On the calculations of the maximum likelihood estimates for the polynomial spline regression model with unknown knots and AR(1) errors

Asymptotic distributions of the maximum likelihood estimators of the regression coefficients and knot points for the polynomial spline regression models with unknown knots and AR(1) errors have been derived by Chan (1989). Chan showed that under some mild conditions the maximum likelihood estimators, after suitable standardization, asymptotically follow normal distributions as n diverges to infinity. For the calculations of the maximum likelihood estimators, iterative methods must be applied. But this is not easy to implement for the model considered. In this paper, we suggested an alternative method to compute the estimates of the regression parameters and knots. It is shown that the estimates obtained by this method are asymptotically equivalent to the maximum likelihood estimates considered by Chan.     

Estimation for u-shaped beta distributions: minimum hellinger distance and related methods

We compare minimum Hellinger distance and minimum Heiiinger disparity estimates for U-shaped beta distributions. Given suitable density estimates, both methods are known to be asymptotically efficient when the data come from the assumed model family, and robust to small perturbations from the model family. Most implementations use kernel density estimates, which may not be appropriate for U-shaped distributions. We compare fixed binwidth histograms, percentile mesh histograms, and averaged shifted histograms. Minimum disparity estimates are less sensitive to the choice of density estimate than are minimum distance estimates, and the percentile mesh histogram gives the best results for both minimum distance and minimum disparity estimates. Minimum distance estimates are biased and a bias-corrected method is proposed. Minimum disparity estimates and bias-corrected minimum distance estimates are comparable to maximum likelihood estimates when the model holds, and give better results than either method of moments or maximum likelihood when the data are discretized or contaminated, Although our re¬sults are for the beta density, the implementations are easily modified for other U-shaped distributions such as the Dirkhlet or normal generated distribution.     

Estimation of parameters in mixtures of inverse gaussian distributions

Iterative procedures are developed for finding maximum likelihood estimates of the parameters of mixtures of two inverse Gaussian distributions. The performance of the estimates based on small samples is studied by simulation experiments. Asymptotic efficiencies relative to estimates based on completely classified samples are also evaluated. Unless the components of the populations are widely separated, the maximum likelihood estimates perform poorly.     

Estimation of exponential regression parameters using binary data

Exponential regression model is important in analyzing data from heterogeneous populations. In this paper we propose a simple method to estimate the regression parameters using binary data. Under certain design distributions, including ellipticaily symmetric distributions, for the explanatory variables, the estimators are shown to be consistent and asymptotically normal when sample size is large. For finite samples, the new estimates were shown to behave reasonably well. They are competitive with the maximum likelihood estimates and more importantly, according to our simulation results, the cost of CPU time for computing new estimates is only 1/7 of that required for computing the usual maximum likelihood estimates. We expect the savings in CPU time would be more dramatic with larger dimension of the regression parameter space.     

Inference in multivariate linear regression models with elliptically distributed errors

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the regression coefficients.     

The competing risks analysis for parallel and series systems using Type-II progressive censoring

Abstract

Recently, the study of the lifetime of systems in reliability and survival analysis in the presence of several causes of failure (competing risks) has attracted attention in the literature. In this paper, series and parallel systems with exponential lifetime for each item of the system are considered. Several causes of failure independently affect lifetime distributions and observations of failure times of the systems are considered under progressive Type-II censored scheme. For series systems, the maximum likelihood estimates of parameters are computed and confidence intervals for parameters of the model are obtained using Fisher information matrix. For parallel systems, the generalized EM algorithm which uses the Newton-Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. Also, the standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. The simulation study confirms the good performance of the introduced approach.     

Semiparametric tests for change-points with epidemic alternatives

The empirical likelihood ratio-based semiparametric tests of change-points with epidemic alternatives are constructed and are proved to have the same limiting null distributions as some well-known tests. The maximum empirical likelihood estimates of the change-points and the epidemic duration are shown to be consistent. Data-based model tests are also provided. The method is applied to a stock market price data and the Nile river data.     

Bayesian Estimation of the Parameters of Bivariate Exponential Distributions

In this article, we develop an empirical Bayesian approach for the Bayesian estimation of parameters in four bivariate exponential (BVE) distributions. We have opted for gamma distribution as a prior for the parameters of the model in which the hyper parameters have been estimated based on the method of moments and maximum likelihood estimates (MLEs). A simulation study was conducted to compute empirical Bayesian estimates of the parameters and their standard errors. We use moment estimators or MLEs to estimate the hyper parameters of the prior distributions. Furthermore, we compare the posterior mode of parameters obtained by different prior distributions and the Bayesian estimates based on gamma priors are very close to the true values as compared to improper priors. We use MCMC method to obtain the posterior mean and compared the same using the improper priors and the classical estimates, MLEs.     

On compounded geometric distributions and their applications

Here, we introduce two-parameter compounded geometric distributions with monotone failure rates. These distributions are derived by compounding geometric distribution and zero-truncated Poisson distribution. Some statistical and reliability properties of the distributions are investigated. Parameters of the proposed distributions are estimated by the maximum likelihood method as well as through the minimum distance method of estimation. Performance of the estimates by both the methods of estimation is compared based on Monte Carlo simulations. An illustration with Air Crash casualties demonstrates that the distributions can be considered as a suitable model under several real situations.     

Estimating the Burr XII Parameters in Constant-Stress Partially Accelerated Life Tests Under Multiple Censored Data

In this article, we present the performance of the maximum likelihood estimates of the Burr XII parameters for constant-stress partially accelerated life tests under multiple censored data. Two maximum likelihood estimation methods are considered. One method is based on observed-data likelihood function and the maximum likelihood estimates are obtained by using the quasi-Newton algorithm. The other method is based on complete-data likelihood function and the maximum likelihood estimates are derived by using the expectation-maximization (EM) algorithm. The variance–covariance matrices are derived to construct the confidence intervals of the parameters. The performance of these two algorithms is compared with each other by a simulation study. The simulation results show that the maximum likelihood estimation via the EM algorithm outperforms the quasi-Newton algorithm in terms of the absolute relative bias, the bias, the root mean square error and the coverage rate. Finally, a numerical example is given to illustrate the performance of the proposed methods.     

Alternative Estimators for Randomized Response Techniques in Multi-Character Surveys

Tail estimates are developed for power law probability distributions with exponential tempering, using a conditional maximum likelihood approach based on the upper-order statistics. Tempered power law distributions are intermediate between heavy power-law tails and Laplace or exponential tails, and are sometimes called “semi-heavy” tailed distributions. The estimation method is demonstrated on simulated data from a tempered stable distribution, and for several data sets from geophysics and finance that show a power law probability tail with some tempering.     

Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model.     

Baysian estimation of the parameters in two modified poission distributions

Posterior distributions and moment are derived for the generalized Poisson and the excess zeroes Poisson distributions.Three examples are presented where both maximum likelihood and posterior estimates are given.     

A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

PITMAN NEARNESS COMPARISONS OF ESTIMATES OF TWO ORDERED NORMAL MEANS

Maximum likelihood estimates of ordered means of two normal distributions having common variance have been shown to be better than the usual maximum likelihood estimates (i.e. corresponding sample means) with respect to Pitman Nearness criterion. The maximum likelihood estimate of common variance taking into consideration the order restriction of the means is shown to have smaller mean square error than the unrestricted maximum likelihood estimate of the common variance. These two estimators have also been compared with respect to Pitman Nearness criterion.     

Fiducial inference for Birnbaum-Saunders distribution

In this paper, we consider fiducial inference for the unknown parameters of the Birnbaum-Saunders distribution. Two generalized fiducial distributions of the parameters are obtained. One is based on the inverse of the structural equation, and the fiducial estimates of the parameters are obtained by a simulation method. The other is based on the method of [Hannig J. Generalized fiducial inference via discretization. Stat. Sinica. 2013;23:489–514], then we use adaptive rejection Metropolis sampling to get the fiducial estimates. We compare the fiducial estimates with the maximum likelihood estimates and Bayesian estimates by simulations. Two real data sets are analysed for illustration.     

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.     

On the simple step-stress model for two-parameter exponential distribution

In this paper, we consider the simple step-stress model for a two-parameter exponential distribution, when both the parameters are unknown and the data are Type-II censored. It is assumed that under two different stress levels, the scale parameter only changes but the location parameter remains unchanged. It is observed that the maximum likelihood estimators do not always exist. We obtain the maximum likelihood estimates of the unknown parameters whenever they exist. We provide the exact conditional distributions of the maximum likelihood estimators of the scale parameters. Since the construction of the exact confidence intervals is very difficult from the conditional distributions, we propose to use the observed Fisher Information matrix for this purpose. We have suggested to use the bootstrap method for constructing confidence intervals. Bayes estimates and associated credible intervals are obtained using the importance sampling technique. Extensive simulations are performed to compare the performances of the different confidence and credible intervals in terms of their coverage percentages and average lengths. The performances of the bootstrap confidence intervals are quite satisfactory even for small sample sizes.     

On parameter estimation of software reliability models

Most software reliability models use the maximum likelihood method to estimate the parameters of the model. The maximum likelihood method assumes that the inter-failure time distributions contribute equally to the likelihood function. Since software reliability is expected to exhibit growth, a weighted likelihood function that gives higher weights to latter inter-failure times compared to earlier ones is suggested. The accuracy of the predictions obtained using the weighted likelihood method is compared with the predictions obtained when the parameters are estimated by the maximum likelihood method on three real datasets. A simulation study is also conducted.     

Consistency of Generalized Maximum Spacing Estimates

General methods for the estimation of distributions can be derived from approximations of certain information measures. For example, both the maximum likelihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kullback–Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distributions is obtained from an approximation based on spacings of an information measure, were used by Ranneby & Ekstrom (1997) (using simple spacings) and Ekstrom (1997b) (using high order spacings) to obtain a class of methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao & Hahn (1999) for the MSP method. In particular, it will be proved that GMSP methods give consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.