A new lifetime distribution for a series-parallel system: properties,applications and estimations under progressive type-II censoring

A compound class of zero truncated Poisson and lifetime distributions is introduced. A specialization is paved to a new three-parameter distribution, called doubly Poisson-exponential distribution, which may represent the lifetime of units connected in a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution. Among its motivations is that its hazard rate function can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods [maximum likelihood, moments, least squares, weighted least squares and Bayes (under linear-exponential and general entropy loss functions) estimations] are used to estimate the involved parameters. The performance of these methods is investigated through a simulation study. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. In addition, confidence intervals, symmetric credible intervals and highest posterior density credible intervals of the parameters are obtained. Finally, an application to a real data set is used to compare the new distribution with other five distributions.     

Improved distribution quantile estimation

A technique for estimating the quantiles or percentiles of a distribution is developed. The parametric form of the distribution is assumed unknown. The estimation procedure is based on a kernel estimator of a probability density function and on aquantile estimator suggested by Harrell and Davis (1982). Simulation studies show that estimation of quantiles in moderately heavyto heavy tails of a distribution is substantially improved by use of the technique.     

A fortran implementation of univariate fourier series density estimation

A set of Fortran-77 subroutines is described which compute a nonparametric density estimator expressed as a Fourier series. In addition, a subroutine is given for the estimation of a cumulative distribution. Performance measures are given based on samples from a Weibull distribution. Due to small size and modest space demands, these subroutines are easily implemented on most small computers.     

Inference with a contrast-based posterior distribution and application in spatial statistics

The likelihood function is often used for parameter estimation. Its use, however, may cause difficulties in specific situations. In order to circumvent these difficulties, we propose a parameter estimation method based on the replacement of the likelihood in the formula of the Bayesian posterior distribution by a function which depends on a contrast measuring the discrepancy between observed data and a parametric model. The properties of the contrast-based (CB) posterior distribution are studied to understand what the consequences of incorporating a contrast in the Bayes formula are. We show that the CB-posterior distribution can be used to make frequentist inference and to assess the asymptotic variance matrix of the estimator with limited analytical calculations compared to the classical contrast approach. Even if the primary focus of this paper is on frequentist estimation, it is shown that for specific contrasts the CB-posterior distribution can be used to make inference in the Bayesian way.The method was used to estimate the parameters of a variogram (simulated data), a Markovian model (simulated data) and a cylinder-based autosimilar model describing soil roughness (real data). Even if the method is presented in the spatial statistics perspective, it can be applied to non-spatial data.     

Unbiased Estimation of Central Moments by using U-statistics

We obtain an estimator of the r th central moment of a distribution, which is unbiased for all distributions for which the first r moments exist. We do this by finding the kernel which allows the r th central moment to be written as a regular statistical functional. The U-statistic associated with this kernel is the unique symmetric unbiased estimator of the r th central moment, and, for each distribution, it has minimum variance among all estimators which are unbiased for all these distributions.     

A target distribution model for nonparametric density estimation

In this paper a model is proposed which represents a wide class of continuous distributions. It is shown how the parameters of this model can be estimated leading to a distribution estimator and a corresponding density estimator. An important property of this estimator is that it can be structured to reflect a priori knowledge of the unknown distribution.

Finally, some examples are shown and some comparisons made with kernel and orthogonal series estimators.     

Pseudo maximum likelihood estimation for the dirichlet-multinomial distribution

Pseudo maximum likelihood estimation (PML) for the Dirich-let-multinomial distribution is proposed and examined in this pa-per. The procedure is compared to that based on moments (MM) for its asymptotic relative efficiency (ARE) relative to the maximum likelihood estimate (ML). It is found that PML, requiring much less computational effort than ML and possessing considerably higher ARE than MM, constitutes a good compromise between ML and MM. PML is also found to have very high ARE when an estimate for the scale parameter in the Dirichlet-multinomial distribution is all that is needed.     

On least squares estimation for categorical data

We consider a multinomial distribution in which the cell probabilities are known arbitrary functions of a vector parameter θ. It is desired to estimate θ by least squares. Three variations of the least squares approach are investigated, and each is found to be equivalent, in the very strong sense of being algebraically identical, to one of the following estimation procedures: maximum likelihood, minimum χ² and minimum modified χ². Two of these results also apply to the multiple hypergeometric distribution.     

Sampling properties of estimators of the log-logistic distribution with application to Canadian precipitation data

We consider the probability-weighted moment and the maximum-likelihood estimators of two parameters in the log-logistic distribution. Quantile estimators are obtained using both methods. The distributional properties of these estimators are studied in large samples, via asymptotic theory, and in small and moderate samples, via Monte Carlo simulation. The distribution is shown to be appropriate for a wide variety of meteorological data.     

A simple method for estimating parameters of the location–scale distribution family

The purpose of this paper is to estimate the parameters of the location–scale distribution family. As a special case, the method is used for estimating the parameters of the normal distribution and Cauchy distribution. For the Cauchy distribution, neither the moment estimation method nor the maximum likelihood estimation method works properly for estimating the parameters. The quantiles for obtaining confidence intervals and point estimates for the parameters of the two-parameter Cauchy distribution are given in the paper. It is shown that the estimators obtained in this paper are unbiased with respect to the median and possess some optimal properties.     

Reliability estimation of generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is introduced in this paper. This lifetime distribution is capable of modelling various shapes of failure rates, and hence various shapes of ageing criteria. The model can be considered as another useful two-parameter generalization of the IED. Statistical and reliability properties of the generalized inverted exponential distribution are derived. Maximum likelihood estimation and least square estimation are used to evaluate the parameters and the reliability of the distribution. Properties of the estimates are also studied.     

Sampling properties of estimators of a Weibull distribution of use in the lumber industry

We consider maximum-likelihood estimators of the three parameters in the Weibull distribution. Motivated by an application regarding the determination of a lower percentile of the strength of dimension lumber, we investigate the sampling properties of these estimators. Consistency is established when the shape parameter is greater than one, including some nonregular cases encountered in fitting lumber data. The joint distribution of the estimators is studied by a Monte Carlo approach. The maximum-likelihood estimator of the 5th percentile is compared with the sample 5th percentile. For the cases considered it was generally found that with sample size 70 the usual asymptotic normality does not hold.     

Estimation of reliability of multicomponent stress–strength for a Kumaraswamy distribution

This article deals with the Bayesian and non Bayesian estimation of multicomponent stress–strength reliability by assuming the Kumaraswamy distribution. Both stress and strength are assumed to have a Kumaraswamy distribution with common and known shape parameter. The reliability of such a system is obtained by the methods of maximum likelihood and Bayesian approach and the results are compared using Markov Chain Monte Carlo (MCMC) technique for both small and large samples. Finally, two data sets are analyzed for illustrative purposes.     

Estimation of Location Parameters Under Nonnormal Errors and Quadratic Loss

Assuming a statistical model in which the joint distribution of the unobservable errors is drawn from independent univariate Student t's that are identically and symmetrically distributed, the sampling performance of traditional robust estimators and a family of Stein-like estimators are compared and evaluated. These results suggest that under thick-tailed distributions, the relative sampling performances and risk characteristics for a range of nonconventional Stein estimators remains approximately the same as in the case of their normal counterparts. The empirical risk implications of misspecifying the error distribution are investigated.     

A Convergence Result for a Class of Asymptotically Linear Estimators

This article provides the distribution of the last exit for strongly consistent estimators. Namely, we consider a small neighborhood of the (almost sure) limit and state the asymptotic distribution of the last time the estimator is outside this neighborhood. Such problems have been considered in the literature by various authors; this article extends these results in a semi-parametric frame. An application to adaptive estimation is provided.     

Asymptotic Normality of Kernel-Type Deconvolution Estimators

Abstract.  We derive asymptotic normality of kernel-type deconvolution estimators of the density, the distribution function at a fixed point, and of the probability of an interval. We consider so-called super smooth deconvolution problems where the characteristic function of the known distribution decreases exponentially, but faster than that of the Cauchy distribution. It turns out that the limit behaviour of the pointwise estimators of the density and distribution function is relatively straightforward, while the asymptotic behaviour of the estimator of the probability of an interval depends in a complicated way on the sequence of bandwidths.     

The Minimum Density Power Divergence Estimation for the Lognormal Density

In this article, we implement the minimum density power divergence estimation for estimating the parameters of the lognormal density. We compare the minimum density power divergence estimator (MDPDE) and the maximum likelihood estimator (MLE) in terms of robustness and asymptotic distribution. The simulations and an example indicate that the MDPDE is less biased than MLE and is as good as MLE in terms of the mean square error under various distributional situations.     

A Three-Parameter Asymmetric Laplace Distribution and Its Extension

ABSTRACT

In this article, a new three-parameter asymmetric Laplace distribution and its extension are introduced. This includes as special case the symmetric Laplace double-exponential distribution. The distribution has established a direct link to estimation of quantile and quantile regression. Properties of the new distribution are presented. Application is made to a flood data modeling example.     

A new estimate of skewness with mean-squared error smaller than that of the sample skewness

Recently, a new procedure for distribution fitting, based on matching of the first two moments, partial and complete, was introduced (Shore, 1995). When the sampling skewness of the fitted distribution is compared to the sample skewness, and both are regarded as estimates of the skewness of the underlying distribution, the mean-squared-error of the former is appreciably lower than that of the latter. In this paper we present some simulation results to support this claim and demonstrate its magnitude. An alternative two-moment distributional fitting procedure, based on a new family of four-parameter distributions, is also introduced and studied. Since three-moment distribution fitting is very common practice in simulation studies, these results may have important implications for the current state-of-the-art of simulation     

Inference about the Ratio of Two Parameters,with Application to Whale Censusing

Inference for the quotient of two parameters estimated separately may be obtained by the delta method. When the distribution of linear transformations involving the numerator and the denominator is available, more exact and elementary methods may be used. Non-Bayesian and Bayesian approaches are developed. The application of the methods to estimating the stock abundance of Northeastern Atlantic minke whales, where the ratio is a raw estimate divided by a measure of observation efficiency, is explained and discussed. The Bayesian approach allows exact inference in quite general situations using only a single, rapidly implemented, one-dimensional numerical integration. A simple analytic approximation is given for the common situation where the joint posterior distribution of the numerator and denominator can be approximated by a normal distribution that gives very little probability to negative values of the denominator. The Bayesian approach also permits the incorporation of model uncertainty (or disagreement) in a natural way, and this was the basis for the conclusions of the International Whaling Commission Scientific Committee at its 1990 meeting.