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.     

Estimation and prediction of Marshall–Olkin extended exponential distribution under progressively type-II censored data

In this paper, we consider Marshall–Olkin extended exponential (MOEE) distribution which is capable of modelling various shapes of failure rates and aging criteria. The purpose of this paper is three fold. First, we derive the maximum likelihood estimators of the unknown parameters and the observed the Fisher information matrix from progressively type-II censored data. Next, the Bayes estimates are evaluated by applying Lindley’s approximation method and Markov Chain Monte Carlo method under the squared error loss function. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. We also compute 95% asymptotic confidence interval and symmetric credible interval along with the coverage probability. Third, we consider one-sample and two-sample prediction problems based on the observed sample and provide appropriate predictive intervals under classical as well as Bayesian framework. Finally, we analyse a real data set to illustrate the results derived.     

The complex Watson distribution and shape analysis

The complex Watson distribution is an important simple distribution on the complex sphere which is used in statistical shape analysis. We describe the density, obtain the integrating constant and provide large sample approximations. Maximum likelihood estimation and hypothesis testing procedures for one and two samples are described. The particular connection with shape analysis is discussed and we consider an application examining shape differences between normal and schizophrenic brains. We make some observations about Bayesian shape inference and finally we describe a more general rotationally symmetric family of distributions.     

Visualization and Bandwidth Matrix Choice

In this study we compare three estimators of the extreme value index: Pickands estimator, the moment estimator and a maximum likelihood estimator. The estimators are explored both theoretically and by Monte Carlo simulation. We obtain two estimators for large quantiles using Pickands and the maximum likelihood estimators. The latter and one based on the moment estimator are then compared through simulation.     

Constrained inference for generalized linear models with incomplete covariate data

Missing data are common in many experiments, including surveys, clinical trials, epidemiological studies, and environmental studies. Unconstrained likelihood inferences for generalized linear models (GLMs) with nonignorable missing covariates have been studied extensively in the literature. However, parameter orderings or constraints may occur naturally in practice, and thus the efficiency of a statistical method may be improved by incorporating parameter constraints into the likelihood function. In this paper, we consider constrained inference for analysing GLMs with nonignorable missing covariates under linear inequality constraints on the model parameters. Specifically, constrained maximum likelihood (ML) estimation is based on the gradient projection expectation maximization approach. Further, we investigate the asymptotic null distribution of the constrained likelihood ratio test (LRT). Simulations study the empirical properties of the constrained ML estimators and LRTs, which demonstrate improved precision of these constrained techniques. An application to contaminant levels in an environmental study is also presented.     

On the modified Burr IV model for hydrological events: development,properties, characterizations and applications

We introduce a new distribution for modeling extreme events about frequency analysis called modified Burr IV (MBIV) distribution. We derive the MBIV distribution on the basis of the generalized Pearson differential equation. The proposed model turns out to be flexible: its density function can be symmetrical, right-skewed, left-skewed, J and bimodal shaped. Its hazard rate has shapes such as bathtub and modified bathtub, increasing, decreasing, and increasing-decreasing-increasing. To show the importance of the MBIV distribution, we establish various mathematical properties such as random number generator, sub-models, moments related properties, inequality measures, reliability measures, uncertainty measures and characterizations. We utilize the maximum likelihood estimation technique to estimate the model parameters. We assess the behavior of the maximum likelihood estimators (MLEs) of the MBIV parameters via a simulation study. Five data sets related to frequency analysis are considered to elucidate the significance of the MBIV distribution. We show that the MBIV model is the best model to analyze data for hydrological events, motivating its high level of adaptability in the applied setting.KEYWORDS: Characterizations, elasticity function, moments, maximum likelihood estimator, reliability     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.     

Shape spaces for prealigned star-shaped objects—studying the growth of plants by principal components analysis

Summary.  We analyse the shapes of star-shaped objects which are prealigned. This is motivated from two examples studying the growth of leaves, and the temporal evolution of tree rings. In the latter case measurements were taken at fixed angles whereas in the former case the angles were free. Subsequently, this leads to different shape spaces, related to different concepts of size, for the analysis. Whereas several shape spaces already existed in the literature when the angles are fixed, a new shape space for free angles, called spherical shape space , needed to be introduced. We compare these different shape spaces both regarding their mathematical properties and in their adequacy to the data at hand; we then apply suitably defined principal component analysis on these. In both examples we find that the shapes evolve mainly along the first principal component during growth; this is the 'geodesic hypothesis' that was formulated by Le and Kume. Moreover, we could link change-points of this evolution to significant changes in environmental conditions.     

Time series models with asymmetric innovations

We consider AR(q) models in time series with asymmetric innovations represented by two families ofdistributions: (i) gamma with support IR : (0, ∞), and (ii) generalized logistic with support IR:(-∞,∞). Since the ML (maximum likelihood) estimators are intractable, we derive the MML (modified maximum likelihood) estimators of the parameters and show that they are remarkably efficient besides being easy to compute. We investigate the efficiency properties of the classical LS (least squares) estimators. Their efficiencies relative to the proposed MML estimators are very low.     

Testing uniformity based on new entropy estimators

In this paper, we first introduce new entropy estimators for distributions with known and bounded supports. Our estimators are obtained by using constrained maximum likelihood estimation of cumulative distribution function for absolutely continuous distributions with known and bounded supports. We prove the consistency of our estimators. Then, we propose uniformity tests based on the proposed entropy estimators and compare their powers with the powers of other tests of uniformity. Our simulation results show that the proposed entropy estimators perform well in estimating entropy and testing uniformity.     

Minimum distance estimators in extreme value distributions

We define minimum distance estimators for the parameters of the extreme value distribution G_o based on the Cramer-von-Mises distance. These estimators are rather robust and consistent, but asymptotically less efficient than the maximum likelihood estimators which are not robust. A small simulation study for finite sample size show that under G_o the finite efficiency of the minimum distance estimators is rather similar to the maximum likelihood ones.     

Analysis of left censored data from the generalized exponential distribution

The generalized exponential distribution proposed by Gupta and Kundu [Gupta, R.D and Kundu, D., 1999, Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173–188.] is an important lifetime distribution in survival analysis. In this paper, we consider the maximum likelihood estimation procedure of the parameters of the generalized exponential distribution when the data are left censored. We obtain the maximum likelihood estimators of the unknown para-meters and the Fisher information matrix. Simulation studies are carried out to observe the performance of the estimators in small sample.     

Bias-corrected maximum semiparametric likelihood estimation under logistic regression models based on case–control data

We consider the corrective approach (Theoretical Statistics, Chapman & Hall, London, 1974, p. 310) and preventive approach (Biometrica 80 (1993) 27) to bias reduction of maximum likelihood estimators under the logistic regression model based on case–control data. The proposed bias-corrected maximum likelihood estimators are based on the semiparametric profile log likelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. We show that the prospective and retrospective analyses on the basis of the corrective approach to bias reduction produce identical bias-corrected maximum likelihood estimators of the odds ratio parameter, but this does not hold when using the preventive approach unless the case and control sample sizes are identical. We present some results on simulation and on the analysis of two real data sets.     

A log-location-scale lifetime distribution with censored data: Regression modeling,residuals, and global influence

In survival analysis applications, the presence of failure rate functions with non monotone shapes is common. Therefore, models that can accommodate such different shapes are needed. In this article, we present a location regression model based on the complementary exponentiated exponential geometric distribution as an alternative to the usual bathtub, increasing, and decreasing failure rates in lifetime data. Assuming censored data, we consider the maximum likelihood inference for analysis, graphical verification for residuals, and test statistics for influential points.     

Empirical likelihood inference for partial functional linear model with missing responses

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.     

Discriminant analysis of survey data

We consider the problem of the effect of sample designs on discriminant analysis. The selection of the learning sample is assumed to depend on the population values of auxiliary variables. Under a superpopulation model with a multivariate normal distribution, unbiasedness and consistency are examined for the conventional estimators (derived under the assumptions of simple random sampling), maximum likelihood estimators, probability-weighted estimators and conditionally unbiased estimators of parameters. Four corresponding sampled linear discriminant functions are examined. The rates of misclassification of these four discriminant functions and the effect of sample design on these four rates of misclassification are discussed. The performances of these four discriminant functions are assessed in a simulation study.     

Inference for IZAWA’S Bivariate Gamma Distribution

Abstract

In this article, we aim to establish some theoretical properties of Izawa’s bivariate gamma distribution having equal shape parameters. First, we propose a procedure to obtain the maximum likelihood estimates and derive an expression for the Fisher information. Simulation studies illuminate the properties of maximum likelihood estimators. We also establish an asymptotic test for independence based on the limiting distribution of maximum likelihood estimators.     

Modelling and analysis of recall-based competing risks data

In this paper we consider the analysis of recall-based competing risks data. The chance of an individual recalling the exact time to event depends on the time of occurrence of the event and time of observation of the individual. In particular, it is assumed that the probability of recall depends on the time elapsed since the occurrence of an event. In this study we consider the likelihood-based inference for the analysis of recall-based competing risks data. The likelihood function is constructed by incorporating the information about the probability of recall. We consider the maximum likelihood estimation of parameters. Simulation studies are carried out to examine the performance of the estimators. The proposed estimation procedure is applied to a real life data set.     

LASSO and shrinkage estimation in Weibull censored regression models

In this paper we address the problem of estimating a vector of regression parameters in the Weibull censored regression model. Our main objective is to provide natural adaptive estimators that significantly improve upon the classical procedures in the situation where some of the predictors may or may not be associated with the response. In the context of two competing Weibull censored regression models (full model and candidate submodel), we consider an adaptive shrinkage estimation strategy that shrinks the full model maximum likelihood estimate in the direction of the submodel maximum likelihood estimate. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Further, we consider a LASSO type estimation strategy and compare the relative performance with the shrinkage estimators. Monte Carlo simulations reveal that when the true model is close to the candidate submodel, the shrinkage strategy performs better than the LASSO strategy when, and only when, there are many inactive predictors in the model. Shrinkage and LASSO strategies are applied to a real data set from Veteran's administration (VA) lung cancer study to illustrate the usefulness of the procedures in practice.     

Bias reduction in the estimation of a shape second-order parameter of a heavy-tailed model

In extreme value theory, the shape second-order parameter is a quite relevant parameter related to the speed of convergence of maximum values, linearly normalized, towards its limit law. The adequate estimation of this parameter is vital for improving the estimation of the extreme value index, the primary parameter in statistics of extremes. In this article, we consider a recent class of semi-parametric estimators of the shape second-order parameter for heavy right-tailed models. These estimators, based on the largest order statistics, depend on a real tuning parameter, which makes them highly flexible and possibly unbiased for several underlying models. In this article, we are interested in the adaptive choice of such tuning parameter and the number of top order statistics used in the estimation procedure. The performance of the methodology for the adaptive choice of parameters is evaluated through a Monte Carlo simulation study.