Distribution Of Maximum Likelihood Estimator Research Articles

Exact inference for exponential distribution with multiply Type-I censored data

ABSTRACTIn this paper, we focus on exact inference for exponential distribution under multiple Type-I censoring, which is a general form of Type-I censoring and represents that the units are terminated at different times. The maximum likelihood estimate of mean parameter is calculated. Further, the distribution of maximum likelihood estimate is derived and it yields an exact lower confidence limit for the mean parameter. Based on a simulation study, we conclude that the exact limit outperforms the bootstrap limit in terms of the coverage probability and average limit. Finally, a real dataset is analyzed for illustration.

OUP accepted manuscript

SummarySeparability is a common simplifying assumption on the covariance structure of spatiotemporal functional data. We present three tests of separability, one a functional extension of the Monte Carlo likelihood method of Mitchell et al. (2006) and two based on quadratic forms. Our tests are based on asymptotic distributions of maximum likelihood estimators and do not require Monte Carlo simulation. The main theoretical contribution of this paper is the specification of the joint asymptotic distribution of these estimators, which can be used to derive many other tests. The main methodological finding is that one of the quadratic form methods, which we call a norm approach, emerges as a clear winner in terms of finite-sample performance in nearly every setting we considered. This approach focuses directly on the Frobenius distance between the spatiotemporal covariance function and its separable approximation. We demonstrate the efficacy of our methods via simulations and application to Irish wind data.

Estimation of parameters of Weibull\u2013Gamma distribution based on progressively censored data

In this paper, the estimation of parameters of a three-parameter Weibull–Gamma distribution based on progressively type-II right censored sample is studied. The maximum likelihood, Bayes, and parametric bootstrap methods are used for estimating the unknown parameters as well as some lifetime parameters reliability function, hazard function and coefficient of variation. Approximate confidence intervals for the unknown parameters as well as reliability function, hazard function and coefficient of variation are constructed based on the s-normal approximation to the asymptotic distribution of maximum likelihood estimators (MLEs), and log-transformed MLEs. In addition, two bootstrap CIs are also proposed. Bayes estimates of the unknown parameters and the corresponding credible intervals are obtained by using the Gibbs within Metropolis–Hasting samplers procedure. Furthermore, the results of Bayes method are obtained under both the balanced squared error loss and balanced linear-exponential loss. Analysis of a simulated data set has also been presented for illustrative purposes. Finally, a Monte Carlo simulation study is carried out to investigate the precision of the Bayes estimates with MLEs and two bootstrap estimates, also to compare the performance of different corresponding CIs considered.

Inference for the shape parameter of lognormal distribution in presence of fuzzy data

Traditional Statistical analysis of lognormal distribution have been proposed for precisely defined crisp data. But there are many other situations in which measurement results from continuous quantities are not precise numbers but more or less fuzzy. This article presents the statistical inference on the shape parameter of lognormal distribution involving experiment whose observations are described in terms of fuzzy data. The maximum likelihood procedure are developed for estimating the unknown parameter. Asymptotic distribution of maximum likelihood estimator is used to construct approximate confidence interval. Also, Bayes estimate and the corresponding highest posterior density credible interval of the unknown parameter are obtained by using Markov Chain Monte Carlo technique. In addition, we describe an estimation method based on moments of lognormal distribution. Extensive simulations are performed to compare the performances of the different proposed methods.

Optimum Step-Stress Accelerated Life Test Plan for Lomax Distribution with an Adaptive Type-II Progressive Hybrid Censoring

The adaptive type -II progressive hybrid censoring has the advantage of saving both t he total test time and the cost of the experiment; also it increases the effi ciency of the statistical analysis. This article discuss es k-level step stress accelerated life tests based on an ad aptive type-II progressive hybrid censoring with product ' s life time following Lomax distribution. The scale param eter of the Lomax failure time distribution at constant levels is assumed to be a log line ar function of the stress level. Maximum likelihood estimators of the model parameters are derive d. Based on normal approximation to the asymptotic distribution of maximum likelihood estimators, t he approximate confidence intervals for model parameters are obtained. The optimal times of changing stress levels are discussed under Doptimality and A-optimality criteria. Such methods maximize the determinant and the trace of Fisher's information matrix for the model parameters. Analysis of t he numerical data has been presented for illustrative proposes.

Optimum Step-Stress Accelerated Life Test Plan for Lomax Distribution with an Adaptive Type-II Progressive Hybrid Censoring

The adaptive type -II progressive hybrid censoring has the advantage of saving both t he total test time and the cost of the experiment; also it increases the effi ciency of the statistical analysis. This article discuss es k-level step stress accelerated life tests based on an ad aptive type-II progressive hybrid censoring with product ' s life time following Lomax distribution. The scale param eter of the Lomax failure time distribution at constant levels is assumed to be a log line ar function of the stress level. Maximum likelihood estimators of the model parameters are derive d. Based on normal approximation to the asymptotic distribution of maximum likelihood estimators, t he approximate confidence intervals for model parameters are obtained. The optimal times of changing stress levels are discussed under Doptimality and A-optimality criteria. Such methods maximize the determinant and the trace of Fisher's information matrix for the model parameters. Analysis of t he numerical data has been presented for illustrative proposes.

Drop fragility of the display of a smart mobile phone: weakest link failure or cumulative shock failure?

It is not unusual for portable devices to be damaged when they are accidentally dropped on hard floors. Shock tests are increasingly being used to evaluate the drop impact response of portable devices. However, the underlying failure mechanisms have not been fully theoretically explored. There are two candidate failure mechanisms that can account for the drop impact fragility of the display in a smart mobile phone during a shock test: weakest link failure and cumulative damage failure. The weakest link theory provides a basis on a Weibull distribution and the cumulative damage theory on an inverse Gaussian distribution. This article proposes a discrimination procedure for the two distribution types. The probability of correct selection is computed using asymptotic results on the ratio of the Maximum Likelihood (ML) to discriminate between the two distributions. Expressions are provided that can be used to compute the asymptotic distributions of ML estimators or their functions when there is model mis-specification. The proposed method is applied to real shock test data for smart mobile phone display modules to determine the underlying failure mechanisms.

Accelerated Life Test Plans Robust to Misspecification of the Stress—Life Relationship

This article discusses methodology for deriving the asymptotic distribution of maximum likelihood estimators of model parameters in accelerated life tests (ALTs) when the form of the stress–life relationship is misspecified. If the practitioner can describe possible departures from an assumed ALT model, often linear in the appropriately transformed accelerating variable, then the results here can be used to obtain ALT plans that provide protection against potential bias and loss in estimation efficiency. General results are derived for log location-scale distributions and time-censored data, whereas specific results are presented for the lognormal and Weibull distributions. In a practical ALT situation, the methodology is applied to derive robust test plans based on minimizing asymptotic bias, asymptotic variance, or asymptotic mean squared error. Test plans are evaluated in terms of bias, efficiency, extrapolation to use level, and expected number failing.

Towards reconciling two asymptotic frameworks in spatial statistics

Two asymptotic frameworks, increasing domain asymptotics and infill asymptotics, have been advanced for obtaining limiting distributions of maximum likelihood estimators of covariance parameters in Gaussian spatial models with or without a nugget effect. These limiting distributions are known to be different in some cases. It is therefore of interest to know, for a given finite sample, which framework is more appropriate. We consider the possibility of making this choice on the basis of how well the limiting distributions obtained under each framework approximate their finite-sample counterparts. We investigate the quality of these approximations both theoretically and empirically, showing that, for certain consistently estimable parameters of exponential covariograms, approximations corresponding to the two frameworks perform about equally well. For those parameters that cannot be estimated consistently, however, the infill asymptotic approximation is preferable.

Who's afraid of reduced-rank parameterizations of multivariate models? Theory and example

Reduced-rank restrictions can add useful parsimony to coefficient matrices of multivariate models, but their use is limited by the daunting complexity of the methods and their theory. The present work takes the easy road, focusing on unifying themes and simplified methods. For Gaussian and non-Gaussian (GLM, GAM, mixed normal, etc.) multivariate models, the present work gives a unified, explicit theory for the general asymptotic (normal) distribution of maximum likelihood estimators (MLE). MLE can be complex and computationally hard, but we show a strong asymptotic equivalence between MLE and a relatively simple minimum (Mahalanobis) distance estimator. The latter method yields particularly simple tests of rank, and we describe its asymptotic behavior in detail. We also examine the method's performance in simulation and via analytical and empirical examples.

Maximum Likelihood Estimation Under Misspecified Lognormal and Weibull Distributions

This article provides expressions for computing the asymptotic distribution of maximum likelihood estimators of lognormal and Weibull parameters when the distribution is misspecified and data are Type I or II censored. The results can be used to derive the asymptotic normal distribution of observed bias in estimation of functions of model parameters such as quantiles. The methods are applied to a Type I censored dataset on locomotive controls. The results are used to derive test plans that are bias-robust for estimating quantiles. Values of the proportions failing that minimize asymptotic mean square error and absolute asymptotic bias are tabulated.

Lognormal and Weibull Accelerated Life Test Plans Under Distribution Misspecification

In this article, we derive expressions for the asymptotic distribution of maximum likelihood estimators of model parameters in accelerated life tests (ALTs) when the model distribution is misspecified. We investigate results for two popular models, namely, the lognormal and Weibull Arrhenius-type ALT models. We propose test plan criteria based on asymptotic bias (ABias) and asymptotic mean squared error (AMSE) to derive ALT test plans. We derive 4:2:1 allocation plans that minimize either ABias/sup 2/ or AMSE. These criteria provide control over estimation bias and variance when the model distribution is misspecified.

Inferring population history from genealogical trees

Inference about population history from DNA sequence data has become increasingly popular. For human populations, questions about whether a population has been expanding and when expansion began are often the focus of attention. For viral populations, questions about the epidemiological history of a virus, e.g., HIV-1 and Hepatitis C, are often of interest. In this paper I address the following question: Can population history be accurately inferred from single locus DNA data? An idealised world is considered in which the tree relating a sample of n non-recombining and selectively neutral DNA sequences is observed, rather than just the sequences themselves. This approach provides an upper limit to the information that possibly can be extracted from a sample. It is shown, based on Kingman's (1982a) coalescent process, that consistent estimation of parameters describing population history (e.g., a growth rate) cannot be achieved for increasing sample size, n. This is worse than often found for estimators of genetic parameters, e.g., the mutation rate typically converges at rate under the assumption that all historical mutations can be observed in the sample. In addition, various results for the distribution of maximum likelihood estimators are presented.

Estimation and Testing for Unit Root Processes with GARCH (1, 1) Errors: Theory and Monte Carlo Evidence

Least squares (LS) and maximum likelihood (ML) estimation are considered for unit root processes with GARCH (1, 1) errors. The asymptotic distributions of LS and ML estimators are derived under the condition α + β < 1. The former has the usual unit root distribution and the latter is a functional of a bivariate Brownian motion, as in Ling and Li [Ling, S., Li, W. K. (1998). Limiting distributions of maximum likelihood estimators for unstable autoregressive moving‐average time series with GARCH errors. Ann. Statist.26:84–125]. Several unit root tests based on LS estimators, ML estimators, and mixing LS and ML estimators, are constructed. Simulation results show that tests based on mixing LS and ML estimators perform better than Dickey–Fuller tests which are based on LS estimators, and that tests based on the ML estimators perform better than the mixed estimators.

Finite sample properties of nonstationary binary response models: A monte carlo analysis

This paper investigates the finite sample distributions of maximum likelihood estimators for nonstationary probit models. This model relates an I (1) variable x_t to a latent I (1) variable y_t^\\ast . The binary variable, y_t , is coded as a one if y_t^\\ast exceeds some threshold, \\alpha , and as a zero otherwise. We find that, analogous to standard OLS models, commonly used tests statistics almost always reject the null hypothesis of no relationship between x_t , and a latent y_t , even when they are, in fact, generated by independent random walks. However, if x_t and y_t^\\ast are cointegrated, the sampling distributions of the parameter estimates are better behaved and standard Z and Wald test statistics are consistent.

Asymptotic distribution of MLE’s of the parameters for a diffusion model with catastrophes and some related inference problems

In this paper the asymptotic distribution of the maximum likelihood estimators (MLE) for a diffusion model with beta catastrophe process is considered. Results are given on the asymptotic distribution of the score function and the MLE when standardized by the information matrix. The test of hypotheses and multivariate t method of simultaneous confidence intervals are also determined.

Asymptotics of maximum likelihood estimator in a two-phase linear regression model

This paper considers two-phase random design linear regression models with arbitrary error densities and where the regression function has a fixed jump at the true change-point. It obtains the consistency and the limiting distributions of maximum likelihood estimators of the underlying parameters in these models. The left end point of the maximizing interval with respect to the change point, herein called the maximum likelihood estimator r ̂ n of the change-point parameter r, is shown to be n-consistent and the underlying likelihood process, as a process in the standardized change-point parameter, is shown to converge weakly to a compound Poisson process. This process obtains maximum over a bounded interval and n( r ̂ n−r) converges weakly to the left end point of this interval. These results are different from those available in the literature for the case of the two-phase linear regression models when jump sizes tend to zero as n tends to infinity.

Polya Posterior Frequency Distributions for Stratified Double Sampling of Categorical Data

Abstract Estimates of category frequencies in stratified double sampling are generally obtained by methods of maximum likelihood assuming a multinomial distribution of data. Although the distribution of estimation errors is asymptotically normal, a reliance on the normal assumption for inference can be problematic for low frequency categories (&amp;lt;10%) and small sample sizes (&amp;lt;1,000). In these situations, a Bayesian approach appears more robust. A vague Dirichlet prior is adopted to compute a Polya posterior distribution. In a simulation study of sampling from eight small-to medium-sized populations, we found that the resampling distributions of maximum likelihood estimates were generally closer to the Polya posteriors than to the assumed normal distributions. Resampling distributions and Polya posteriors deviated frequently and significantly from a normal distribution. A Polya posterior was often orders of magnitude faster to compute than a resampling distribution. FOR. SCI. 48(3):569–581.

A method for approximating the density of maximum-likelihood and maximum a posteriori estimates under a Gaussian noise model

The performance of maximum-likelihood (ML) and maximum a posteriori (MAP) estimates in non-linear problems at low data SNR is not well predicted by the Cramér-Rao or other lower bounds on variance. In order to better characterize the distribution of ML and MAP estimates under these conditions, we derive a point approximation to density values of the conditional distribution of such estimates. In an example problem, this approximate distribution captures the essential features of the distribution of ML estimates in the presence of Gaussian-distributed noise.

On the asymptotic distribution of maximum likelihood estimators with doubly censored data

A method of estimating lifetime parameters from doubly censored data is given on the basis of the maximum likelihood principle. Consistency and asymptotic normality of the proposed estimator is established. When the observation window is defined by the interval of time between the first and the pth events in a homogeneous Poisson process, asymptotic variances and efficiencies are analyzed assuming exponential lifetime distribution.