Observed Fisher Information Matrix Research Articles

Inference for dependent competing risks from bivariate Kumaraswamy distribution under generalized progressive hybrid censoring

In this paper, competing risks model is considered when causes of failure are dependent. When latent failure times are distributed by the Marshall-Olkin bivariate Kumaraswamy model, inference for the unknown model parameters is studied under a generalized progressive hybrid censoring. Maximum likelihood estimates of unknown parameters are established, and the associated existence and uniqueness are provided. The approximate confidence intervals are constructed via the observed Fisher information matrix. Moreover, Bayes estimates and the credible intervals of the unknown parameters are also presented based a flexible Gamma-Dirichlet prior, and the importance sampling method is used to compute associated estimates. Simulation study and a lifetime example are given for illustration purposes.

Inference on a progressive type I interval-censored truncated normal distribution

In this paper, we consider the problem of making statistical inference for a truncated normal distribution under progressive type I interval censoring. We obtain maximum likelihood estimators of unknown parameters using the expectation-maximization algorithm and in sequel, we also compute corresponding midpoint estimates of parameters. Estimation based on the probability plot method is also considered. Asymptotic confidence intervals of unknown parameters are constructed based on the observed Fisher information matrix. We obtain Bayes estimators of parameters with respect to informative and non-informative prior distributions under squared error and linex loss functions. We compute these estimates using the importance sampling procedure. The highest posterior density intervals of unknown parameters are constructed as well. We present a Monte Carlo simulation study to compare the performance of proposed point and interval estimators. Analysis of a real data set is also performed for illustration purposes. Finally, inspection times and optimal censoring plans based on the expected Fisher information matrix are discussed.

Inference for exponential competing risks data under generalized progressive hybrid censoring

In this paper, a competing risks model based on a generalized progressive hybrid censoring is considered. When the latent lifetime distributions of failure causes are exponential distributed and partially observed, maximum likelihood estimates for unknown parameters are established and the associated asymptotic confidence interval estimates are provided by using approximate theory via the observed Fisher information matrix. Moreover, Bayes point estimates and the highest posterior density credible intervals of unknown parameters are also considered, and the importance sampling procedure is used to approximate corresponding estimates. Finally, a real-life example and simulation study are presented for illustration.

Analysis of interval-censored competing risks data under missing causes

ABSTRACT In this article, interval-censored competing risks data are analyzed when some of the causes of failure are missing. The vertical modeling approach has been proposed here. This approach utilizes the data to extract information to the maximum possible extent especially when some causes of failure are missing. The maximum likelihood estimates of the model parameters are obtained. The asymptotic confidence intervals for the model parameters are constructed using approaches based on observed Fisher information matrix, and parametric bootstrap. A simulation study is considered in detail to assess the performance of the point and interval estimators. It is observed that the proposed analysis performs better than the complete case analysis. This establishes the fact that the our methodology is an extremely useful technique for interval-censored competing risks data when some of the causes of failure are missing. Such analysis seems to be quite useful for smaller sample sizes where complete case analysis may have a significant impact on the inferential procedures. Through Monte Carlo simulations, the effect of a possible model misspecification is also assessed on the basis of the cumulative incidence function. For illustration purposes, three datasets are analyzed and in all cases the conclusion appears to be quite realistic.

Parameter estimation of regression model with AR(p) error terms based on skew distributions with EM algorithm

In the linear regression model, the errors are usually assumed to be uncorrelated. However, in real-life data, this assumption is not often plausible. In this study, first, we will assume that the errors of the regression model have autoregressive structure. This type of regression models has been considered before. However, in those papers under this assumption usually, the symmetric distributions are used as error distribution. The main contribution of this work is to use skew distributions instead of symmetric distributions as error distribution in regression models with autoregressive errors. We provide expectation maximization algorithm to compute the maximum likelihood estimates for the parameters. The performances of the proposed estimators are demonstrated with a simulation study and a real data example. We also provide the confidence intervals using the observed Fisher information matrix for the corresponding estimators.

Inferences Under Step-Stress Partially Accelerated Life Tests for Linear Exponential Distribution Based on Progressive Type-II Censored Data

In this paper, estimation problem for the parameters of linear exponential distribution based on step-stress partially accelerated life test under progressive Type-II censoring schemes is considered. To obtain the estimation for the distribution parameters and acceleration factor the maximum likelihood and Bayesian methods are used for this purpose. In addition, observed Fisher information matrix to find the confidence intervals of the model parameters is derived. By using Markov chain Monte Carlo method, approximate Bayes estimates under loss functions are obtained. Finally, a simulation study and a numerical example for illustration are conducted.

Estimation issues in the Exponential–Logarithmic model under hybrid censoring

In this paper we develop both frequentist and Bayesian estimation methodologies for parameters of an Exponential–Logarithmic Distribution under Type-I hybrid censoring. In frequentist approach, it is observed that the Maximum Likelihood Estimators (MLEs) do not have closed form expressions. We use both the EM and SEM algorithms to compute the MLEs and using the missing information principle obtain the observed Fisher information matrix which is then used to construct the asymptotic confidence intervals. Further, two bootstrap interval estimates are proposed for the unknown parameters. Under squared error loss and LINEX loss functions, we obtain Bayes estimates of the unknown parameters assuming independent gamma and beta priors using the Lindley method, Tierney–Kadane method and the importance sampling procedure. The problem of prediction is also explored. A real life data set as well as simulated data have been analyzed for illustrative purposes.

Estimation and Prediction for a Progressively First-Failure Censored Inverted Exponentiated Rayleigh Distribution

We discuss inverted exponentiated Rayleigh distribution under progressive first-failure censoring. Maximum likelihood and Bayes estimates of unknown parameters are obtained. An expectation–maximization algorithm is used for computing maximum likelihood estimates. Asymptotic intervals are constructed from the observed Fisher information matrix. Bayes estimates of unknown parameters are obtained under the squared error loss function. We construct highest posterior density intervals based on importance sampling. Different predictors and prediction intervals of censored observations are discussed. A Monte Carlo simulations study is performed to compare different methods. Finally, three real data sets are analyzed for illustration purposes.

Inference for dependence competing risks from bivariate exponential model under generalized progressive hybrid censoring with partially observed failure causes

Inference are considered for the dependence competing risks model by using the Marshal-Olkin bivariate exponential distribution. Under generalized progressively hybrid censoring with partially observed failure causes, the maximum likelihood estimators are established, and the approximate confidence intervals are also constructed via the observed Fisher information matrix. Moreover, Bayes estimates and highest probability density credible intervals are presented and the importance sampling technique is used to compute corresponding results. Finally, the numerical analysis is proposed for illustration.

Inference for an Inverted Exponentiated Pareto Distribution Under Progressive Censoring

In this paper, estimation of unknown parameters of an inverted exponentiated Pareto distribution is considered under progressive Type-II censoring. Maximum likelihood estimates are obtained from the expectation–maximization algorithm. We also compute the observed Fisher information matrix. In the sequel, asymptotic and bootstrap-p intervals are constructed. Bayes estimates are derived using the importance sampling procedure with respect to symmetric and asymmetric loss functions. Highest posterior density intervals of unknown parameters are constructed as well. The problem of one- and two-sample prediction is discussed in Bayesian framework. Optimal plans are obtained with respect to two information measure criteria. We assess the behavior of suggested estimation and prediction methods using a simulation study. A real dataset is also analyzed for illustration purposes. Finally, we present some concluding remarks.

Inference for Weibull Competing Risks Data Under Generalized Progressive Hybrid Censoring

A competing risks model is considered under a generalized progressive hybrid censoring. When the latent failure times are Weibull distributed, maximum-likelihood estimates for the unknown model parameters are established where the associated existence and uniqueness are shown. An asymptotic distribution of the maximum-likelihood estimators is used to construct approximate confidence intervals via the observed fisher information matrix. Moreover, Bayes point estimates and the highest probability density credible intervals of unknown parameters are also presented, and the Gibbs sampling technique is used to approximate corresponding estimates. Simulation studies and real-life example are presented for illustration purpose.

Adaptive Radar Detectors Based on the Observed FIM

Modified versions of Rao, Wald, and Durbin tests are considered exploiting an estimator of the Fisher Information Matrix (FIM) in place of the exact one. They are asymptotically equivalent (under some technical conditions) to the standard counterparts and rely on the use of the Observed FIM (OFIM), which is proportional to the negative Hessian of the log-likelihood. The developed framework is applied to the problem of adaptive radar detection of a point-like target in homogeneous or partially-homogeneous interference. Remarkably, for both the scenarios, it is shown that Rao, Wald, and Durbin tests with OFIM are statistically equivalent to the Generalized Likelihood Ratio Test (GLRT) for the specific detection problem (namely Kelly's detector for the homogeneous environment and the Adaptive Coherence Estimator (ACE) [S. Kraut and L. L. Scharf, “The CFAR adaptive subspace detector is a scale-invariant GLRT,” IEEE Trans. Signal Process., vol. 47, no. 9, pp. 2538-2541, Sep. 1999.], also known as Adaptive Normalized Matched Filter (ANMF) [E. Conte, M. Lops, and G. Ricci, “Asymptotically optimum radar detection in compound-gaussian clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. 31, no. 2, pp. 617-625, Apr. 1995], for the partially-homogeneous scenario). This provides a new interpretation of the mentioned GLRTs laying the foundations for a better understanding of their theoretical validity.

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.

Inference of progressively censored competing risks data from Kumaraswamy distributions

A competing risks model based on Kumaraswamy distribution is discussed under progressive censoring. When the latent lifetime model of failure causes features different and common parameters, maximum likelihood estimates for unknown parameters are established where the existence and uniqueness of the estimates are provided, and the approximate confidence intervals are also constructed via the observed fisher information matrix. Moreover, Bayes estimates and associated highest posterior density credible intervals are also obtained based on Monte-Carlo Markov chain sampling methods. In addition, to test the equivalence of parameters between the competing risks, likelihood ratio test is also proposed. Finally, simulation studies and real-life example are presented for illustration purpose. • Inference for different and common competing risks parameters is considered. • Maximum likelihood estimators are established. • Confidence intervals are proposed via observed fisher information matrix. • Bayesian estimates are derived via MCMC sampling method. • Data example on football game proportion time is illustrated.

Weighted Weibull distribution: Bivariate and multivariate cases

Gupta and Kundu (Statistics 43 (2009) 621–643) introduced a new class of weighted exponential distribution and established its several properties. The probability density function of the proposed weighted exponential distribution is unimodal and it has an increasing hazard function. Following the same line Shahbaz, Shahbaz and Butt (Pak. J. Stat. Oper. Res. VI (2010) 53–59) introduced weighted Weibull distribution, and we derive several new properties of this weighted Weibull distribution. The main aim of this paper is to introduce bivariate and multivariate distributions with weighted Weibull marginals and establish their several properties. It is shown that the hazard function of the weighted Weibull distribution can have increasing, decreasing and inverted bathtub shapes. The proposed multivariate model has been obtained as a hidden truncation model similarly as the univariate weighted Weibull model. It is observed that to compute the maximum likelihood estimators of the unknown parameters for the proposed $p$-variate distribution, one needs to solve $(p+2)$ non-linear equations. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters. We obtain the observed Fisher information matrix, which can be used for constructing asymptotic confidence intervals. One data analysis has been performed for illustrative purposes, and it is observed that the proposed EM algorithm is very easy to implement, and the performance is quite satisfactory.

Inference for four-unit hybrid system with masked data under partially acceleration life test

ABSTRACTIn this paper, we investigate parameter inference for four-unit hybrid systems based on Type-II progressively hybrid censored and masked data from constant-stress partially accelerated life test. It is assumed that the unit lifetime in systems has exponential hazard rate. The maximum likelihood estimations (MLEs) for unknown parameters and acceleration factor are obtained. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLEs are derived. Approximate confidence intervals (CIs) and studentized-t bootstrap CIs for the parameters are presented based on normal approximation to the asymptotic distribution of MLEs and bootstrap method, respectively. A Monte Carlo simulation study is carried out to investigate the performance of the MLEs and to compare the effects of two different CIs in terms of interval length and coverage percentage for the parameters.

Classical and Bayesian estimation in log-logistic distribution under random censoring

This article deals with the classical and Bayesian estimation of the parameters of log-logistic distribution using random censorship model. The maximum likelihood estimators and the asymptotic confidence intervals based on observed Fisher information matrix of the parameters are derived. Bayes estimators of the parameters under generalized entropy loss function using independent gamma priors are obtained. For Bayesian computation, Tierney–Kadane’s approximation and Markov chain Monte Carlo (MCMC) methods are used. Also, the highest posterior credible intervals of the parameters based on MCMC method are constructed. A Monte Carlo simulation study is carried out to compare the behavior of various estimators developed in this article. Finally, a real data analysis is performed for illustration purposes.

Asymptotic and finite-sample properties of estimators based on stochastic gradients

Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires careful tuning of key parameters. Here, we introduce implicit stochastic gradient descent procedures, which involve parameter updates that are implicitly defined. Intuitively, implicit updates shrink standard stochastic gradient descent updates. The amount of shrinkage depends on the observed Fisher information matrix, which does not need to be explicitly computed; thus, implicit procedures increase stability without increasing the computational burden. Our theoretical analysis provides the first full characterization of the asymptotic behavior of both standard and implicit stochastic gradient descent-based estimators, including finite-sample error bounds. Importantly, analytical expressions for the variances of these stochastic gradient-based estimators reveal their exact loss of efficiency. We also develop new algorithms to compute implicit stochastic gradient descent-based estimators for generalized linear models, Cox proportional hazards, M-estimators, in practice, and perform extensive experiments. Our results suggest that implicit stochastic gradient descent procedures are poised to become a workhorse for approximate inference from large data sets.

Parameter Estimation for a Burr XII Distribution under Progressive Censoring

SYNOPTIC ABSTRACTThis article deals with the problem of estimating unknown parameters of a Burr Type XII distribution with the data that are progressive Type-II censored. The maximum likelihood estimators are derived using an EM algorithm. Approximate confidence intervals based on the observed Fisher information matrix and bootstrap intervals of the unknown parameters are obtained. Bayes estimators are derived under different loss functions by making use of the Tierney and Kadane method and importance sampling procedure. Samples obtained from the importance sampling procedure are further used to construct the highest posterior density intervals of unknown parameters. A simulation study is conducted to study the performance of proposed estimators. Finally, a real life data and a simulated data are analyzed for illustration.

Estimation Methods for the Generalized Inverted Exponential Distribution Under Type II Progressively Hybrid Censoring with Application to Spreading of Micro-Drops Data

In this article, we consider the statistical inferences of the unknown parameters of a generalized inverted exponential distribution based on the Type II progressively hybrid censored sample. By applying the expectation–maximization (EM) algorithm, the maximum likelihood estimators are developed for estimating the unknown parameters. The observed Fisher information matrix is obtained using the missing information principle, and it can be used for constructing asymptotic confidence intervals. By applying the bootstrapping technique, the confidence intervals for the parameters are also derived. Bayesian estimates of the unknown parameters are obtained using the Lindley’s approximation. Monte Carlo simulations are implemented and observations are given. Finally, a real data set representing the spread factor of micro-drops is analyzed to illustrative purposes.