Uniqueness Of Maximum Likelihood Estimates Research Articles

Reliability estimation for Kumaraswamy distribution under block progressive type-II censoring

In this article, estimates of reliability performance characteristics are discussed when life test is conducted upon various test facilities under block progressive type-II censoring. When the differences in different test facilities are taken into account and lifetime of products follows a distribution with Kumaraswamy hazard function, different estimates are obtained from classical and hierarchical Bayesian frameworks, respectively. The existence and uniqueness of maximum likelihood estimators of model parameters are established, and approximate confidence intervals are obtained as well based on asymptotic theory and delta method. Moreover, associated Bayes estimators are also evaluated using a hybrid Metropolis–Hasting sampling in hierarchical framework. Finally, performances of the proposed approaches are compared via extensive simulation study, and two real data examples are also presented for application viewpoint.

Inference for a general family of exponentiated distributions under ranked set sampling with partially observed complementary competing risks data

ABSTRACT Ranked set sampling (RSS) acts as an efficient way for collecting failure information due to its ability of saving testing time and cost, and this paper discusses statistical inference for complementary competing risks model under a modified RSS scheme called the maximum ranked set sampling procedure with unequal samples (MRSSU). When the lifetimes of causes of failure are characterized by a general family of exponentiated distributions with partially observed failure causes, parameter estimation is explored from classical likelihood and Bayesian approaches. Existence and uniqueness of maximum likelihood estimators for model parameters are established, and approximate confidence intervals are constructed in consequence. With respect to general flexible priors, Bayes point and interval estimates are constructed, and associated Monte-Carlo sampling is proposed for complex posterior computation. In addition, when there is extra restriction information available, likelihood and Bayes estimates are also proposed in this regard. Extensive simulation studies are conducted to investigate the performance of different methods, and a real-life example is carried out to demonstrate the applications of our results.

Bayesian Estimation and Prediction for Inverse Power Maxwell Distribution with Applications to Tax Revenue and Health Care Data

This article discusses the problem of estimating the unknown model parameters as well as prediction of future observations from inverse power Maxwell distribution. The maximum likelihood method is applied for estimating the model parameters using Newton-Rapson iterative procedures. The existence and uniqueness of maximum likelihood estimates are established using Cauchy-Schwartz inequality. Approximate confidence intervals are constructed using Fisher information matrix. Using independent gamma informative priors, the Bayes estimates of unknown model parameters are obtained under squared error and Linex loss functions. Two approximation techniques namely: Lindley’s approximation and Metropolis-Hastings within Gibbs sampler algorithm have been employed to derive the Bayes estimators and also to construct the associate highest posterior density credible intervals. Based on the informative (observed) sample, Bayesian prediction, predictive density, and predictive intervals are derived for future observation and decision. The performance of proposed methods are evaluated though a Monte Carlo simulation experiment. Two real-life datasets related to tax revenue and heath are incorporated to show the practical utility of proposed methodology in real phenomenon.

Reliability estimation for bathtub-shaped distribution under block progressive censoring

In this article, estimates of reliability performance characteristics are discussed when life test is conducted upon various test facilities under block progressive Type-II censoring. When the differences in different test facilities are taken into account and the lifetime of products follows a bathtub-shaped hazard rate distribution, different estimates are obtained from classical and hierarchical Bayesian frameworks, respectively. Existence and uniqueness of maximum likelihood estimators of model parameters are established, and approximate confidence intervals are obtained as well based on asymptotic theory and delta method. Moreover, associated Bayes estimators are also evaluated using a hybrid Metropolis–Hasting sampling in hierarchical framework. Finally, performances of the proposed approaches are compared via extensive simulation study, and a real data example is also presented for application viewpoint.

On partially observed competing risks model for Chen distribution under generalized progressive hybrid censoring

In this paper, we discuss the inference for the competing risks model when the failure times follow Chen distribution. With assumption of two causes of failures, which are partially observed, are considered as independent. The existence and uniqueness of maximum likelihood estimates for model parameters are obtained under generalized progressive hybrid censoring. Also, we discussed the classical and Bayesian inferences of the model parameters under the assumption of restricted and nonrestricted parameters. Performance of classical point and interval estimators are compared with Bayesian point and interval estimators by conducting extensive simulation study. In addition to that, for illustration purpose, a real life example is discussed. Finally, some concluding remarks, regarding the presented model, are made.

Statistical inference of Lomax distribution based on adaptive progressive Type-II hybrid censored competing risks data

In this article, statistical inference is taken into account about two-parameter Lomax distribution under adaptive progressive Type-II hybrid censored data in combination with competing risks model. Frequency and Bayesian estimators under both symmetric and asymmetric loss functions are obtained for unknown parameters as well as cause-special reliability and hazard functions. Furthermore, the existence and uniqueness of maximum likelihood estimators are given. The corresponding confidence and credible intervals are also constructed based on asymptotic theory, delta method and MCMC technique. According to the simulation results, the performance of all the proposed point and interval estimates is evaluated. Finally, an example by analyzing real experimental data is given to illustrate all the deductive process established in this article.

Estimation for Weibull Parameters with Generalized Progressive Hybrid Censored Data

In this paper, the interest is in estimating the Weibull products when the available data is obtained via generalized progressive hybrid censoring. The testing scheme conducts products of interest under a more flexible way and allows collecting failure data in efficient and adaptable experimental scenarios than traditional lifetime testing. When the latent lifetime of products follows Weibull distribution, classical and Bayesian inferences are considered for unknown parameters. The existence and uniqueness of maximum likelihood estimates are established, and approximate confidence intervals are also constructed via asymptotic theory. Bayes point estimates as well as the credible intervals of the parameters are obtained, and correspondingly, Monte Carlo sampling technique is also provided for complex posterior computation. Extensive numerical analysis is carried out, and the results show that the generalized progressive hybrid censoring is an adaptive procedure in practical lifetime experiment, both proposed classical and Bayesian inferential approaches perform satisfactorily, and the Bayesian results are superior to conventional likelihood estimates.

Inference for a general family of inverted exponentiated distributions with partially observed competing risks under generalized progressive hybrid censoring

In this paper, statistical inference for a competing risks model is discussed when latent failure times belong to a general family of inverted exponentiated distributions. Based on a generalized progressive hybrid censored data with partially observed failure causes, estimations for unknown parameters are presented under nonrestricted and restricted parameter cases from classic and Bayesian perspectives, respectively. The existence and uniqueness of maximum likelihood estimators of the unknown parameters are established, and the associated approximate confidence intervals are also constructed via Fisher information matrix. In sequel, the Bayes estimators and credible intervals of the parameters are also obtained as well. Finally, the performance of different estimators are evaluated using Monte Carlo simulations and a real data set is also analyzed for illustration.

Inference for partially observed competing risks model for Kumaraswamy distribution under generalized progressive hybrid censoring

ABSTRACT In this paper, inference for a competing risks model is studied when latent failure times follow Kumaraswamy distribution and causes of failure are partially observed. Under generalized progressive hybrid censoring, existence and uniqueness of maximum likelihood estimators of model parameters are established. The confidence intervals are obtained by using asymptotic distribution theory. We further compute Bayes estimators along with credible intervals. In addition, inference is also discussed when there is order restricted shape parameters. The performance of all estimates is investigated using Monte-Carlo simulations. Finally, analysis of a real data set is presented for illustration purposes.

Exponential intervened Poisson distribution

We introduce a new family of distributions using intervened power series distribution. It is a rich family containing distributions such as Marshall–Olkin extended families of distributions, families of distributions generated through truncated negative binomial, and families of distributions generated through truncated binomial distribution. Also, the proposed distribution is a generalization of the families of distributions generated through zero truncated Poisson distribution. As a particular case, we study exponential intervened Poisson (EIP) distribution. This is a generalization of exponential-Poisson distribution. The shape properties of the pdf and hazard rate are proved. The model identifiability is established. The expression for moments, mean deviation, distribution of order statistics, and residual life function of EIP distribution are derived. The unknown parameters of the distribution are estimated using the method of maximum likelihood. The existence and uniqueness of maximum likelihood estimates (MLEs) are proved. Simulation study is carried out to show the performance of MLEs. EIP distribution is fitted to two real data sets and it is shown that the distribution is more appropriate than other competitive models. The adequacy of the EIP model for the data set is established using parametric bootstrap approach.

Estimation and testing for separable variance–covariance structures

The statistical analysis of data for a p‐variate response observed repeatedly on q occasions or of spatiotemporal data recorded at p locations by q times for n individuals may require that constraints be imposed on the modeling of the variance–covariance structure of the underlying process, not because of the repeated‐measures or spatiotemporal nature of the data but because there is not enough data otherwise to estimate the model parameters. Besides stationarity and isotropy, separability is an interesting option for that purpose because it reduces the number of variance‐covariance parameters to estimate, from pq(pq + 1)/2 to the Kronecker product of two matrices with p(p + 1)/2 and q(q + 1)/2 parameters. Originally, in the late 1980s, separability of the variance–covariance structure was assumed. Under this model, combined with the normality assumption on the underlying distribution, novel theoretical developments were thus made. The question of estimation of the parameters of a separable variance–covariance structure, more particularly by maximum likelihood, was raised from the early 1990s on, the question of testing for this structure being effectively addressed several years later. The existence and uniqueness of maximum likelihood estimators for the matrix normal distribution (i.e., the doubly multivariate normal distribution characterized by a simply separable variance–covariance structure) have been and remain questions of interest, as shown by recent results. Below, the reader is guided throughout the field of study of the separable variance–covariance structures as the author provides a fair treatment of the topic, its components, extensions (e.g., double separability), and future perspectives.This article is categorized under Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms

Inverted exponentiated Weibull distribution with applications to lifetime data

In this paper, we introduce the inverted exponentiated Weibull (IEW) distribution which contains exponentiated inverted Weibull distribution, inverse Weibull (IW) distribution, and inverted exponentiated distribution as submodels. The proposed distribution is obtained by the inverse form of the exponentiated Weibull distribution. In particular, we explain that the proposed distribution can be interpreted by Marshall and Olkin`s book (Lifetime Distributions: Structure of Non-parametric, Semiparametric, and Parametric Families, 2007, Springer) idea. We derive the cumulative distribution function and hazard function and calculate expression for its moment. The hazard function of the IEW distribution can be decreasing, increasing or bathtub-shaped. The maximum likelihood estimation (MLE) is obtained. Then we show the existence and uniqueness of MLE. We can also obtain the Bayesian estimation by using the Gibbs sampler with the Metropolis-Hastings algorithm. We also give applications with a simulated data set and two real data set to show the flexibility of the IEW distribution. Finally, conclusions are mentioned.

Modified inverse moment estimation: its principle and applications

In this survey, we present a modified inverse moment estimation of parameters and its applications. We use a specific model to demonstrate its principle and how to apply this method in practice. The estimation of unknown parameters is considered. A necessary and sufficient condition for the existence and uniqueness of maximum-likelihood estimates of the parameters is obtained for the classical maximum likelihood estimation. Inverse moment and modified inverse moment estimators are proposed and their properties are studied. Monte Carlo simulations are conducted to compare the performances of these estimators. As far as the biases and mean squared errors are concerned, modified inverse moment estimator works the best in all cases considered for estimating the unknown parameters. Its performance is followed by inverse moment estimator and maximum likelihood estimator, especially for small sample sizes.

On the existence of maximum likelihood estimates for weighted logistic regression

ABSTRACTThe problems of existence and uniqueness of maximum likelihood estimates for logistic regression were completely solved by Silvapulle in 1981 and Albert and Anderson in 1984. In this paper, we extend the well-known results by Silvapulle and by Albert and Anderson to weighted logistic regression. We analytically prove the equivalence between the overlap condition used by Albert and Anderson and that used by Silvapulle. We show that the maximum likelihood estimate of weighted logistic regression does not exist if there is a complete separation or a quasicomplete separation of the data points, and exists and is unique if there is an overlap of data points. Our proofs and results for weighted logistic apply to unweighted logistic regression.

Birnbaum–Saunders distribution based on Laplace kernel and some properties and inferential issues

For the Birnbaum–Saunders distribution based on Laplace kernel, we discuss the shape characteristics of density and hazard functions. We show the existence and uniqueness of maximum likelihood estimates. Simple modified moment estimators are proposed and compared with maximum likelihood estimates.

Some Notes on Parameter Estimation for Generalized Exponential Distribution

This article deals with the problem of estimating parameters of the generalized exponential distribution based on a complete sample. We consider the case when both the parameters are unknown. We first prove the existence and uniqueness of maximum likelihood estimators (MLEs) of the parameters for general sample size. Then we propose the inverse moment estimators (IMEs) of the parameters of the generalized exponential distribution. The precisions of MLEs and IMEs are compared through numerical simulations. Besides, two methods for constructing interval estimators for the parameters are proposed and their precisions are investigated through numerical simulations.

MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS IN RAYLEIGH DISTRIBUTION WITH RANDOMLY CENSORED DATA

In this paper,existence and uniqueness of maximum likelihood estimates of the parameters in Rayleigh distribution with randomly censored data are studied.Furthermore,the strong and weak consistency,as well as the asymptotic normality of the maximum likelihood estimates are obtained under mild conditions.

Maximum likelihood estimation for random sequential adsorption

The existence and uniqueness of maximum likelihood estimators for the time and range parameters in random sequential adsorption models are investigated.

On maximum likelihood estimation for log-linear models with non-ignorable non-response

Non-response boundary solutions occur in log-linear models with non-ignorable non-response. We prove existence and uniqueness of maximum likelihood estimates under weak conditions, and their consistency and asymptotic normality. This contrasts with findings for boundary solutions in log-linear models for sparse tables.

System Availability in the Presence of Estimating Common-cause Time-varying Failure Rates

This study presents a method for calculating the availability of a system depicted by availability block diagram, with identically distributed components, in the presence of estimating common cause hazard, we use the Marshall and Olkin formulation of the multivariate exponential distribution. That is, the components are subject to failure by Poisson failure processes that govern simultaneous failure of a specific subset of the components. A model is proposed for the analysis of systems subject to common-cause failures that are not considered to have a constant rate but that are assumed to obey a uniqueness of maximum likelihood estimators of the 2-parameter Weibull distribution. The method for calculating the system availability requires that a procedure exists for determining the system availability from component availabilities, under the statistically independent component assumption. The study includes an example to illustrate the method.