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In this paper, we consider the problem of estimating stress-strength reliability for inverse Weibull lifetime models having the same shape parameters but different scale parameters. We obtain the maximum likelihood estimator and its asymptotic distribution. Since the classical estimator doesn’t hold explicit forms, we propose an approximate maximum likelihood estimator. The asymptotic confidence interval and two bootstrap intervals are obtained. Using the Gibbs sampling technique, Bayesian estimator and the corresponding credible interval are obtained. The Metropolis-Hastings algorithm is used to generate random variates. Monte Carlo simulations are conducted to compare the proposed methods. Analysis of a real dataset is performed.

We discuss a new way of constructing pointwise confidence intervals for the distribution function in the current status model. The confidence intervals are based on the smoothed maximum likelihood estimator, using local smooth functional theory and normal limit distributions. Bootstrap methods for constructing these intervals are considered. Other methods to construct confidence intervals, using the non-standard limit distribution of the (restricted) maximum likelihood estimator, are compared with our approach via simulations and real data applications.

An early example of a compound decision problem of Robbins (1951) is employed to illustrate some features of the development of empirical Bayes methods. Our primary objective is to draw attention to the constructive role that the nonparametric maximum likelihood estimator for mixture models introduced by Kiefer & Wolfowitz (1956) can play in these developments.

In this paper, we consider a set of linear constraints on the coefficients of the random panel data model. Furthermore, Bayesian approach based on Markov Chain Monte Carlo (MCMC) is employed to making inferences on the model coefficients subject to this constraints.

We derive a first-order bias-corrected maximum likelihood estimator for the negative binomial dispersion parameter. This estimator is compared, in terms of bias and efficiency, with the maximum likelihood estimator investigated by Piegorsch (1990, Biometrics 46, 863-867), the moment and the maximum extended quasi-likelihood estimators investigated by Clark and Perry (1989, Biometrics 45, 309-316), and a double-extended quasi-likelihood estimator. The bias-corrected maximum likelihood estimator has superior bias and efficiency properties in most instances. For ease of comparison we give results for the two-parameter negative binomial model. However, an example involving negative binomial regression is given.

Background Group sequential designs are increasingly employed to allow trials to stop early with statistical rigor. While existing work focuses on intention-to-treat effect on clinical endpoints, the properties of mediation analysis (commonly conducted in psychological trials to understand a causal mechanism) remain unknown under group sequential designs. Methods Considering a group sequential design with one interim analysis for early stopping for efficacy, we conduct a simulation study to evaluate existing analysis techniques when the treatment effect on a continuous outcome is partially or fully mediated by a continuous intermediate variable measuring a casual mechanism. We study the probability of rejecting the null hypotheses on the total effect (i.e., intention-to-treat effect), direct effect and indirect effect, respectively. We examine the bias of maximum likelihood estimator for these effects. We investigate if the penalized (and conditional) maximum likelihood estimator has smaller bias than the maximum likelihood estimator when a trial stopped (did not stop) early. Results The presence of an intermediate variable reduces the power of a group sequential design when sample size calculation ignores the causal mechanism, though type I error control remains unaffected. The maximum likelihood estimator is unbiased only for the mediator-outcome path, impacting the properties of mediation analysis since existing methods typically rely on it to estimate the pathways. The penalized maximum likelihood estimator for other pathways has similar bias to the stage-one maximum likelihood estimator, while the conditional maximum likelihood estimator shows negligible or smaller bias than the usual maximum likelihood estimator for estimating the total and the direct effects only. Conclusions Mediation analysis needs additional consideration in group sequential designs. As with fixed trial designs, the sample size calculation of group sequential designs should account for the total variability underlying a causal mechanism when the treatment effect is hypothesized to be mediated by an intermediate variable, or risk the overall power to detect an intention-to-treat (total) effect being lower than the nominal value. We suggest reporting several estimators and acknowledging that they may be biased for some mediation pathways. More research is needed to develop methods for the analysis of indirect effect under group sequential designs.

Purpose The Hurst exponent has been very important in telling the difference between fractal signals and explaining their significance. For estimators of the Hurst exponent, accuracy and efficiency are two inevitable considerations. The main purpose of this study is to raise the execution efficiency of the existing estimators, especially the fast maximum likelihood estimator (MLE), which has optimal accuracy. Design/methodology/approach A two-stage procedure combining a quicker method and a more accurate one to estimate the Hurst exponent from a large to small range will be developed. For the best possible accuracy, the data-induction method is currently ideal for the first-stage estimator and the fast MLE is the best candidate for the second-stage estimator. Findings For signals modeled as discrete-time fractional Gaussian noise, the proposed two-stage estimator can save up to 41.18 per cent the computational time of the fast MLE while remaining almost as accurate as the fast MLE, and even for signals modeled as discrete-time fractional Brownian motion, it can also save about 35.29 per cent except for smaller data sizes. Originality/value The proposed two-stage estimation procedure is a novel idea. It can be expected that other fields of parameter estimation can apply the concept of the two-stage estimation procedure to raise computational performance while remaining almost as accurate as the more accurate of two estimators.

This paper investigates asymptotic properties of the maximum likelihood estimator and the quasi-maximum likelihood estimator for the spatial autoregressive model. The rates of convergence of those estimators may depend on some general features of the spatial weights matrix of the model. It is important to make the distinction with different spatial scenarios. Under the scenario that each unit will be influenced by only a few neighboring units, the estimators may have √n-rate of convergence and be asymptotically normal. When each unit can be influenced by many neighbors, irregularity of the information matrix may occur and various components of the estimators may have different rates of convergence.

We establish the exponential nonuniform Berry–Esseen bound for the maximum likelihood estimator of unknown drift parameter in an ultraspherical Jacobi process using the change of measure method and precise asymptotic analysis techniques. As applications, the optimal uniform Berry–Esseen bound and optimal Cramér-type moderate deviation for the corresponding maximum likelihood estimator are obtained.