Hideki Nagatsuka

A consistent method of estimation for the three-parameter Weibull distribution

In this paper, we propose a new method for the estimation of parameters of the three-parameter Weibull distribution. The method is based on a data transformation, which avoids the problem of unbounded likelihood. In the proposed method, under mild conditions, the estimates always exist uniquely in the entire parameter space, and the estimators also have consistency over the entire parameter space. Through Monte Carlo simulations, we further show that the proposed method performs better than some existing methods in terms of bias and root mean squared error (RMSE). Finally, two examples based on real data sets are presented to illustrate the proposed method.

Parameter Estimation of the Shape Parameter of the Castillo–Hadi Model

Abstract In Castillo and Hadi [Castillo, E., Hadi, A. S. (1995). Modeling lifetime data with application to fatigue models. J. Amer. Statist. Assoc. 90:1041–1054], a new model for the analysis of lifetime data in the presence of a covariate is presented. The model is derived based on physical and statistical considerations. In this article, we focus on estimation of the power parameter of this model, and propose the new methods for estimation based on the location-scale-free transformation.

An Efficient Method of Parameter and Quantile Estimation for the Three-Parameter Weibull Distribution Based on Statistics Invariant to Unknown Location Parameter

The three-parameter Weibull distribution is widely used in life testing and reliability analysis. In this article, we propose an efficient method for the estimation of parameters and quantiles of the three-parameter Weibull distribution, which avoids the problem of unbounded likelihood, by using statistics invariant to unknown location. Through a Monte Carlo simulation study, we show that the proposed method performs well compared to other prominent methods based on bias and MSE. Finally, we present two illustrative examples.

A consistent method of estimation for the parameters of the three-parameter inverse Gaussian distribution

In this paper, we propose a consistent method of estimation for the parameters of the three-parameter inverse Gaussian distribution. We then discuss some properties of these estimators and show by means of a Monte Carlo simulation study that the proposed estimators perform better than some other prominent estimators in terms of bias and root mean squared error. Finally, we present two real-life examples to illustrate the method of inference developed here.

A Consistent Method of Estimation For The Three-Parameter Gamma Distribution

For the three-parameter gamma distribution, it is known that the method of moments as well as the maximum likelihood method have difficulties such as non-existence in some range of the parameters, convergence problems, and large variability. For this reason, in this article, we propose a method of estimation based on a transformation involving order statistics from the sample. In this method, the estimates always exist uniquely over the entire parameter space, and the estimators also have consistency over the entire parameter space. The bias and mean squared error of the estimators are also examined by means of a Monte Carlo simulation study, and the empirical results show the small-sample superiority in addition to the desirable large sample properties.

A Study of Estimation for the Three-Parameter Weibull Distribution Based on Doubly Type-II Censored Data Using a Least Squares Method

A least squares procedure for parameter estimation of the three-parameter Weibull distribution based on the doubly type-2 censored samples are proposed. The estimate of the shape parameter can be obtained by minimizing a function of only one parameter and the estimates of the location and scale parameters can be obtained by two closed-formed formulas, by using the proposed method. The proposed method has advantages compared to the conventional estimators: (1) the bias of the estimators can be smaller for small samples, (2) the estimates can be obtained with higher probability, (3) the estimates are much easier to obtain computationally.

Parameter and quantile estimation for the three-parameter lognormal distribution based on statistics invariant to unknown location

Lognormal distribution is one of the popular distributions used for modelling positively skewed data, especially those encountered in economic and financial data. In this paper, we propose an efficient method for the estimation of parameters and quantiles of the three-parameter lognormal distribution, which avoids the problem of unbounded likelihood, by using statistics that are invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared to other prominent methods in terms of both bias and mean-squared error. Finally, we present two illustrative examples.

A method for estimating parameters and quantiles of the three-parameter inverse Gaussian distribution based on statistics invariant to unknown location

The inverse Gaussian (IG) distribution, also known as the Wald distribution, is a long-tailed positively skewed distribution and a well-known lifetime distribution. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter IG distribution, which is based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared with other prominent methods in terms of bias and variance. Finally, we present two illustrative examples.

Parameter Estimation of Multivariate Distributions under Order Restrictions of the Parameters: An Extension of Isotonic Regression

Abstract In this work, we propose a method of parameter estimation of continuous multivariate distributions under simple order restrictions on parameters. The method is an extension of the isotonic regression and relaxes the restriction the isotonic regression has. A Monte Carlo simulation study shows that the proposed estimators perform better than the conventional estimators in terms of root mean squared error. Finally, we present two examples to illustrate the method of inference developed here.

Analysis for a trend on the optimal arrangements in a multi-state consecutive-k-out-of-n:F system

In traditional reliability theory, both the system and its components are allowed to take two possible states: either working or failed. In the binary context, a system with n components in sequence is called a consecutive-k-out-of-n:F system if the system fails whenever at least k consecutive components in the system fail. Many studies have appeared over the years dealing with the reliability evaluation of binary consecutive-k-out-of-n:F system. For consecutive-k-out-of-n:F system, one of the most important problem in this system is the optimal arrangement problem wherein the solution is obtained by the arrangement of components with maximum system reliability. On the other hand, in the multi-state consecutive-k-out-of-n:F system, both the components and the system are allowed to be in M+1 possible state. One of the important problem in the multi-state consecutive-k-out-of-n:F system is the optimal arrangement problem wherein the solution is obtained by the arrangement of components with maximum expectation of system states. (Note: k means the number of consecutive components, whose failed components lead to the system failure. Though the number is not a scalar and should be expressed as a vector (k1,k2,…,kM).) In general, the optimal arrangements depend on the values of component state probabilities. It is known, however, that for some case, optimal arrangements do not depend on the values of component state probabilities but on the ranks of component state probabilities. In this study, we consider optimal component arrangement for a multi-state consecutive-k-out-of-n:F system which maximize the expectation of the system state when these components are arranged to the positions in the system. First, we investigated characteristics of optimal arrangements in multi-state consecutive-k-out-of-n:F systems when M=3 and maximum kl (l=1,2,3) is 2, by using analytical approach. We proposed characteristics of optimal arrangement in the systems when n=3, 4 and 5. From these results, we obtained characteristics and trends of conditional optimal arrangements with M=3 and maximum kls are 2. Next, we investigated characteristics of optimal arrangements in multi-state consecutive-k-out-of-n:F systems by using heuristic approach. From these results, we obtained a trend of optimal arrangements with other parameters.