Empirical Likelihood Method for General Additive-Multiplicative Hazard Models

An empirical likelihood-based inferential procedure is developed for a class of general additive-multiplicative hazard models. The proposed log-empirical likelihood ratio test statistic for the parameter vector is shown to have a chi-squared limiting distribution. The result can be used to make inference about the entire parameter vector as well as any linear combination of it. The asymptotic power of the proposed test statistic under contiguous alternatives is discussed. The method is illustrated by extensive simulation studies and a real example.     

An Approximation for the Test of the Equality of the Smallest Eigenvalues of a Covariance Matrix

A limiting distribution of the likelihood ratio statistic for the test of the equality of the q smallest eigenvalues of a covariance matrix is obtained. This distribution can be used as an alternative to the chi-squared distribution which is usually used with this test. It is shown that this new method yields reasonable significance levels for those situations in which the chi-squared approximation is inadequate.     

A Discrete Analog of the Generalized Exponential Distribution

In this article, we attempt to introduce a discrete analog of the generalized exponential distribution of Gupta and Kundu (1999 Gupta , R. D. and Kundu , D. ( 1999 ). Generalized exponential distributions . Austral. NZ J. Statist. 41 ( 2 ): 173 – 188 .[Crossref], [Web of Science ®] , [Google Scholar]). This new discrete generalized exponential (DGE(α, p)) distribution can be viewed as another generalization of the geometric distribution and it is more flexible in data modeling. We shall first study some basic distributional and moment properties of this family of new distributions. Then, we will reveal their structural properties and applications and also investigate estimation of their parameters. Finally, we shall discuss their convolution properties and arrive at some characterizations in the special cases DGE(2, p) and DGE(3, p).     

Lack-of-Fit Tests for Generalized Linear Models via Splines

Cubic B-splines are used to estimate the nonparametric component of a semiparametric generalized linear model. A penalized log-likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one df. The smoothing parameter is determined by giving a specified value for its asymptotically expected value under the null hypothesis. A simulation study is conducted to evaluate its power performance; a real-life dataset is used to illustrate its practical use.     

A Multivariate Generalized Poisson Regression Model

A multivariate generalized Poisson regression model based on the multivariate generalized Poisson distribution is defined and studied. The regression model can be used to describe a count data with any type of dispersion. The model allows for both positive and negative correlation between any pair of the response variables. The parameters of the regression model are estimated by using the maximum likelihood method. Some test statistics are discussed, and two numerical data sets are used to illustrate the applications of the multivariate count data regression model.     

Assessing local influence for elliptical linear models under equality constraints

ABSTRACT

In this article, we study the local influence for the elliptical linear regression model under equality constraints. We first obtain the parameter estimators of this model using the penalized log-likelihood function and iterative techniques. Then we obtain the diagnostics under the perturbations of constant variance, responses, and explanatory variables in the spirit of Cook (1986 Cook, R.D. (1986). Assessment of local influence. J. Royal Stat. Soc. Ser. B 48(2):133–169. [Google Scholar]). Finally, a numerical example on the data set of the salinity of water is given to illustrate the theoretical results.     

Estimation of a change point in the variance function based on the χ2-distribution

ABSTRACT

Let us consider that the variance function or its νth derivative in a regression model has a change/discontinuity point at an unknown location. To use the local polynomial fits, the log-variance function which break the positivity is targeted. The location and the jump size of the change point are estimated based on a one-sided kernel-weighted local-likelihood function which is provided by the χ²-distribution. The whole structure of the log-variance function is then estimated using the data sets split by the estimated location. Asymptotic results of the proposed estimators are described. Numerical works demonstrate the performances of the methods with simulated and real examples.     

The logistic-X family of distributions and its applications

ABSTRACT

The logistic distribution has a prominent role in the theory and practice of statistics. We introduce a new family of continuous distributions generated from a logistic random variable called the logistic-X family. Its density function can be symmetrical, left-skewed, right-skewed, and reversed-J shaped, and can have increasing, decreasing, bathtub, and upside-down bathtub hazard rates shaped. Further, it can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon entropy, and order statistics. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. We also investigate the properties of one special model, the logistic-Fréchet distribution, and illustrate its importance by means of two applications to real data sets.     

On testing hypotheses with divergence statistics

Using divergence measures based on entropy functions, a procedure to test statistical hypotheses is proposed. Replacing the parameters by suitable estimators in the expresion of the divergence measure, the test statistics are obtained. Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators are considered, so they can be used to construct confidence intervals and to test statistical hypotheses based on one or more samples. These results can also be applied to multinomial populations. Tests of goodness of fit and tests of homogeneity can be constructed.     

Implementation of recursive nonparametric kernel estimation and a monte carlo study on its finite sample properties

The recursive estimator for the conditional mean of a nonparametric regression model with independent observations was thoroughly explored by Ahmad and Lin (1976), and Singh and Ullah (1986). Their studies are mainly concerned with the estimator's asymptotic behaviour. However, they do not include much discussion on the strategy of computing the estimates. In this paper, we provide a convenient implementation of the recursive estimator and examine its finite sample properties through simulation studies. Our study has demonstrated that for relatively short length of recursive updating, the estimates are generally equivalent to their fixed window width counterparts However, we found that substantial recursive updating can seriously lower the estimator's efficiency even though it is a consistent estimator.