A note on comparison and improvement of estimators based on likelihood

The problem of estimation of a parameter of interest in the presence of a nuisance parameter, which is either location or scale, is considered. Three estimators are taken into account: usual maximum likelihood (ML) estimator, maximum integrated likelihood estimator and the bias-corrected ML estimator. General results on comparison of these estimators w.r.t. the second-order risk based on the mean-squared error are obtained. Possible improvements of basic estimators via the notion of admissibility and methodology given in Ghosh and Sinha [A necessary and sufficient condition for second order admissibility with applications to Berkson's bioassay problem. Ann Stat. 1981;9(6):1334–1338] are considered. In the recent paper by Tanaka et al. [On improved estimation of a gamma shape parameter. Statistics. 2014; doi:10.1080/02331888.2014.915842], this problem was considered for estimating the shape parameter of gamma distribution. Here, we perform more accurate comparison of estimators for this case as well as for some other cases.     

Pooled estimators for the shape parameter of the three parameter gamma distribution

The aim of the paper is to study the pooled estimator of the shape parameter of the three parameter gamma distribution when k independent samples are available. Sufficient conditions for the existence of the pooled estimator are given and the small as well as the large sample properties are studied. The harmonic mean of the k estimators of the independent samples is proposed in the place of the pooled estimator, in the case in which the latter does not exist.     

Linear approximate ML estimation in scaled Type I generalized logistic distribution based on Type-II censored samples

The scaled (two-parameter) Type I generalized logistic distribution (GLD) is considered with the known shape parameter. The ML method does not yield an explicit estimator for the scale parameter even in complete samples. In this article, we therefore construct a new linear estimator for scale parameter, based on complete and doubly Type-II censored samples, by making linear approximations to the intractable terms of the likelihood equation using least-squares (LS) method, a new approach of linearization. We call this as linear approximate maximum likelihood estimator (LAMLE). We also construct LAMLE based on Taylor series method of linear approximation and found that this estimator is slightly biased than that based on the LS method. A Monte Carlo simulation is used to investigate the performance of LAMLE and found that it is almost as efficient as MLE, though biased than MLE. We also compare unbiased LAMLE with BLUE based on the exact variances of the estimators and interestingly this new unbiased LAMLE is found just as efficient as the BLUE in both complete and Type-II censored samples. Since MLE is known as asymptotically unbiased, in large samples we compare unbiased LAMLE with MLE and found that this estimator is almost as efficient as MLE. We have also discussed interval estimation of the scale parameter from complete and Type-II censored samples. Finally, we present some numerical examples to illustrate the construction of the new estimators developed here.     

Pivotal inference for the inverse Rayleigh distribution based on general progressively Type-II censored samples

In this paper, we consider the problem of estimating the scale parameter of the inverse Rayleigh distribution based on general progressively Type-II censored samples and progressively Type-II censored samples. The pivotal quantity method is used to derive the estimator of the scale parameter. Besides, considering that the maximum likelihood estimator is tough to obtain for this distribution, we derive an explicit estimator of the scale parameter by approximating the likelihood equation with Taylor expansion. The interval estimation is also studied based on pivotal inference. Then we conduct Monte Carlo simulations and compare the performance of different estimators. We demonstrate that the pivotal inference is simpler and more effective. The further application of the pivotal quantity method is also discussed theoretically. Finally, two real data sets are analyzed using our methods.     

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLE's are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters.

Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.     

Estimation of common location parameter of several heterogeneous exponential populations based on generalized order statistics

In this article, several independent populations following exponential distribution with common location parameter and unknown and unequal scale parameters are considered. From these populations, several independent samples of generalized order statistics (gos) are drawn. Under the setup of gos, the problem of estimation of common location parameter is discussed and various estimators of common location parameter are derived. The authors obtained maximum likelihood estimator (MLE), modified MLE and uniformly minimum variance unbiased estimator of common location parameter. Furthermore, under scaled-squared error loss function, a general inadmissibility result of invariant estimator is proposed. The derived results are further reduced for upper record values which is a special case of gos. Finally, simulation study and real life example are reported to show the performances of various competing estimators in terms of percentage risk improvement.     

Approximate MLE for the scale parameter of the generalized exponential distribution under random censoring

In this paper, we consider the maximum likelihood estimator (MLE) of the scale parameter of the generalized exponential (GE) distribution based on a random censoring model. We assume the censoring distribution also follows a GE distribution. Since the estimator does not provide an explicit solution, we propose a simple method of deriving an explicit estimator by approximating the likelihood function. In order to compare the performance of the estimators, Monte Carlo simulation is conducted. The results show that the MLE and the approximate MLE are almost identical in terms of bias and variance.     

The conditional MLE in the two-parameter geometric distribution and its competitors

The conditional maximum likelihood estimator of the shape parameter in the two-parameter geometric distribution is introduced and explored. The estimator is compared with the unconditional maximum likelihood estimator and the uniformly minimum variance unbiased estimator.     

Estimation of the parameters of a truncated gamma distribution

This paper deals with the estimation of the parameters of a truncated gamma distribution over (0,τ), where τ is assumed to be a real number. We obtain a necessary and sufficient condition for the existence of the maximum likelihood estimator(MLE). The probability of nonexistence of MLE is observed to be positive. A simulation study indicates that the modified maximum likelihood estimator and the mixed estimator, which exist with probability one,are to be preferred over MLE. The bias, the mean square error, and the probability of nearness form a basis of our simulation study.     

Estimation of P(Y < X) for Weibull Distribution Under Progressive Type-II Censoring

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.     

Confidence Intervals for the Current Status Model

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.     

Inference of R = P[X < Y] for skew normal distribution

This article studies the estimation of R = P[X < Y] when X and Y are two independent skew normal distribution with different parameters. When the scale parameter is unknown, the maximum likelihood estimator of R is proposed. The maximum likelihood estimator, uniformly minimum variance unbiased estimator, Bayes estimation, and confidence interval of R are obtained when the common scale parameter is known. In the general case, the maximum likelihood estimator of R is also discussed. To compare the different proposed methods, Monte Carlo simulations are performed. At last, the analysis of a real dataset has been presented for illustrative purposes too.     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

Exact confidence intervals for the shape parameter of the gamma distribution

In this paper exact confidence intervals (CIs) for the shape parameter of the gamma distribution are constructed using the method of Bølviken and Skovlund [Confidence intervals from Monte Carlo tests. J Amer Statist Assoc. 1996;91:1071–1078]. The CIs which are based on the maximum likelihood estimator or the moment estimator are compared to bootstrap CIs via a simulation study.     

Objective Bayesian Analysis of Skew‐t Distributions

Abstract. We study the Jeffreys prior and its properties for the shape parameter of univariate skew‐t distributions with linear and nonlinear Student's t skewing functions. In both cases, we show that the resulting priors for the shape parameter are symmetric around zero and proper. Moreover, we propose a Student's t approximation of the Jeffreys prior that makes an objective Bayesian analysis easy to perform. We carry out a Monte Carlo simulation study that demonstrates an overall better behaviour of the maximum a posteriori estimator compared with the maximum likelihood estimator. We also compare the frequentist coverage of the credible intervals based on the Jeffreys prior and its approximation and show that they are similar. We further discuss location‐scale models under scale mixtures of skew‐normal distributions and show some conditions for the existence of the posterior distribution and its moments. Finally, we present three numerical examples to illustrate the implications of our results on inference for skew‐t distributions.     

Efficient estimation of the PDF and the CDF of the inverse Rayleigh distribution

In this paper, we consider the estimation of the probability density function and the cumulative distribution function of the inverse Rayleigh distribution. In this regard, the following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood (ML) estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the mean integrated squared error. As the result of simulation studies and real data applications indicate, when the sample size is not very small the ML estimator performs better than the others.     

Umvu estimation for the cumulative hazard function in a pareto distribution of the first kind

The uniformly minimum variance unbiased estimator of the cumulative hazard function in the Pareto distribution of the first kind is derived. The variance of the estimator is also obtained in an analytic form, and for some cases its values are compared numerically with mean square errors of the maximum likelihood estimator.     

ASYMPTOTIC LIKELIHOOD APPROXIMATIONS USING A PARTIAL LAPLACE APPROXIMATION

Elimination of a nuisance variable is often non‐trivial and may involve the evaluation of an intractable integral. One approach to evaluate these integrals is to use the Laplace approximation. This paper concentrates on a new approximation, called the partial Laplace approximation, that is useful when the integrand can be partitioned into two multiplicative disjoint functions. The technique is applied to the linear mixed model and shows that the approximate likelihood obtained can be partitioned to provide a conditional likelihood for the location parameters and a marginal likelihood for the scale parameters equivalent to restricted maximum likelihood (REML). Similarly, the partial Laplace approximation is applied to the t‐distribution to obtain an approximate REML for the scale parameter. A simulation study reveals that, in comparison to maximum likelihood, the scale parameter estimates of the t‐distribution obtained from the approximate REML show reduced bias.     

On the performance of some new Liu parameters for the gamma regression model

The maximum likelihood (ML) method is used to estimate the unknown Gamma regression (GR) coefficients. In the presence of multicollinearity, the variance of the ML method becomes overstated and the inference based on the ML method may not be trustworthy. To combat multicollinearity, the Liu estimator has been used. In this estimator, estimation of the Liu parameter d is an important problem. A few estimation methods are available in the literature for estimating such a parameter. This study has considered some of these methods and also proposed some new methods for estimation of the d. The Monte Carlo simulation study has been conducted to assess the performance of the proposed methods where the mean squared error (MSE) is considered as a performance criterion. Based on the Monte Carlo simulation and application results, it is shown that the Liu estimator is always superior to the ML and recommendation about which best Liu parameter should be used in the Liu estimator for the GR model is given.     

Improved maximum-likelihood estimation of the shape parameter in the Nakagami distribution

We develop and evaluate analytic and bootstrap bias-corrected maximum-likelihood estimators for the shape parameter in the Nakagami distribution. This distribution is widely used in a variety of disciplines, and the corresponding estimator of its scale parameter is trivially unbiased. We find that both ‘corrective’ and ‘preventive’ analytic approaches to eliminating the bias, to O(n ^?2), are equally, and extremely, effective and simple to implement. As a bonus, the sizeable reduction in bias comes with a small reduction in the mean-squared error. Overall, we prefer analytic bias corrections in the case of this estimator. This preference is based on the relative computational costs and the magnitudes of the bias reductions that can be achieved in each case. Our results are illustrated with two real-data applications, including the one which provides the first application of the Nakagami distribution to data for ocean wave heights.