Closed-Form Estimators for the Gamma Distribution Derived From Likelihood Equations

It is well-known that maximum likelihood (ML) estimators of the two parameters in a gamma distribution do not have closed forms. This poses difficulties in some applications such as real-time signal processing using low-grade processors. The gamma distribution is a special case of a generalized gamma distribution. Surprisingly, two out of the three likelihood equations of the generalized gamma distribution can be used as estimating equations for the gamma distribution, based on which simple closed-form estimators for the two gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estimators. The study consolidates this conjecture by establishing the asymptotic behaviors of the new estimators. In addition, the closed-forms enable bias-corrections to these estimators. The bias-correction significantly improves the small-sample performance.     

A new two-parameter lifetime distribution with decreasing,increasing or upside-down bathtub-shaped failure rate

In this paper, we introduce a generalization of the Bilal distribution, where a new two-parameter distribution is presented. We show that its failure rate function can be upside-down bathtub shaped. The failure rate can also be decreasing or increasing. A comprehensive mathematical treatment of the new distribution is provided. The estimation by maximum likelihood is discussed, and a closed-form expression for Fisher’s information matrix is obtained. Asymptotic interval estimators for both of the two unknown parameters are also given. A simulation study is conducted and applications to real data sets are presented.     

Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions

This paper deals with the Bayesian estimation of generalized exponential distribution in the proportional hazards model of random censorship under asymmetric loss functions. It is well known for the two-parameter lifetime distributions that the continuous conjugate priors for parameters do not exist; we assume independent gamma priors for the scale and the shape parameters. It is observed that the closed-form expressions for the Bayes estimators cannot be obtained; we propose Tierney–Kadane's approximation and Gibbs sampling to approximate the Bayes estimates. Monte Carlo simulation is carried out to observe the behavior of the proposed methods and one real data analysis is performed for illustration. Bayesian methods are compared with maximum likelihood and it is observed that the Bayes estimators perform better than the maximum-likelihood estimators in some cases.     

Estimation of parameters and the mean life of a mixed failure time distribution

This paper deals with estimation of parameters and the mean life of a mixed failure time distribution that has a discrete probability mass at zero and an exponential distribution with mean O for positive values. A new sampling scheme similar to Jayade and Prasad (1990) is proposed for estimation of parameters. We derive expressions for biases and mean square errors (MSEs) of the maximum likelihood estimators (MLEs). We also obtain the uniformly minimum variance unbiased estimators (UMVUEs) of the parameters. We compare the estimator of O and mean life fj based on the proposed sampling scheme with the estimators obtained by using the sampling scheme of Jayade and Prasad (1990).     

Estimation of the location and scale parameters of the extreme value distmbution based on multiply type-II censored samples

In this paper, we consider the problem of estimating the location and scale parameters of an extreme value distribution based on multiply Type-II censored samples. We first describe the best linear unbiased estimators and the maximum likelihood estimators of these parameters. After observing that the best linear unbiased estimators need the construction of some tables for its coefficients and that the maximum likelihood estimators do not exist in an explicit algebraic form and hence need to be found by numerical methods, we develop approximate maximum likelihood estimators by appropriately approximating the likelihood equations. In addition to being simple explicit estimators, these estimators turn out to be nearly as efficient as the best linear unbiased estimators and the maximum likelihood estimators. Next, we derive the asymptotic variances and covariance of these estimators in terms of the first two single moments and the product moments of order statistics from the standard extreme value distribution. Finally, we present an example in order to illustrate all the methods of estimation of parameters discussed in this paper.     

On second-order improved estimation of a gamma scale parameter

This paper concludes our comprehensive study on point estimation of model parameters of a gamma distribution from a second-order decision theoretic point of view. It should be noted that efficient estimation of gamma model parameters for samples ‘not large’ is a challenging task since the exact sampling distributions of the maximum likelihood estimators and its variants are not known. Estimation of a gamma scale parameter has received less attention from the earlier researchers compared to shape parameter estimation. What we have observed here is that improved estimation of the shape parameter does not necessarily lead to improved scale estimation if a natural moment condition (which is also the maximum likelihood restriction) is satisfied. Therefore, this work deals with the gamma scale parameter estimation as a separate new problem, not as a by-product of the shape parameter estimation, and studies several estimators in terms of second-order risk.     

Parameter estimation of state space models for univariate observations

This paper contributes to the problem of estimation of state space model parameters by proposing estimators for the mean, the autoregressive parameters and the noise variances which, contrarily to maximum likelihood, may be calculated without assuming any specific distribution for the errors. The estimators suggested widen the scope of the application of the generalized method of moments to some heteroscedastic models, as in the case of state-space models with varying coefficients, and give sufficient conditions for their consistency. The paper includes a simulation study comparing the proposed estimators with maximum likelihood estimators. Finally, these methods are applied to the calibration of the meteorological radar and estimation of area rainfall.     

Maximum likelihood vs. maximum goodness of fit estimation of the three-parameter Weibull distribution

The use of statistics based on the empirical distribution function is analysed for estimation of the scale, shape, and location parameters of the three-parameter Weibull distribution. The resulting maximum goodness of fit (MGF) estimators are compared with their maximum likelihood counterparts. In addition to the Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling statistics, some related empirical distribution function statistics using different weight functions are considered. The results show that the MGF estimators of the scale and shape parameters are usually more efficient than the maximum likelihood estimators when the shape parameter is smaller than 2, particularly if the sample size is large.     

Estimation of parameters and reliability from generalized life models

Simultaneous estimation of parameters with p (≥ 2) components, where each component has a generalized life distribution, is considered under a sum of squared error loss function. Improved estimators are obtained which dominate the maximum likelihood and the niinimum mean square estimators. Robustness of the improved estimators is shown even when the component distributions are dependent. The result is extended to the estimation of the system reliability when the components are connected in series. Several numerical studies are performed to demonstrate the risk improvement and the Pitman closeness of the new estimators.     

An extension of the half-normal distribution

In this paper we introduce a new distribution, namely, the slashed half-normal distribution and it can be seen as an extension of the half-normal distribution. It is shown that the resulting distribution has more kurtosis than the ordinary half-normal distribution. Moments and some properties are derived for the new distribution. Moment estimators and maximum likelihood estimators can computed using numerical procedures. Results of two real data application are reported where model fitting is implemented by using maximum likelihood estimation. The applications illustrate the better performance of the new distribution.     

Estimation of the parameters of the gamma distribution by means of a maximal invariant

The use of a scale invariance criterion allows estimation of the shape parameter of the two parameter gamma distribution without estimating the scale parameter. Simulation experiments are used to show that the resulting estimators of both parameters are better than the usual maximum likelihood estimators in terms of both bias and mean square error. Approximately unbiased versions of the maximal invariant based estimators are derived and are shown to be as good as approximately unbiased versions of the usual maximum likelihood estimators     

Maximum-likelihood and closed-form estimators of epidemiologic measures under misclassification

There is a large literature on estimation under misclassification. The present paper reviews epidemiologic inference under misclassification in the multiway contingency-table setting, and addresses a few controversial issues. In the 1990s, claims of inefficiency of early closed-form estimators of odds ratios under misclassification arose from misapplication of the estimators to studies with internal validation. In reality, these estimators are maximum likelihood (ML) and hence efficient under the external-validation assumptions used for their derivation. For the internal-validation case, a new closed-form estimator is derived that incorporates the nondifferentiality constraint into the predictive-value (“direct” or “inverse-matrix”) estimator. Results are presented in a general framework that applies to misclassification in models for multiway tables, and that allows the target parameter to be any measure of association or effect.     

On the Estimation of Parameters of Kumaraswamy-G Distributions

Cordeiro and de Castro proposed a new family of generalized distributions based on the Kumaraswamy distribution (denoted as Kw-G). Nadarajah et al. showed that the density function of the new family of distributions can be expressed as a linear combination of the density of exponentiated family of distributions. They derived some properties of Kw-G distributions and discussed estimation of parameters using the maximum likelihood (ML) method. Cheng and Amin and Ranneby introduced a new method of estimating parameters based on Kullback–Leibler divergence (the maximum spacing (MSP) method). In this article, the estimates of parameters of Kw-G distributions are obtained using the MSP method. For some special Kw-G distributions, the new estimators are compared with ML estimators. It is shown by simulations and a real data application that MSP estimators have better properties than ML estimators.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

Extreme value index estimator using maximum likelihood and moment estimation

ABSTRACT

When a distribution function is in the max domain of attraction of an extreme value distribution, its tail can be well approximated by a generalized Pareto distribution. Based on this fact we use a moment estimation idea to propose an adapted maximum likelihood estimator for the extreme value index, which can be understood as a combination of the maximum likelihood estimation and moment estimation. Under certain regularity conditions, we derive the asymptotic normality of the new estimator and investigate its finite sample behavior by comparing with several classical or competitive estimators. A simulation study shows that the new estimator is competitive with other estimators in view of average bias, average MSE, and coefficient of variance of the new device for the optimal selection of the threshold.     

A new bivariate distribution with weighted exponential marginals and its multivariate generalization

Gupta and Kundu (Statistics 43:621–643, 2009) recently introduced a new class of weighted exponential distribution. It is observed that the proposed weighted exponential distribution is very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the weighted exponential marginals. Different properties of this new bivariate distribution have been investigated. This new family has three unknown parameters, and it is observed that the maximum likelihood estimators of the unknown parameters can be obtained by solving a one-dimensional optimization procedure. We obtain the asymptotic distribution of the maximum likelihood estimators. Small simulation experiments have been performed to see the behavior of the maximum likelihood estimators, and one data analysis has been presented for illustrative purposes. Finally we discuss the multivariate generalization of the proposed model.     

Statistical Inference of Type-II Progressively Hybrid Censored Data with Weibull Lifetimes

In this article, we discuss the maximum likelihood estimators and approximate maximum likelihood estimators of the parameters of the Weibull distribution with two different progressively hybrid censoring schemes. We also present the associated expressions of the expected total test time and the expected effective sample size which will be useful for experimental planning purpose. Finally, the efficiency of the point estimation of the parameters based on the two progressive hybrid censoring schemes are compared and the merits of each censoring scheme are discussed.     

Estimating the parameters of a doubly truncated normal distribution

This paper deals with the maximum likelihood estimation of parameters for a doubly truncated normal distribution when the truncation points are known. We prove, in this case, that the MLEs are nonexistent (become infinite) with positive probability. For estimators that exist with probability one, the class of Bayes modal estimators or modified maximum likelihood estimators is explored. Another useful estimating procedure, called mixed estimation, is proposed. Simulations compare the behavior of the MLEs, the modified MLEs, and the mixed estimators which reveal that the MLE, in addition to being nonexistent with positive probability, behaves poorly near the upper boundary of the interval of its existence. The modified MLEs and the mixed estimators are seen to be remarkably better than the MLE     

A novel weighted likelihood estimation with empirical Bayes flavor

We propose a novel approach to estimation, where a set of estimators of a parameter is combined into a weighted average to produce the final estimator. The weights are chosen to be proportional to the likelihood evaluated at the estimators. We investigate the method for a set of estimators obtained by using the maximum likelihood principle applied to each individual observation. The method can be viewed as a Bayesian approach with a data-driven prior distribution. We provide several examples illustrating the new method and argue for its consistency, asymptotic normality, and efficiency. We also conduct simulation studies to assess the performance of the estimators. This straightforward methodology produces consistent estimators comparable with those obtained by the maximum likelihood method. The method also approximates the distribution of the estimator through the “posterior” distribution.     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.