Penalized maximum likelihood estimation in the modified extended Weibull distribution

We address the issue of performing inference on the parameters that index the modified extended Weibull (MEW) distribution. We show that numerical maximization of the MEW log-likelihood function can be problematic. It is even possible to encounter maximum likelihood estimates that are not finite, that is, it is possible to encounter monotonic likelihood functions. We consider different penalization schemes to improve maximum likelihood point estimation. A penalization scheme based on the Jeffreys’ invariant prior is shown to be particularly useful. Simulation results on point estimation, interval estimation, and hypothesis testing inference are presented. Two empirical applications are presented and discussed.     

Towards a unified definition of maximum likelihood

A unified definition of maximum likelihood (ml) is given. It is based on a pairwise comparison of probability measures near the observed data point. This definition does not suffer from the usual inadequacies of earlier definitions, i.e., it does not depend on the choice of a density version in the dominated case. The definition covers the undominated case as well, i.e., it provides a consistent approach to nonparametric ml problems, which heretofore have been solved on a more less ad hoc basis. It is shown that the new ml definition is a true extension of the classical ml approach, as it is practiced in the dominated case. Hence the classical methodology can simply be subsumed. Parametric and nonparametric examples are discussed.     

Bias-reduced maximum likelihood estimation of the zero-inflated Poisson distribution

Using the techniques developed by Subrahmaniam and Ching’anda (1978), we study the robustness to nonnormality of the linear discriminant functions. It is seen that the LDF procedure is quite robust against the likelihood ratio rule. The latter yields in all cases much smaller overall error rates; however, the disparity between the error rates of the LDF and LR procedures is not large enough to warrant the recommendation to use the more complicated LR procedure.     

Product limit estimates: a generalized maximum likelihood study

The generalized maximum likelihood estimate (GMLE) assumptions are studied for four product-limit estimates (PLE): Censoring PLE (Kaplan-Meier estimate), truncation PLE, censoring-truncation PLE, and the degenerated PLE - the empirical distribution function. This paper shows that all the PLE's are also the GMLE's even if they are derived from partial likelihoods by natural parameterization techniques. However, a counter example is given to show that Kiefer Wolfowitz's assumption (1956) for consistency of GMLE can hardly be satisfied for un-dominated case.     

On the existence of maximum likelihood estimates for weighted logistic regression

The problems of existence and uniqueness of maximum likelihood estimates for logistic regression were completely solved by Silvapulle in 1981 and Albert and Anderson in 1984. In this paper, we extend the well-known results by Silvapulle and by Albert and Anderson to weighted logistic regression. We analytically prove the equivalence between the overlap condition used by Albert and Anderson and that used by Silvapulle. We show that the maximum likelihood estimate of weighted logistic regression does not exist if there is a complete separation or a quasicomplete separation of the data points, and exists and is unique if there is an overlap of data points. Our proofs and results for weighted logistic apply to unweighted logistic regression.     

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.     

Skewness of maximum likelihood estimators in dispersion models

We introduce the dispersion models with a regression structure to extend the generalized linear models, the exponential family nonlinear models (Cordeiro and Paula, 1989) and the proper dispersion models (Jørgensen, 1997a). We provide a matrix expression for the skewness of the maximum likelihood estimators of the regression parameters in dispersion models. The formula is suitable for computer implementation and can be applied for several important submodels discussed in the literature. Expressions for the skewness of the maximum likelihood estimators of the precision and dispersion parameters are also derived. In particular, our results extend previous formulas obtained by Cordeiro and Cordeiro (2001) and Cavalcanti et al. (2009). A simulation study is performed to show the practice importance of our results.     

Bias reduction for the maximum likelihood estimator of the doubly-truncated Poisson distribution

ABSTRACT

We derive an analytic expression for the bias of the maximum likelihood estimator of the parameter in a doubly-truncated Poisson distribution, which proves highly effective as a means of bias correction. For smaller sample sizes, our method outperforms the alternative of bias correction via the parametric bootstrap. Bias is of little concern in the positive Poisson distribution, the most common form of truncation in the applied literature. Bias appears to be the most severe in the doubly-truncated Poisson distribution, when the mean of the distribution is close to the right (upper) truncation.     

Bias-Corrected maximum likelihood estimation of the parameters of the weighted Lindley distribution

The two-parameter weighted Lindley distribution is useful for modeling survival data, whereas its maximum likelihood estimators (MLEs) are biased in finite samples. This motivates us to construct nearly unbiased estimators for the unknown parameters. We adopt a “corrective” approach to derive modified MLEs that are bias-free to second order. We also consider an alternative bias-correction mechanism based on Efron’s bootstrap resampling. Monte Carlo simulations are conducted to compare the performance between the proposed and two previous methods in the literature. The numerical evidence shows that the bias-corrected estimators are extremely accurate even for very small sample sizes and are superior than the previous estimators in terms of biases and root mean squared errors. Finally, applications to two real datasets are presented for illustrative purposes.     

On the consistency of the maximum spacing method

The main result of this paper is a consistency theorem for the maximum spacing method, a general method of estimating parameters in continuous univariate distributions, introduced by Cheng and Amin (J. Roy. Statist. Soc. Ser. A 45 (1983) 394–403) and independently by Ranneby (Scand. J. Statist. 11 (1984) 93–112). This main result generalizes a theorem of Ranneby (Scand. J. Statist. 11 (1984) 93–112). Also, some examples are given, which shows that this estimation method works also in cases where the maximum likelihood method breaks down.     

Local maximum likelihood estimation and inference

Local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique. It extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces. Associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment. This paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation. The approach is then applied to least squares nonparametric regression and to nonparametric logistic regression. Our experiences in these two settings show that the general idea outlined here is powerful and encouraging.     

The sliced inverse regression algorithm as a maximum likelihood procedure

It is shown that the sliced inverse regression procedure proposed by Li corresponds to the maximum likelihood estimate where the observations in each slice are samples of multivariate normal distributions with means in an affine manifold.     

A note on bias reduction of maximum likelihood estimates for the scalar skew t distribution

The skew t distribution is a flexible parametric family to fit data, because it includes parameters that let us regulate skewness and kurtosis. A problem with this distribution is that, for moderate sample sizes, the maximum likelihood estimator of the shape parameter is infinite with positive probability. In order to try to solve this problem, Sartori (2006) has proposed using a modified score function as an estimating equation for the shape parameter. In this note we prove that the resulting modified maximum likelihood estimator is always finite, considering the degrees of freedom as known and greater than or equal to 2.     

Analytic bias correction for maximum likelihood estimators when the bias function is non-constant

Recently, many articles have obtained analytical expressions for the biases of various maximum likelihood estimators, despite their lack of closed-form solution. These bias expressions have provided an attractive alternative to the bootstrap. Unless the bias function is “flat,” however, the expressions are being evaluated at the wrong point(s). We propose an “improved” analytical bias-adjusted estimator, in which the bias expression is evaluated at a more appropriate point (at the bias adjusted estimator itself). Simulations illustrate that the improved analytical bias-adjusted estimator can eliminate significantly more bias than the simple estimator, which has been well established in the literature.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

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.     

The maximum likelihood estimators in the growth curve model with serial covariance structure

The maximum likelihood estimators of unknown parameters in the growth curve model with serial covariance structure under some conditions are derived in the paper.     

Modern likelihood inference for the maximum/minimum of a bivariate normal vector

ABSTRACT

We consider the use of modern likelihood asymptotics in the construction of confidence intervals for the parameter which determines the skewness of the distribution of the maximum/minimum of an exchangeable bivariate normal random vector. Simulation studies were conducted to investigate the accuracy of the proposed methods and to compare them to available alternatives. Accuracy is evaluated in terms of both coverage probability and expected length of the interval. We furthermore illustrate the suitability of our proposals by means of two data sets, consisting of, respectively, measurements taken on the brains of 10 mono-zygotic twins and measurements of mineral content of bones in the dominant and non-dominant arms for 25 elderly women.     

More on the four-parameter kappa distribution

The generalized extreme-value has been the distribution of choice for modeling available maxima (or minima) data since theory has shown it to be the limiting form of the distribution of extremes. However, fits to finite samples are not always adequate. Hosking (1994) and Parida (1999) suggest the four-parameter Kappa distribution as an alternative. Hosking (1994) developed an L-moment procedure for estimation. Some compromises must be made in practice however, as seen in Parida (1999). L-moment estimators of the four-parameter Kappa distribution are not always computable nor feasible. A simulation study in this paper quantifies the extent of each problem. Maximum likelihood is investigated as an alternative method of estimation and a simulation study compares the performance of both methods of estimation. Finally, further benefits of maximum likelihood are shown when wind speeds From the Tropical Pacific are examined and the weekly maxima for 10 buoys in the area are analyzed.