Asymptotic Normality Of Maximum Likelihood Estimator Research Articles

Inference and optimal censoring scheme for a competing-risks model with type-II progressive censoring

ABSTRACT For statistical inference of competing risks under type-II progressive censoring, lifetimes are modeled by an inverted exponential Rayleigh distribution, which allows the use of a non-monotonic hazard function. Maximum-likelihood estimators for all parameters exist and are unique. The Newton-Raphson algorithm and maximum stochastic expectation each provide estimates. Confidence intervals result from the Fisher matrix and the asymptotic normality of maximum-likelihood estimators. For small samples, the Bootstrap estimators of the parameters do not need to be asymptotically normal. In addition, the Monte Carlo method with the Metropolis-Hastings algorithm and importance sampling allow for Bayesian estimation, with the associated highest posterior-density intervals. The Bayesian method takes account of prior information, contrary to the frequentist method. The Bootstrap method improves the precision of the estimation, especially in the case of small sample sizes. The estimated range obtained by Bootstrap is between 20% and 60% smaller than that obtained by maximum likelihood. Frequentist and Bayesian estimations using the inverted exponentiated Rayleigh distribution under type-II progressive censoring allow for fitting empirical mouse mortality data and obtaining parameter estimates of this distribution. A quantile-dependent criterion and a quantile-independent criterion are used to determine the optimal censoring and to design the experiment.

Bayesian and Non-Bayesian Inference for Weibull Inverted Exponential Model under Progressive First-Failure Censoring Data

In this article, the estimation of the parameters and the reliability and hazard functions for Weibull inverted exponential (WIE) distribution is considered based on progressive first-failure censoring (PFFC) data. For non-Bayesian inference, maximum likelihood (ML) estimators are acquired; meanwhile, their existence is verified. Via asymptotic normality of ML estimators and delta method, the corresponding confidence intervals (CIs) of the parameters and the reliability and hazard functions are constructed. For Bayesian inference, Lindley’s approximation and Markov chain Monte Carlo (MCMC) techniques are proposed to obain the Bayes estimators and the corresponding credible intervals (CRIs). To this end, both symmetric and asymmetric loss functions are used. A large number of Monte Carlo simulations are implemented to evaluate the efficiency of the developed methods. Eventually, a numerical example is analyzed for illustrative purposes.

On classical and Bayesian asymptotics in stochastic differential equations with random effects having mixture normal distributions

Delattre et al. (2013) considered a system of stochastic differential equations (SDEs) in a random effects setup. Under the independent and identical (iid) situation, and assuming normal distribution of the random effects, they established weak consistency of the maximum likelihood estimators (MLEs) of the population parameters of the random effects. In this article, respecting the increasing importance and versatility of normal mixtures and their ability to approximate any standard distribution, we consider the random effects having mixture of normal distributions and prove asymptotic results associated with the MLEs in both independent and identical (iid) and independent but not identical (non-iid) situations. Besides, we consider iid and non-iid setups under the Bayesian paradigm and establish posterior consistency and asymptotic normality of the posterior distribution of the population parameters, even when the number of mixture components is unknown and treated as a random variable. Although ours is an independent work, we later noted that Delattre et al. (2013) also assumed the SDE setup with normal mixture distribution of the random effect parameters but considered only the iid case and proved only weak consistency of the MLE under an extra, strong assumption as opposed to strong consistency that we are able to prove without the extra assumption. Furthermore, they did not deal with asymptotic normality of MLE or the Bayesian asymptotics counterpart which we investigate in details. Ample simulation experiments and application to a real, stock market data set reveal the importance and usefulness of our methods even for small samples.

Inference of accelerated dependent competing risks model for Marshall–Olkin bivariate Weibull distribution with nonconstant parameters

In this paper, we discuss the statistical inference of constant-stress accelerated dependent competing risks model under Type-II hybrid censoring schemes. The dependency structure is modeled by a Marshall–Olkin bivariate Weibull distribution. Both the shape and the scale parameters in the model are assumed to be dependent on the stress levels through a log-linear relationship. The maximum likelihood estimates (MLEs) of the model parameters are derived. Confidence intervals (CIs) of the model parameters are constructed based on the asymptotic normality of MLEs and the bias-corrected percentile bootstrap method. Bayes estimates with the squared error loss function and the highest posterior density (HPD) credible intervals (CIs) are obtained by using an importance sampling method. In addition, the estimates for the accelerated coefficients and the reliability under normal use stress level at mission time are derived. A Monte Carlo simulation study is used to evaluate the performance of the proposed statistical inference methods. A real data example is used to illustrate the methodologies proposed in this paper and to compare the proposed model with the copula model.

Exact Inference on Weibull Parameters With Multiply Type-I Censored Data

In reliability engineering, the Weibull distribution and censoring are widely employed. The multiple Type-I censoring is the general form of Type-I censoring and represents that all the test units are terminated at different times. In this paper, the exact inference on Weibull parameters under multiple Type-I censoring is discussed, as the existing studies are limited. The least-square estimates (LSE) and maximum likelihood estimates (MLE) of Weibull parameters are presented. For the problem that the MLE does not always exist and has no closed-form, the approximate maximum likelihood estimate (AMLE) is proposed. Concerning the confidence interval (CI) for Weibull parameters, the pivotal quantity based on the LSE is developed to construct it. Another CI is obtained using the asymptotic normality of MLE. By the missing information principle, the observed Fisher information matrix is derived to replace the commonly approximate one through evaluating the second order derivatives of log-likelihood function with the MLE. As a comparison, the CI via the bootstrap method is provided. Monte Carlo simulation demonstrates that the proposed AMLE is convenient and superior to the MLE. And the proposed CI based on pivotal quantity outperforms others. Besides, the derived observed Fisher information matrix is also better. Finally, a real example is analyzed to illustrate the application of proposed methods.

Point and interval estimation under progressive type-I interval censoring with random removal

This work considers point and interval estimation based on data from a life test under progressive type-I interval censoring with random removal. The asymptotic properties of the maximum likelihood estimators (MLEs) are established under appropriate regularity conditions. Asymptotic confidence intervals and $$\beta $$-content $$\gamma $$-level tolerance interval are obtained by using the asymptotic normality of MLEs. A simulation study is undertaken to assess the performance of the MLEs, confidence intervals and tolerance interval. Lastly, the minimum sample size required to achieve a desired $$\beta $$-content $$\gamma $$-level tolerance interval is determined.

Statistical Analysis in Constant-stress Accelerated Life Tests for Generalized Exponential Distribution with Progressive Type-I Hybrid Censoring

Based on progressive Type-I hybrid censored data, statistical analysis in constant-stress accelerated life test (CS-ALT) for generalized exponential (GE) distribution is discussed. The maximum likelihood estimates (MLEs) of the parameters and the reliability function are obtained with EM algorithm, as well as the observed Fisher information matrix, the asymptotic variance-covariance matrix of the MLEs, and the asymptotic unbiased estimate (AUE) of the scale parameter. Confidence intervals (CIs) for the parameters are derived using asymptotic normality of MLEs and percentile bootstrap (Boot-p) method. Finally, the point estimates and interval estimates of the parameters are compared separately through the Monte-Carlo method.

The Two-Piece Normal-Laplace Distribution

This article considers the two-piece normal-Laplace (TPNL) distribution, a split skew distribution consisting of a normal part, and a Laplace part. The distribution is indexed by three parameters, representing location, scale, and shape. As illustrated with several examples, the TPNL family of distributions provides a useful alternative to other families of asymmetric distributions on the real line. However, because the likelihood function is not well behaved, standard theory of maximum-likelihood (ML) estimation does not apply to the TPNL family. In particular, the likelihood function can have multiple local maxima. We provide a procedure for computing ML estimators, and prove consistency and asymptotic normality of ML estimators, using non standard methods.

Step-Down Analysis for Changes in the Covariance Matrix and Other Parameters

Step-down diagnostic analysis to infer which means have changed (after detection of a shift in the mean vector) is familiar in multivariate statistical process control. We show how to include in the diagnostic analysis the elements of the covariance matrix and other higher order moments and give practical examples. We use the asymptotic normality of maximum-likelihood estimators to classify the parameters that describe the prior- and post-shift distributions into those for which there is significant evidence of a shift and those for which such evidence is lacking. This diagnostic information can be useful in guiding the search for the cause(s) of the shift. For practical applications, the choice of parameterization may affect the ease of interpretation of the diagnostic results. It may be more informative to use standard deviations and correlation coefficients instead of variances and covariances, and this approach is illustrated in our analysis and examples. We also consider application to distributions other than the multivariate normal and give an example with a multivariate gamma distribution. Although illustrated with independent and identically distributed observations, the proposed methodology is more general.

Inference for Log-Gamma Distribution Based on Progressively Type-II Censored Data

We discuss the maximum likelihood estimates (MLEs) of the parameters of the log-gamma distribution based on progressively Type-II censored samples. We use the profile likelihood approach to tackle the problem of the estimation of the shape parameter κ. We derive approximate maximum likelihood estimators of the parameters μ and σ and use them as initial values in the determination of the MLEs through the Newton–Raphson method. Next, we discuss the EM algorithm and propose a modified EM algorithm for the determination of the MLEs. A simulation study is conducted to evaluate the bias and mean square error of these estimators and examine their behavior as the progressive censoring scheme and the shape parameter vary. We also discuss the interval estimation of the parameters μ and σ and show that the intervals based on the asymptotic normality of MLEs have very poor probability coverages for small values of m. Finally, we present two examples to illustrate all the methods of inference discussed in this paper.

Consistency and asymptotic normality of MLE of parameter vector in a randomly censored GLM with incomplete information

The generalized linear model (GLM) based on the observed data with incomplete information in the case of random censorship is defined. Under the given conditions, the existence and uniqueness of the solution on the likelihood equations with respect to the parameter vector β of the model are discussed, and the consistency and asymptotic normality of the maximum likelihood estimator (MLE)\(\hat \beta _n \) are proved.

Point and Interval Estimation for Bivariate Normal Distribution Based on Progressively Type-II Censored Data

ABSTRACT The maximum likelihood estimates (MLEs) of parameters of a bivariate normal distribution are derived based on progressively Type-II censored data. The asymptotic variances and covariances of the MLEs are derived from the Fisher information matrix. Using the asymptotic normality of MLEs and the asymptotic variances and covariances derived from the Fisher information matrix, interval estimation of the parameters is discussed and the probability coverages of the 90% and 95% confidence intervals for all the parameters are then evaluated by means of Monte Carlo simulations. To improve the probability coverages of the confidence intervals, especially for the correlation coefficient, sample-based Monte Carlo percentage points are determined and the probability coverages of the 90% and 95% confidence intervals obtained using these percentage points are evaluated and shown to be quite satisfactory. Finally, an illustrative example is presented.

Reliability analysis of accelerated life-test data from a repairable system

This paper analyzes accelerating testing of a repairable item modeled by a nonhomogeneous Poisson process with covariates. We extensively analyze a single accelerating variable with two stress levels, and derive closed-form maximum likelihood (ML) solutions. These closed-form solutions provide: (1) an easier way to obtain point estimates of the unknown parameters under usual operating conditions, and (2) a way to obtain confidence intervals on the process parameters and function thereof which are more accurate than those based on asymptotic normality of ML estimates. We analyze the circumstances in which a drawback to closed-form estimation arises, and guides the extent that our procedures may equally apply. An example application drawn from a real situation of accelerated testing is presented, and numerical estimates based on our procedures are derived and discussed. Theoretical and simulation results show that estimation procedures based on the power-law process and regression methods can be a flexible, useful tool for reliability analysis of a repairable item. >

A model-based approach for estimation of two-dimensional maximum entropy power spectra

A stochastic model-based approach is presented for estimation of the two-dimensional maximum entropy power spectrum (MEPS) from given finite uniform array data. The method consists of fitting an appropriate two-dimensional noncausal Gaussian-Markov random field (GMRF) model to the given data using the maximum likelihood (ML) technique for parameter estimation. The nonlinear criterion function used for ML estimation is similar in structure to the function arising in the deterministic approach of Lang and McClellan. The model-based approach provides new insights into the two-dimensional MEPS estimation problem. For example, using the asymptotic normality of ML estimates, we derive simultaneous confidence bands for the estimated MEPS. It turns out that when the true correlations are generated by a noncausal GMRF model, the two-dimensional MEPS can be obtained by solving linear equations. This approach also suggests techniques for realizing two-dimensional GMRF models from the given correlation data. Several numerical examples are given to illustrate the usefulness of the approach.

Consistency and Asymptotic Normality of MLE's for Exponential Models

Conditions are given for the strong consistency and asymptotic normality of the MLE (maximum likelihood estimator) for multiparameter exponential models. Because of the special structure assumed, the conditions are less restrictive than required by general theorems in this area. The technique involves certain convex functions on Euclidean spaces that arise naturally in the present context. Some examples are considered; among them, the multinomial distribution. Some convexity and continuity properties of multivariate cumulant generating functions are also discussed.