a study of the generalized tukey lambda family

The Tukey lambda family of distributions together with its extensions have played an important role in statistical practice. In this paper a con¬tinuously defined two-parameter generalization of this family, which holds promise of a variety of additional applications, is variously studied. The coefficients of skewness and kurtosis and the density shapes of its members are examined and the family is related to the classical Pearsonian system of distributions.     

Generalized Gumbel distribution

A generalization of the Gumbel distribution is presented to deal with general situations in modeling univariate data with broad range of skewness in the density function. This generalization is derived by considering a logarithmic transformation of an odd Weibull random variable. As a result, the generalized Gumbel distribution is not only useful for testing goodness-of-fit of Gumbel and reverse-Gumbel distributions as submodels, but it is also convenient for modeling and fitting a wide variety of data sets that are not possible to be modeled by well-known distributions. Skewness and kurtosis shapes of the generalized Gumbel distribution are illustrated by constructing the Galton’s skewness and Moor’s kurtosis plane. Parameters are estimated by using maximum likelihood method in two different ways due to the fact that the reverse transformation of the proposed distribution does not change its density function. In order to illustrate the flexibility of this generalization, wave and surge height data set is analyzed, and the fitness is compared with Gumbel and generalized extreme value distributions.     

Generalized modulus power transformations

Several power transformations proposed in the past are examined to find out the type of distributions that they can normalize, and a general family of transformations "the Generalized Modulus Power Transformation- (GEMPT), is proposed, The GEMPT will remove skewness and kurtosis and induce normality from a broad class of distributions, which we investigate, implying certain limitations for all power transformations. The use of GEMPT is illustrated and shown to iead to a better approximation to a normal distribution in an example in which the response is expected to follow a rectangular hyperbola.     

Exploratory model analysis using cdf knotting with applications to distinguishability,limiting forms,and moment ratios to the three parameter weibull family

An exploratory model analysis device we call CDF knotting is introduced. It is a technique we have found useful for exploring relationships between points in the parameter space of a model and global properties of associated distribution functions. It can be used to alert the model builder to a condition we call lack of distinguishability which is to nonlinear models what multicollinearity is to linear models. While there are simple remedial actions to deal with multicollinearity in linear models, techniques such as deleting redundant variables in those models do not have obvious parallels for nonlinear models. In some of these nonlinear situations, however, CDF knotting may lead to alternative models with fewer parameters whose distribution functions are very similar to those of the original overparameterized model. We also show how CDF knotting can be exploited as a mathematical tool for deriving limiting distributions and illustrate the technique for the 3-parameterWeibull family obtaining limiting forms and moment ratios which correct and extend previously published results. Finally, geometric insights obtained by CDF knotting are verified relative to data fitting and estimation.     

The exponentiated weibull family: some properties and a flood data application

The exponentiated Weibull family, a Weibull extension obtained by adding a second shape parameter, consists of regular distributions with bathtub shaped, unimodal and a broad variety of monotone hazard rates. It can be used for modeling lifetime data from reliability, survival and population studies, various extreme value data, and for constructing isotones of the tests of the composite hypothesis of exponentiality. The structural analysis of the family in this paper includes study of its skewness and kurtosis properties, density shapes and tail character, and the associated extreme value and extreme spacings distributions. Its usefulness in modeling extreme value data is illustrated using the floods of the Floyd River at James, Iowa.     

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     

The spokane heart study: weibull regression and coronary artery disease

Coronary artery calcium is a marker of coronary artery disease and measures the progression of atherosclerosis. It is measured by electron beam computed tomography, and the measured amount of coronary artery calcium is highly skewed to the right and left censored. The distribution of coronary artery calcium appears to be Weibull. We propose a Weibull regression model and we analyze the data using these techniques. Our analysis is based on data from the Spokane Heart Study, which is a cohort of about a thousand subjects that are assessed every two years for coronary artery calcium and risk factors of coronary artery disease. The major focus of the heart study is to determine the natural history of atherosclerosis in its early phase, and we analyze the data as a cross-sectional study with 859 subjects. We would also like to highlight the use of Weibull regression techniques in situations like this, where we have extreme right skewed data. Our main emphasis will be on examining the effect of the traditional risk factors of age, gender, lipid profile (cholesterol and HDL), patient history of lipid abnormality, hypertension, and smoking, and other family history risks on coronary artery calcium. We found that the most important factors influencing the disease were age, sex, and patient history of smoking and lipid abnormality.     

Applications of a new power normal family

The main purpose of this paper is to give an algorithm to attain joint normality of non-normal multivariate observations through a new power normal family introduced by the author (Isogai, 1999). The algorithm tries to transform each marginal variable simultaneously to joint normality, but due to a large number of parameters it repeats a maximization process with respect to the conditional normal density of one transformed variable given the other transformed variables. A non-normal data set is used to examine performance of the algorithm, and the degree of achievement of joint normality is evaluated by measures of multivariate skewness and kurtosis. Besides the above topic, making use of properties of our power normal family, we discuss not only a normal approximation formula of non-central F distributions in the frame of regression analysis but also some decomposition formulas of a power parameter, which appear in a Wilson-Hilferty power transformation setting.     

Probability density function of quotient of order statistics from the pareto,power and weibull distributions

This paper deals with the probability density functions of quotient of order statistics. We use the Mellin transform technique, to find the distribution of the quotient Z= X/Xwhere X.,X(i < j) are the ith and jth order statistics from the Pareto, Power and Weibull distributions     

Bayesian analysis of life-testing problems involving the weibull distribution

Credible and highest posterior density intervals for the reliability function and parameters of a two-parameter Weibull process are obtained and the estimates compared with their corresponding classical counterparts.     

Extremal analysis of short series with outliers: sea-levels and athletics records

The analysis of extreme values is often required from short series which are biasedly sampled or contain outliers. Data for sea-levels at two UK east coast sites and data on athletics records for women's 3000 m track races are shown to exhibit such characteristics. Univariate extreme value methods provide a poor quantification of the extreme values for these data. By using bivariate extreme value methods we analyse jointly these data with related observations, from neighbouring coastal sites and 1500 m races respectively. We show that using bivariate methods provides substantial benefits, both in these applications and more generally with the amount of information gained being determined by the degree of dependence, the lengths and the amount of overlap of the two series, the homogeneity of the marginal characteristics of the variables and the presence and type of the outlier.     

Bayesian analysis of the linear regression model with an edgeworth series prior distribution

The purpose of the present investigation 1s to observe the effect of departure from normahty of the prior distribution of regresslon parameters on the Bayman analysis of a h e a r regresslon model Assuming an Edgeworth serles prior distribution for the regresslon coefficients and gamma prior for the disturbances precision, the expressions for the posterlor distribution, posterlor mean and Bayes risk under a quadratic loss function are obtalned The results of a numerical evaluation are also analyzed     

On Smooth Statistical Tail Functionals

Many estimators of the extreme value index of a distribution function F that are based on a certain number k _n of largest order statistics can be represented as a statistical tail function al, that is a functional T applied to the empirical tail quantile function Q _n. We study the asymptotic behaviour of such estimators with a scale and location invariant functional T under weak second order conditions on F . For that purpose first a new approximation of the empirical tail quantile function is established. As a consequence we obtain weak consistency and asymptotic normality of T ( Q _n) if T is continuous and Hadamard differentiable, respectively, at the upper quantile function of a generalized Pareto distribution and k _pn tends to infinity sufficiently slowly. Then we investigate the asymptotic variance and bias. In particular, those functionals T re characterized that lead to an estimator with minimal asymptotic variance. Finally, we introduce a method to construct estimators of the extreme value index with a made-to-order asymptotic behaviour     

An analysis of quantile measures of kurtosis: center and tails

The consequences of substituting the denominator Q ₃(p)  −  Q ₁(p) by Q ₂  −  Q ₁(p) in Groeneveld’s class of quantile measures of kurtosis (γ ₂(p)) for symmetric distributions, are explored using the symmetric influence function. The relationship between the measure γ ₂(p) and the alternative class of kurtosis measures κ₂(p) is derived together with the relationship between their influence functions. The Laplace, Logistic, symmetric Two-sided Power, Tukey and Beta distributions are considered in the examples in order to discuss the results obtained pertaining to unimodal, heavy tailed, bounded domain and U-shaped distributions. The authors thank the referee for the careful review.     

Limited information bayesian analysis of a structural coefficient in a simultaneous equations system

An analytical expression is obtained for the marginal posterior density for a structural coefficient in a simultaneous equations system based on a limited information Bayesian analysis. A con- ditional posterior density is obtained given reduced form para- meters. This conditional posterior density is in univariate student t form. Numerical examples suggest that the conditional density hasa tighter distribution around the posterior mean than the unconditional density when the correlation between the endo- genous variables and the structural error term is high.     

Robust Bayesian analysis of generalized inverted family of distributions

ABSTRACT

In the current study we develop the robust Bayesian inference for the generalized inverted family of distributions (GIFD) under an ε-contamination class of prior distributions for the shape parameter α, with different possibilities of known and unknown scale parameter. We used Type II censoring and Bartholomew sampling scheme (1963) for the following derivations under the squared-error loss function (SELF) and linear exponential (LINEX) loss function : ML-II Bayes estimators of the i) parameters; ii) Reliability function and; iii) Hazard function. We also present simulation study and analysis of a real data set.     

Skew generalized extreme value distribution: Probability-weighted moments estimation and application to block maxima procedure

ABSTRACT

Following the work of Azzalini (1985 Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Stat. 12:171–178.[Web of Science ®] , [Google Scholar] and 1986 Azzalini, A. (1986). Further results on a class of distributions which includes the normal ones. Statistica 46:199–208. [Google Scholar]) on the skew-normal distribution, we propose an extension of the generalized extreme value (GEV) distribution, the SGEV. This new distribution allows for a better fit of maxima and can be interpreted as both the distribution of maxima when maxima are taken on dependent data and when maxima are taken over a random block size. We propose to estimate the parameters of the SGEV distribution via the probability-weighted moment method. A simulation study is presented to provide an application of the SGEV on block maxima procedure and return level estimation. The proposed method is also implemented on a real-life data.     

Detection of a pair of outliers in a sample from a Gumbel distribution with known scale parameter

Test procedures on outlier detection problems for Gumbel distribution are rarely available. Hence, a test statistic is proposed here for detection of a pair of upper and lower outliers from a Gumbel distribution with known scale parameter. The critical values of the statistic are obtained and some examples are also given to highlight the use of the statistic. The advantage of the proposed statistic is that the scale parameter, though assumed to be known is not explicitly involved in the determination of the critical values.