Statistical Inference Based on Pooled Data: A Moment-Based Estimating Equation Approach

We consider statistical inference on parameters of a distribution when only pooled data are observed. A moment-based estimating equation approach is proposed to deal with situations where likelihood functions based on pooled data are difficult to work with. We outline the method to obtain estimates and test statistics of the parameters of interest in the general setting. We demonstrate the approach on the family of distributions generated by the Box-Cox transformation model, and, in the process, construct tests for goodness of fit based on the pooled data.     

Bayesian analysis for partially complete time and type of failure data

In this paper, we consider the Bayesian analysis of competing risks data, when the data are partially complete in both time and type of failures. It is assumed that the latent cause of failures have independent Weibull distributions with the common shape parameter, but different scale parameters. When the shape parameter is known, it is assumed that the scale parameters have Beta–Gamma priors. In this case, the Bayes estimates and the associated credible intervals can be obtained in explicit forms. When the shape parameter is also unknown, it is assumed that it has a very flexible log-concave prior density functions. When the common shape parameter is unknown, the Bayes estimates of the unknown parameters and the associated credible intervals cannot be obtained in explicit forms. We propose to use Markov Chain Monte Carlo sampling technique to compute Bayes estimates and also to compute associated credible intervals. We further consider the case when the covariates are also present. The analysis of two competing risks data sets, one with covariates and the other without covariates, have been performed for illustrative purposes. It is observed that the proposed model is very flexible, and the method is very easy to implement in practice.     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

Estimation of exponential regression parameters using binary data

Exponential regression model is important in analyzing data from heterogeneous populations. In this paper we propose a simple method to estimate the regression parameters using binary data. Under certain design distributions, including ellipticaily symmetric distributions, for the explanatory variables, the estimators are shown to be consistent and asymptotically normal when sample size is large. For finite samples, the new estimates were shown to behave reasonably well. They are competitive with the maximum likelihood estimates and more importantly, according to our simulation results, the cost of CPU time for computing new estimates is only 1/7 of that required for computing the usual maximum likelihood estimates. We expect the savings in CPU time would be more dramatic with larger dimension of the regression parameter space.     

Matched case–control data analyses with missing covariates

We consider methods for analysing matched case–control data when some covariates ( W ) are completely observed but other covariates ( X ) are missing for some subjects. In matched case–control studies, the complete-record analysis discards completely observed subjects if none of their matching cases or controls are completely observed. We investigate an imputation estimate obtained by solving a joint estimating equation for log-odds ratios of disease and parameters in an imputation model. Imputation estimates for coefficients of W are shown to have smaller bias and mean-square error than do estimates from the complete-record analysis.     

A Monte Carlo comparison of GMM and QMLE estimators for short dynamic panel data models with spatial errors

We suggest a generalized spatial system GMM (SGMM) estimation for short dynamic panel data models with spatial errors and fixed effects when n is large and T is fixed (usually small). Monte Carlo studies are conducted to evaluate the finite sample properties with the quasi-maximum likelihood estimation (QMLE). The results show that, QMLE, with a proper approximation for initial observation, performs better than SGMM in general cases. However, it performs poorly when spatial dependence is large. QMLE and SGMM perform better for different parameters when there is unknown heteroscedasticity in the disturbances and the data are highly persistent. Both estimates are not sensitive to the treatment of initial values. Estimation of the spatial autoregressive parameter is generally biased when either the data are highly persistent or spatial dependence is large. Choices of spatial weights matrices and the sign of spatial dependence do affect the performance of the estimates, especially in the case of the heteroscedastic disturbance. We also give empirical guidelines for the model.     

Statistical inference of Marshall-Olkin bivariate Weibull distribution with three shocks based on progressive interval censored data

There are several failure modes may cause system failed in reliability and survival analysis. It is usually assumed that the causes of failure modes are independent each other, though this assumption does not always hold. Dependent competing risks modes from Marshall-Olkin bivariate Weibull distribution under Type-I progressive interval censoring scheme are considered in this paper. We derive the maximum likelihood function, the maximum likelihood estimates, the 95% Bootstrap confidence intervals and the 95% coverage percentages of the parameters when shape parameter is known, and EM algorithm is applied when shape parameter is unknown. The Monte-Carlo simulation is given to illustrate the theoretical analysis and the effects of parameters estimates under different sample sizes. Finally, a data set has been analyzed for illustrative purposes.     

Estimation and prediction for an inverted exponentiated Rayleigh distribution under hybrid censoring

In this paper we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution when it is known that data are hybrid Type I censored. The maximum likelihood and Bayes estimates are derived. In sequel interval estimates are also constructed. We further consider one- and two-sample prediction of future observations and also obtain prediction intervals. The performance of proposed methods of estimation and prediction is studied using simulations and an illustrative example is discussed in support of the suggested methods.     

Bayesian inference of the inverse Weibull mixture distribution using type-I censoring

A large number of models have been derived from the two-parameter Weibull distribution including the inverse Weibull (IW) model which is found suitable for modeling the complex failure data set. In this paper, we present the Bayesian inference for the mixture of two IW models. For this purpose, the Bayes estimates of the parameters of the mixture model along with their posterior risks using informative as well as the non-informative prior are obtained. These estimates have been attained considering two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the former case, Bayes estimates are obtained under three loss functions while for the latter case only the squared error loss function is used. Simulation study is carried out in order to explore numerical aspects of the proposed Bayes estimators. A real-life data set is also presented for both cases, and parameters obtained under case when shape parameter is known are tested through testing of hypothesis procedure.     

On estimating parameters of a progressively censored lognormal distribution

We consider the problem of making statistical inference on unknown parameters of a lognormal distribution under the assumption that samples are progressively censored. The maximum likelihood estimates (MLEs) are obtained by using the expectation-maximization algorithm. The observed and expected Fisher information matrices are provided as well. Approximate MLEs of unknown parameters are also obtained. Bayes and generalized estimates are derived under squared error loss function. We compute these estimates using Lindley's method as well as importance sampling method. Highest posterior density interval and asymptotic interval estimates are constructed for unknown parameters. A simulation study is conducted to compare proposed estimates. Further, a data set is analysed for illustrative purposes. Finally, optimal progressive censoring plans are discussed under different optimality criteria and results are presented.     

Optimal sample size allocation for accelerated life test under progressive type-II censoring with competing risks

In this article, we study the optimization problem of sample size allocation when the competing risks data are from a progressive type-II censoring in a constant-stress accelerated life test with multiple levels. The failure times of the individual causes are assumed to be statistically independent and exponentially distributed with different parameters. We obtain the estimates of the unknown parameters through a maximum likelihood method, and also derive the Fisher information matrix. We propose three optimization criteria and two search scenarios to obtain the sample size allocation at each stress level. Some numerical results are studied to illustrate the usage of the proposed methods.     

On discrimination and classification with multivariate repeated measures data

We study the problem of classification with multiple q-variate observations with and without time effect on each individual. We develop new classification rules for populations with certain structured and unstructured mean vectors and under certain covariance structures. The new classification rules are effective when the number of observations is not large enough to estimate the variance–covariance matrix. Computational schemes for maximum likelihood estimates of required population parameters are given. We apply our findings to two real data sets as well as to a simulated data set.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

A Profile Conditional Likelihood Approach for the Semiparametric Transformation Regression Model with Missing Covariates

We propose a profile conditional likelihood approach to handle missing covariates in the general semiparametric transformation regression model. The method estimates the marginal survival function by the Kaplan-Meier estimator, and then estimates the parameters of the survival model and the covariate distribution from a conditional likelihood, substituting the Kaplan-Meier estimator for the marginal survival function in the conditional likelihood. This method is simpler than full maximum likelihood approaches, and yields consistent and asymptotically normally distributed estimator of the regression parameter when censoring is independent of the covariates. The estimator demonstrates very high relative efficiency in simulations. When compared with complete-case analysis, the proposed estimator can be more efficient when the missing data are missing completely at random and can correct bias when the missing data are missing at random. The potential application of the proposed method to the generalized probit model with missing continuous covariates is also outlined.     

Bayesian inference and prediction of order statistics for a Type-II censored Weibull distribution

This paper describes the Bayesian inference and prediction of the two-parameter Weibull distribution when the data are Type-II censored data. The aim of this paper is twofold. First we consider the Bayesian inference of the unknown parameters under different loss functions. The Bayes estimates cannot be obtained in closed form. We use Gibbs sampling procedure to draw Markov Chain Monte Carlo (MCMC) samples and it has been used to compute the Bayes estimates and also to construct symmetric credible intervals. Further we consider the Bayes prediction of the future order statistics based on the observed sample. We consider the posterior predictive density of the future observations and also construct a predictive interval with a given coverage probability. Monte Carlo simulations are performed to compare different methods and one data analysis is performed for illustration purposes.     

Robust inference in generalized linear models for longitudinal data

The author develops a robust quasi‐likelihood method, which appears to be useful for down‐weighting any influential data points when estimating the model parameters. He illustrates the computational issues of the method in an example. He uses simulations to study the behaviour of the robust estimates when data are contaminated with outliers, and he compares these estimates to those obtained by the ordinary quasi‐likelihood method.     

Estimation for u-shaped beta distributions: minimum hellinger distance and related methods

We compare minimum Hellinger distance and minimum Heiiinger disparity estimates for U-shaped beta distributions. Given suitable density estimates, both methods are known to be asymptotically efficient when the data come from the assumed model family, and robust to small perturbations from the model family. Most implementations use kernel density estimates, which may not be appropriate for U-shaped distributions. We compare fixed binwidth histograms, percentile mesh histograms, and averaged shifted histograms. Minimum disparity estimates are less sensitive to the choice of density estimate than are minimum distance estimates, and the percentile mesh histogram gives the best results for both minimum distance and minimum disparity estimates. Minimum distance estimates are biased and a bias-corrected method is proposed. Minimum disparity estimates and bias-corrected minimum distance estimates are comparable to maximum likelihood estimates when the model holds, and give better results than either method of moments or maximum likelihood when the data are discretized or contaminated, Although our re¬sults are for the beta density, the implementations are easily modified for other U-shaped distributions such as the Dirkhlet or normal generated distribution.     

Discriminant Analysis for the von Mises-Fisher Distribution

The von Mises-Fisher distribution is widely used for modeling directional data. In this article, we derive the discriminant rules based on this distribution to assign objects into pre-existing classes. We determine a distance between two von Mises-Fisher populations and we calculate estimates of the misclassification probabilities. We also analyze the behavior of the distance between two von Mises-Fisher populations and of the estimates of the misclassification probabilities when we modify the parameters of the populations or the samples size or the dimension of the sphere. Finally, we present an example with real spherical data available in the literature.     

Fiducial inference for Birnbaum-Saunders distribution

In this paper, we consider fiducial inference for the unknown parameters of the Birnbaum-Saunders distribution. Two generalized fiducial distributions of the parameters are obtained. One is based on the inverse of the structural equation, and the fiducial estimates of the parameters are obtained by a simulation method. The other is based on the method of [Hannig J. Generalized fiducial inference via discretization. Stat. Sinica. 2013;23:489–514], then we use adaptive rejection Metropolis sampling to get the fiducial estimates. We compare the fiducial estimates with the maximum likelihood estimates and Bayesian estimates by simulations. Two real data sets are analysed for illustration.     

Percentile-based Empirical Distribution Function Estimates for Performance Evaluation of Healthcare Providers

Hierarchical models are widely-used to characterize the performance of individual healthcare providers. However, little attention has been devoted to system-wide performance evaluations, the goals of which include identifying extreme (e.g., top 10%) provider performance and developing statistical benchmarks to define high-quality care. Obtaining optimal estimates of these quantities requires estimating the empirical distribution function (EDF) of provider-specific parameters that generate the dataset under consideration. However, the difficulty of obtaining uncertainty bounds for a square-error loss minimizing EDF estimate has hindered its use in system-wide performance evaluations. We therefore develop and study a percentile-based EDF estimate for univariate provider-specific parameters. We compute order statistics of samples drawn from the posterior distribution of provider-specific parameters to obtain relevant uncertainty assessments of an EDF estimate and its features, such as thresholds and percentiles. We apply our method to data from the Medicare End Stage Renal Disease (ESRD) Program, a health insurance program for people with irreversible kidney failure. We highlight the risk of misclassifying providers as exceptionally good or poor performers when uncertainty in statistical benchmark estimates is ignored. Given the high stakes of performance evaluations, statistical benchmarks should be accompanied by precision estimates.