Observed Fisher Information Matrix Research Articles

Analysis of Burr Type-XII Distribution Under Step Stress Partially Accelerated Life Tests with Type-I and Adaptive Type-II Progressively Hybrid Censoring Schemes

In this paper, we investigate the maximum likelihood estimation of the unknown parameters of the Burr Type-XII distribution and the acceleration factor based on two different progressively hybrid censoring schemes, namely, Type-I progressive hybrid censoring scheme (T-I PHCS) proposed by Kundu and Joarder (Comput Stat Data Anal 50:2509–2528, 2006) and adaptive Type-II progressive hybrid censoring scheme (AT-II PHCS) introduced by Ng et al. (Nav Res Logist 56:687–698, 2009) under step-stress partially accelerated life test model. The observed Fisher information matrix is obtained to construct an approximate confidence interval for the unknown parameters. The performances of the estimators of the model parameters using the above mentioned progressively hybrid censoring schemes are evaluated and compared in terms of the mean squared errors and relative errors through a Monte Carlo simulation study.

Robust background normalization method for one-channel microarrays

Abstract Background Microarray technology, aims to measure the amount of changes in transcripted messages for each gene by RNA via quantifying the colour intensity on the arrays. But due to the different experimental conditions, these measurements can include both systematic and random erroneous signals. For this reason, we present a novel gene expression index, called multi-RGX (Multiple-probe Robust Gene Expression Index) for one-channel microarrays. Methods Multi-RGX, different from other gene expression indices, considers the long-tailed symmetric (LTS) density, covering a wider range of distributions for modelling gene expressions on the log-scale, resulting in robust inference and it takes into account both probe and gene specific intensities. Furthermore, we derive the covariance-variance matrix of model parameters from the observed Fisher information matrix and test the performance of the multi-RGX method in three different datasets. Results Our method is found to be a promising method regarding its alternatives in terms of accuracy and computational time. Conclusion Multi-RGX gives accurate results with respect to its alternatives, with a reduction in computational cost.

A NOTE ON FERGUSSON AND PLATEN: “APPLICATION OF MAXIMUM LIKELIHOOD ESTIMATION TO STOCHASTIC SHORT RATE MODELS”

In a very recent and interesting paper, Fergusson and Platen (2015) investigate the applicability of the maximum likelihood (ML) method for estimating the parameters of some of the most popular stochastic models for the short interest rate. One of the main results of this paper is the analytical expression of the so-called observed Fisher information matrix for the Vasicek model at the ML point. However, in such a matrix some entries are not derived correctly and one entry is left unspecified. In the following, we provide the correct analytical expression of that matrix.

Estimation and prediction for a Burr type-III distribution with progressive censoring

ABSTRACTWe consider estimation of unknown parameters and reliability characteristics of a Burr type-III distribution under progressive censoring. Predictive estimates for censored observations and the associated prediction intervals are also obtained. We derive maximum-likelihood estimators of unknown quantities using the EM algorithm and then also obtain the observed Fisher information matrix. We provide various Bayes estimators for unknown parameters under the squared error loss function. Highest posterior density and asymptotic intervals are also constructed. We evaluate performance of proposed methods using simulations. Finally, an illustrative example is presented in support of the methods discussed.

A new distribution function with bounded support: the reflected generalized Topp-Leone power series distribution

In this paper we introduce a new flexible class of distributions with bounded support, called reflected Generalized Topp-Leone Power Series (rGTL-PS), obtained by compounding the reflected Generalized Topp-Leone (van Drop and Kotz, 2006) and the family of Power Series distributions. The proposed class includes, as special cases, some new distributions with limited support such as the rGTL-Logarithmic, the rGTL-Geometric, the rGTL-Poisson and rGTL-Binomial. This work is an attempt to partially fill a gap regarding the presence, in the literature, of continuous distributions with bounded support, which instead appear to be very useful in many real contexts, included the reliability. Some properties of the class, including moments, hazard rate and quantile are investigated. Moreover, the maximum likelihood estimators of the parameters are examined and the observed Fisher information matrix provided. Finally, in order to show the usefulness of the new class, some applications to real data are reported.

Estimation and prediction for Chen distribution with bathtub shape under progressive censoring

ABSTRACTWe consider estimation of the unknown parameters of Chen distribution [Chen Z. A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statist Probab Lett. 2000;49:155–161] with bathtub shape using progressive-censored samples. We obtain maximum likelihood estimates by making use of an expectation–maximization algorithm. Different Bayes estimates are derived under squared error and balanced squared error loss functions. It is observed that the associated posterior distribution appears in an intractable form. So we have used an approximation method to compute these estimates. A Metropolis–Hasting algorithm is also proposed and some more approximate Bayes estimates are obtained. Asymptotic confidence interval is constructed using observed Fisher information matrix. Bootstrap intervals are proposed as well. Sample generated from MH algorithm are further used in the construction of HPD intervals. Finally, we have obtained prediction intervals and estimates for future observations in one- and two-sample situations. A numerical study is conducted to compare the performance of proposed methods using simulations. Finally, we analyse real data sets for illustration purposes.

Statistical inference of for Weibull distribution under Type-II progressively hybrid censored data

ABSTRACTType-II progressively hybrid censoring is a mixture of Type-II progressive and hybrid censoring schemes. In this paper, we consider the estimation of when X and Y are two independent Weibull distributions with different scale parameters, but having the same shape parameter, and the data obtained from both distributions are Type-II progressively hybrid censored. Different methods for estimating are applied. The maximum likelihood estimator and the approximate maximum likelihood estimator of are proposed. The observed Fisher information matrix is computed and it is used to construct an asymptotic confidence interval for . Also, two bootstrap confidence intervals are proposed. Bayes estimate of and the associated credible interval are also obtained under the assumption of independent suitable priors. Monte Carlo simulations are performed to compare the efficiency of the proposed estimators. One data analysis has been presented for illustrative purposes.

Reflected Generalized Beta Inverse Weibull Distribution: definition and properties

In this paper we study a broad class of distribution functions which is defined by means of reflected generalized beta distribution. This class includes that of Beta-generated distribution as a special case. In particular, we use this class to extend the Inverse Weibull distribution in order to obtain the Reflected Generalized Beta Inverse Weibull Distribution.For this new distribution, moments, entropy, order statistics and a reliability measure are derived. The link between the Inverse Weibull and the Dagum distribution is generalized. Then the maximum likelihood estimators of the parameters are examined and the observed Fisher information matrix provided. Finally, the usefulness of the model is illustrated by means of an application to real data.

Delta method and bootstrap in linear mixed models to estimate a proportion when no event is observed: application to intralesional resection in bone tumor surgery.

Resecting bone tumors requires good cutting accuracy to reduce the occurrence of local recurrence. This issue is considerably reduced with a navigated technology. The estimation of extreme proportions is challenging especially with small or moderate sample sizes. When no success is observed, the commonly used binomial proportion confidence interval is not suitable while the rule of three provides a simple solution. Unfortunately, these approaches are unable to differentiate between different unobserved events. Different delta methods and bootstrap procedures are compared in univariate and linear mixed models with simulations and real data by assuming the normality. The delta method on the z-score and parametric bootstrap provide similar results but the delta method requires the estimation of the covariance matrix of the estimates. In mixed models, the observed Fisher information matrix with unbounded variance components should be preferred. The parametric bootstrap, easier to apply, outperforms the delta method for larger sample sizes but it may be time costly. Copyright © 2016 John Wiley & Sons, Ltd.

A bivariate Birnbaum–Saunders regression model

In this work, we propose a bivariate Birnbaum–Saunders regression model through the use of bivariate Sinh-normal distribution. The proposed regression model has its marginal as the Birnbaum–Saunders regression model of Rieck and Nedelman (1991), which has been discussed extensively by various authors with natural applications in survival and reliability studies. This bivariate regression model can be used to analyze correlated log-lifetimes of two units, in which the dependence structure between observations arises from the bivariate normal distribution.The main aim of this paper is to propose a bivariate Birnbaum–Saunders regression model and discuss some of its properties. Specifically, we have developed the moment estimation, the maximum likelihood estimation and the observed Fisher information matrix. Hypothesis testing is also performed by the use of the asymptotic normality of the maximum-likelihood estimators. Finally, the results of simulation studies as well as an application to a real data set are presented to illustrate the model and all the inferential methods developed here.

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.

Accelerated Degradation Analysis for the Quality of a System Based on the Gamma Process

As most systems these days are highly reliable with long lifetimes, failures of systems become rare; consequently, traditional failure time analysis may not be able to provide a precise assessment of the system reliability. In this regard, a degradation measure, as a percentage of the initial value, is an alternate way of describing the system health. This paper presents accelerated degradation analysis that characterizes the health and quality of systems with monotonic and bounded degradation. The maximum likelihood estimates (MLEs) of the model parameters are derived, based on a gamma process, time-scale transformation, and a power link function for associating the covariates. Then, methods of estimating the reliability, the mean and median lifetime, the conditional reliability, and the remaining useful life of systems under normal use conditions are all described. Moreover, approximate confidence intervals for the parameters of interest are developed based on the observed Fisher information matrix. A model validation metric with exact power is introduced. A Monte Carlo simulation study is carried out for evaluating the performance of the proposed methods. For an illustration of the proposed model, and the methods of inference developed here, a numerical example involving light intensity of light emitting diodes (LED) is analyzed.

Graphical Representation of Optimal Time for a Step-Stress Accelerated Life Test Design Using Frechet Distribution

The article provides an approach of getting optimal time through graph for Simple step stress accelerated test of inverse weibull distribution. In this we estimate parameters using log linear relationship by maximum likelihood method. Along with this, asymptotic variance and covariance matrix of the estimators are given. Comparison between expected and observed Fisher Information matrix is also shown. Furthermore, confidence interval coverage of the estimators is also presented for checking the precession of estimator. This approach is illustrated with an example using software.

Analysis of dependent competing risks in the presence of progressive hybrid censoring using Marshall–Olkin bivariate Weibull distribution

The lifetime of subjects in reliability and survival analysis in the presence of several causes of failure (i.e., competing risks) has attracted attention in the literature. Most studies have simplified the computations by assuming that the causes are independent, though this does not hold. Dependent competing risks under progressively hybrid censoring condition using a Marshall–Olkin bivariate Weibull distribution is investigated. Maximum likelihood and approximated maximum likelihood estimators are developed for estimating the unknown parameters. Asymptotic distributions of the maximum likelihood estimators are used to construct approximate confidence intervals using the observed Fisher information matrix. Based on a simulation and real applications, it is illustrated that when a parametric distributional assumption is nearly true, a close approximation could be achieved by deliberately censoring the number of subjects and the study duration using Type-II progressively hybrid censoring, which might help to save time and money in research studies.

Parameter Estimation for Lomax Distribution under Type II Censoring

In this paper, The problem of estimating unknown paramaters of Lomax distribution is considered under the assumption that samples are type-II censoring. The maximum likelihood estimates are developed for unknown paramaters using EM algorithm and NR method. We obtain the observed Fisher information matrix using the missing information principle . A numerical study is performed to compare the proposed estimate.

Inference about the Burr Type III Distribution under Type-II Hybrid Censored Data

This paper presents the statistical inference on the parameters of the Burr type III distribution, when the data are Type-II hybrid censored. The maximum likelihood estimators are developed for the unknown parameters using the EM algorithm method. We provided the observed Fisher information matrix using the missing information principle which is useful for constructing the asymptotic confidence intervals. The Bayesian estimates of the unknown parameters under the assumption of independent gamma priors are obtained using two approximations, namely Lindley's approximation and the Markov Chain Monte Carlo technique. Monte Carlo simulations are performed to observe the behavior of the proposed methods and a real dataset representing is used to illustrate the derived results.

Estimation of the bivariate generalized Lomax distribution parameters based on censored samples

Recently it is observed that the generalized exponential distribution can be used quite effectively to analyze lifetime data in one dimension. This paper extends Marshall and Olkin's bivariate exponential model to the Generalized Bivariate Lomax (BGL) Distribution. The cumulative distribution function, the probability density function and the Marginal distribution of the BGL distribution are reached. The maximum likelihood estimation procedure is derived for the estimation of the BGL parameters based on Censored Samples when all parameters are unknown and also obtain the observed Fisher information matrix. A special case of the distribution of the BGL distribution is reached in a closed form when one of the parameters is known. Simulation study was analyzed, and a numerical comparison is made between the proposed estimation procedure of the BGL distribution and Block-Basu estimation technique.

Markov Chain Monte Carlo to Study the Estimation of the Coefficient of Variation

The coefficient of variation (CV ) of a population is defined as the ratio of the population standard deviation to the population mean. It is regarded as a measure of stability or uncertainty, and can indicate the relative dispersion of data in the population to the population mean. In this article, based on the upper record values, we study the behavior of the CV of a random variable that follows a Lomax distribution. Specifically, we compute the maximum likelihood estimations (MLEs) and the confidence intervals of CV based on the observed Fisher information matrix using asymptotic distribution of the maximum likelihood estimator and also by using the bootstrapping technique. In addition, we propose to apply Markov Chain Monte Carlo (MCMC) techniques to tackle this problem, which allows us to construct the credible intervals. A numerical example based on a real data is presented to illustrate the implementation of the proposed procedure. Finally, Monte Carlo simulations are performed to observe the behavior of the proposed methods.

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.

Statistical Analysis of the Log-Normal Distribution under Type-II Progressive Hybrid Censoring Schemes

In this article, we obtain the maximum likelihood estimators (MLEs) and approximate maximum likelihood estimators (AMLEs) of the parameters, from a two-parameter log-normal distribution based on the adaptive Type-II progressive hybrid censoring scheme, which was introduced by Ng et al. (2009) for life testing or reliability experiment. In order to compare the results, we calculate corresponding estimators of the Type-II progressive hybrid censoring scheme. In particular, we provide computational formulas of the expected total test time and the expected number of failures for each scheme. We also compute the observed Fisher information matrix and use them to obtain the asymptotic confidence intervals. A simulation study carries out to evaluate the bias and mean square error of the MLEs and AMLEs from the two above-mentioned schemes. Finally, we present a numerical example to illustrate the methods of inference discussed here.