Asymptotic Distribution Of Maximum Likelihood Estimator Research Articles

Bayesian estimation for geometric process with the Weibull distribution

In this paper, we focus on Bayesian estimation of the parameters in the geometric process (GP) in which the first occurrence time of an event is assumed to have Weibull distribution. The Bayesian estimators are derived based on both symmetric (Squared Error) and asymmetric (General Entropy, LINEX) loss functions. Since the Bayesian estimators of unknown parameters cannot be obtained analytically, Lindley’s approximation and the Markov Chain Monte Carlo (MCMC) methods are applied to compute the Bayesian estimates. Furthermore, by using the MCMC methods, credible intervals of the parameters are constructed. Maximum likelihood (ML) estimators are also derived for unknown parameters. The confidence intervals of the parameters are obtained based on an asymptotic distribution of ML estimators. Moreover, the performances of the proposed Bayesian estimators are compared with the corresponding ML, modified moment and modified maximum likelihood estimators through an extensive simulation study. Finally, analyses of two different real data sets are presented for illustrative purposes.

Modelling and Assessment of Survival Probability of Shock Model with Two Kinds of Shocks

The study aims at modelling and assessment of survival probability of a component experiencing two kinds of shocks namely, damage shock and fatal shock. Shocks are occurring randomly in time as events of a Poisson Process. The two cases of fixed/random threshold of components are studied. Survival probabilities of proposed models are derived. Maximum likelihood estimators (MLEs) of survival probabilities are obtained using the data from life testing experiments. Fisher information and asymptotic distribution of MLEs of parameters are obtained when a constant threshold is considered. Computation and comparison of estimators of two cases (constant threshold and random threshold) are made through simulation studies. The study recommends the consideration of threshold as a random variable.

Inferences on stress–strength reliability based on ranked set sampling data in case of Lindley distribution

ABSTRACTIn this study, we consider point and interval estimation of stress–strength reliability based on ranked set sampling when the distribution of the stress and the strength are both Lindley. Firstly, maximum likelihood (ML) estimator of R is obtained. Then, we find asymptotic distribution of ML estimator of R to construct the asymptotic confidence interval. Furthermore, bootstrap confidence intervals of R are constructed using two different resampling methods. The performances of proposed methods are compared with their simple random sampling counterparts via an extensive Monte-Carlo simulation study. At the end of the study, a real data set is analysed for illustrative purposes.

Reliability estimation of k-unit series system based on progressiverly censored data

In this article, we consider a k-unit series system with component lifetime distribution to be a member of the scale family of distributions. We discuss estimation of the scale parameter and estimation of reliability function of the family based on progressively Type-II censored sample. The maximum likelihood estimator (MLE) of the scale parameter is derived using Expectation-Maximization (EM) algorithm and is used to estimate reliability function. Confidence intervals are constructed using asymptotic distribution of MLE. β-expectation tolerance interval for lifetime of the system is obtained. We consider half-logistic distribution as a member of the scale family and study performance of the MLE, reliability estimate and confidence interval using simulation experiments.

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.

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.

Estimation for Distributions with Monotone Failure Rate

In this paper, we shall investigate a problem analogous to the one treated in Prakasa Rao [6]. We shall now suppose that there is a sample of $n$ independent observations from a distribution $F$ with monotone failure rate $r$ and the problem is to obtain a maximum likelihood estimator (MLE) of $r$ and its asymptotic distribution. Grenander [4] and Marshall and Proschan [5] have obtained the MLE of $r$ and the latter showed that these estimators are consistent. We obtain the asymptotic distribution of this estimator. The estimation problem is reduced at first to that of a stochastic process and the asymptotic distribution of MLE is obtained by means of theorems on convergence of distributions of stochastic processes. Methods used in this paper are similar to those in Prakasa Rao [6] and therefore, proofs are given only at places where they differ from the proofs of that paper. We shall consider the case of distributions with increasing failure rate (IFR) in detail. Results in the case of distributions with decreasing failure rate (DFR) are analogous to those of IFR and we shall state the main result in Section 7. Sections 2 and 3 deal with the definition and properties of distributions with monotone failure rate $r$. Some results related to the asymptotic properties of the MLE of $r$ are given in Section 4. The problem is reduced to that of a stochastic process in Section 5. The asymptotic distribution of the MLE is obtained in Section 6.

Estimation of the Location of the Cusp of a Continuous Density

Chernoff and Rubin [1] and Rubin [5] investigated the problem of estimation of the location of a discontinuity in density. They have shown that the maximum likelihood estimator (MLE) is hyper-efficient under some regularity conditions on the density and that asymptotically the estimation problem is equivalent to that of a non-stationary process with unknown center of non-stationarity. We have obtained here similar results for a family of densities $f(x, \vartheta)$ which are continuous with a cusp at the point $\vartheta$. In this connection, it is worth noting that Daniels [2] has obtained a modified MLE for the family of densities $f(x, \vartheta) = C(\lambda) \exp \{-|x - \vartheta|^\lambda\}$, for $\lambda$ such that $\frac{1}{2} < \lambda < 1$, where $C(\lambda)$ is a constant depending on $\lambda$ and he has shown that this estimator is asymptotically efficient. In this paper, we shall show that the MLE of $\vartheta$ is hyper-efficient for the family of densities $f(x, \vartheta)$ given by \begin{equation*}\begin{align*}\tag{1.1}\log f(x, \vartheta) = \epsilon(x, \vartheta)|x - \vartheta|^\lambda + g(x, \vartheta) \text{for} |x| \leqq A \\ = g(x,\vartheta)\quad\text{for} |x| > A \\ \end{align*}\end{equation*} where $A$ is a constant greater than zero, \begin{equation*}\begin{align*}\tag{1.2}\epsilon (x,\vartheta) = \beta(\vartheta) \text{if} x < \vartheta \\ = \gamma(\vartheta) \text{if} x > \vartheta, \\ \end{align*}\end{equation*}\begin{equation*}\tag{1.3}0 < \lambda < \frac{1}{2} \text{and}\end{equation*}\begin{equation*}\tag{1.4}\vartheta\varepsilon (a, b) \text{where} -A < a < b < A,\end{equation*} under some regularity conditions on $f(x, \vartheta)$ and we shall derive the asymptotic distribution of the MLE implicitly. Section 2 contains the regularity conditions imposed on the family of densities $f(x, \vartheta)$. Section 3 contains some results related to the asymptotic properties of the MLE. The estimation problem is reduced to that of a stochastic process in Section 4. The asymptotic distribution of MLE is obtained in Section 5.