sequence. We use the efficient score vector of the partial likelihood function. This has several advantages. First, the alternative value of the parameter does not have to be estimated; hence, we have a procedure that has a simple structure with only one parameter estimation using all available observations. This is in contrast with the generalized likelihood ratio-based change point tests. The efficient score vector is used in various ways. As a vector, its components correspond to the different components of the multinomial logistic regression model’s parameter vector. Using its quadratic form a test can be defined, where the presence of a change in any or all parameters is tested for. If there are too many parameters one can test for any subset while treating the rest as nuisance parameters. Our motivating example is a DNA sequence of four categories, and our test result shows that in the published data the distribution of the four categories is not stationary.     

A gradient-based algorithm for semiparametric models with missing covariates

In the parametric regression model, the covariate missing problem under missing at random is considered. It is often desirable to use flexible parametric or semiparametric models for the covariate distribution, which can reduce a potential misspecification problem. Recently, a completely nonparametric approach was developed by [H.Y. Chen, Nonparametric and semiparametric models for missing covariates in parameter regression, J. Amer. Statist. Assoc. 99 (2004), pp. 1176–1189; Z. Zhang and H.E. Rockette, On maximum likelihood estimation in parametric regression with missing covariates, J. Statist. Plann. Inference 47 (2005), pp. 206–223]. Although it does not require a model for the covariate distribution or the missing data mechanism, the proposed method assumes that the covariate distribution is supported only by observed values. Consequently, their estimator is a restricted maximum likelihood estimator (MLE) rather than the global MLE. In this article, we show the restricted semiparametric MLE could be very misleading in some cases. We discuss why this problem occurs and suggest an algorithm to obtain the global MLE. Then, we assess the performance of the proposed method via some simulation experiments.