Generalized additive models for location, scale and shape

Summary.  A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y , as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.     

Slashed generalized Rayleigh distribution

We introduce a new family of distributions suitable for fitting positive data sets with high kurtosis which is called the Slashed Generalized Rayleigh Distribution. This distribution arises as the quotient of two independent random variables, one being a generalized Rayleigh distribution in the numerator and the other a power of the uniform distribution in the denominator. We present properties and carry out estimation of the model parameters by moment and maximum likelihood (ML) methods. Finally, we conduct a small simulation study to evaluate the performance of ML estimators and analyze real data sets to illustrate the usefulness of the new model.     

Slashed generalized exponential distribution

In this paper, we introduce an extension of the generalized exponential (GE) distribution, making it more robust against possible influential observations. The new model is defined as the quotient between a GE random variable and a beta-distributed random variable with one unknown parameter. The resulting distribution is a distribution with greater kurtosis than the GE distribution. Probability properties of the distribution such as moments and asymmetry and kurtosis are studied. Likewise, statistical properties are investigated using the method of moments and the maximum likelihood approach. Two real data analyses are reported illustrating better performance of the new model over the GE model.     

A Generalized Class of Ratio Type Exponential Estimators of Population Mean Under Linear Transformation of Auxiliary Variable

Recently, Shabbir and Gupta [Shabbir, J. and Gupta, S. (2011). On estimating finite population mean in simple and stratified random sampling. Communications in Statistics-Theory and Methods, 40(2), 199–212] defined a class of ratio type exponential estimators of population mean under a very specific linear transformation of auxiliary variable. In the present article, we propose a generalized class of ratio type exponential estimators of population mean in simple random sampling under a very general linear transformation of auxiliary variable. Shabbir and Gupta's [Shabbir, J. and Gupta, S. (2011). On estimating finite population mean in simple and stratified random sampling. Communications in Statistics-Theory and Methods, 40(2), 199–212] class of estimators is a particular member of our proposed class of estimators. It has been found that the optimal estimator of our proposed generalized class of estimators is always more efficient than almost all the existing estimators defined under the same situations. Moreover, in comparison to a few existing estimators, our proposed estimator becomes more efficient under some simple conditions. Theoretical results obtained in the article have been verified by taking a numerical illustration. Finally, a simulation study has been carried out to see the relative performance of our proposed estimator with respect to some existing estimators which are less efficient under certain conditions as compared to the proposed estimator.     

Ultrastructural elliptical models

Dolby's (1976) ultrastructural model with no replications is investigated within the class of the elliptical distributions. General asymptotic results are given for the sample covariance matrix S in the presence of incidental parameters. These results are used to study the asymptotic behaviour of some estimators of the slope parameter, unifying and extending existing results in the literature. In particular, under some regularity conditions they are shown to be consistent and asymptotically normal. For the special case of the structural model, some asymptotic relative efficiencies are also reported which show that generalized least squares and the method of moment estimators can be highly inefficient under nonnormality.     

On detection of outliers in symmetric normal models

Extensions of recent results for detection of mean slippage type outliers from i.i.d. multivariate normal and elliptically symmetric distributions are made to symmetric case, that is, when the observations are equicorrelated. The main tool used is Wijsman's (1967) representation theorem. The results obtained can be viewed as a robustness property of the use of Mardia's multivariate kurtosis as a locally optimal test statistic to detect outliers against equicorrelated distributions.     

Shape properties of the g-and-h and johnson families

A substantial part of examining the properties of a distributional family consists of considering shape properties. It is important that this examination is sufficiently thorough to enable understanding of the behaviour of the family, its comparison with others, and to assist in developing future families. The g-and-h distributions and the Johnson system are examined here in these terms     

On the efficiency of using the sample kurtosis in selecting optimal lpestimators

This paper examines the efficiency of thesample kurtosisin obtaining L_P estimates as an estimates of central tendency for symmetric distributions. Moreover, guidelines are established for determining an optimal value of P based on the kurtosis of the error distribution.     

Tables to Facilitate Fitting Tadikamalla and Johnson's LB Distributions

This paper is concerned with estimating the parameters of Tadikamalla-Johnson's L_B distribution using the first four moments. Tables of the parameters of the L_B distribution are given for selected values of skewness (0.0(0.05) 1.0(0.1)2.0) and corresponding available values of kurtosis at intervals of 0.2. The construction and use of these tables is explained with a numerical example.     

Testing Variability in the Two-Sample Case

A number of robust methods for testing variability have been reported in previous literature. An examination of these procedures for a wide variety of populations confirms their general robustness. Shoemaker's improvement of the F test extends that test use to a realistic variety of population shapes. However, a combination of the Brown–Forsythe and O'Brien methods based on testing kurtosis is shown to be conservative for a wide range of sample sizes and population distributions. The composite test is also shown to be more powerful in most conditions than other conservative procedures.