Govind S. Mudholkar

Control Procedures for Residuals Associated With Principal Component Analysis

This paper is concerned with the treatment of residuals associated with principal component analysis. These residuals are the difference between the original observations and the predictions of them using less than a full set of principal components. Specifically, procedures are proposed for testing the residuals associated with a single observation vector and for an overall test for a group of observations. In this development, it is assumed that the underlying covariance matrix is known; this is reasonable for many quality control applications where the proposed procedures may be quite useful in detecting outliers in the data. A numerical example is included.

The exponentiated Weibull family: a reanalysis of the bus-motor-failure data

The Weibull family with survival function exp{-(y/σ)α}, for α > 0 and y ≥ 0, is generalized by introducing an additional shape parameter θ. The space of shape parameters α > 0 and θ > 0 can be divided by boundary line α = 1 and curve (αθ = 1 into four regions over which the hazard function is, respectively, increasing, bathtub-shaped, decreasing, and unimodal. The new family is suitable for modeling data that indicate nonmonotone hazard rates and can be adopted for testing goodness of fit of Weibull as a submodel. The usefulness and flexibility of the family is illustrated by reanalyzing five classical data sets on bus-motor failures from Davis that are typical of data in repair–reuse situations and Efrons datapertainingtoahead-and-neck-cancerclinical trial. These illustrative datainvolvecensoring and indicate bathtub, unimodal, and increasing but possibly non-Weibull hazard-shape models.

A Generalization of the Weibull Distribution with Application to the Analysis of Survival Data

Abstract The Weibull distribution, which is frequently used for modeling survival data, is embedded in a larger family obtained by introducing an additional shape parameter. This generalized family not only contains distributions with unimodal and bathtub hazard shapes, but also allows for a broader class of monotone hazard rates. Furthermore, the distributions in this family are analytically tractable and computationally manageable. The modeling and analysis of survival data using this family is discussed and illustrated in terms of a lifetime dataset and the results of a two-arm clinical trial.

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.

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.

The epsilon–skew–normal distribution for analyzing near-normal data

Abstract A family of asymmetric distributions, which first appeared in Fechner (1897, Kollectivmasslehre. Leipzig, Engleman) is reparameterized using a skewness parameter e and named the epsilon–skew–normal family. It is denoted by ESN(θ,σ,e). Its basic properties such as the relationship between the mean and mode, and its higher-order moments are examined. They are used to obtain simple estimators of the parameters measuring the location θ, the scale σ, and the skewness e. The maximum likelihood estimates are derived and it is shown that the estimators of θ and σ are asymptotically independent. The estimators reduce properly to the normal case when e=0. The ESN(θ,σ,e) can be used both as a model and as a prior distribution in Bayesian analysis. The posterior distributions in both cases are unimodal, and the modes are available in closed form.

Benefits and disadvantages of joint hypermobility among musicians.

BACKGROUND Joint hypermobility is considered to be both an advantage and a disadvantage. However, the degree of hypermobility in members of particular occupations requiring intense physical activity and the nature of the association between symptoms referable to specific joints and their hypermobility are unknown. METHODS We interviewed 660 musicians (300 women and 360 men) about work-related symptoms such as joint pain and swelling and examined them for joint hypermobility according to a standard protocol. We then determined the relation between the mobility of their fingers, thumbs, elbows, knees, and spine and any symptoms referable to these regions. RESULTS Five of the 96 musicians (5 percent) with hypermobility of the wrists, mostly instrumentalists who played the flute, violin, or piano, had pain and stiffness in this region, whereas 100 of the 564 musicians (18 percent) without such hypermobility had symptoms (P = 0.001). Hypermobility of the elbow was associated with symptoms in only 1 of 208 musicians (< 1 percent), whereas 7 of 452 (2 percent) without this hypermobility had symptoms (P = 0.45). Among the 132 musicians who had hypermobile knees, 6 (5 percent) had symptoms, whereas only 1 of 528 (< 1 percent) with normal knees had symptoms (P < 0.001). Of the 462 musicians who had normal mobility of the spine, 50 (11 percent) had symptoms involving the back, as compared with 46 of the 198 musicians (23 percent) who had hypermobility of the spine (P < 0.001). CONCLUSIONS Among musicians who play instruments requiring repetitive motion, hypermobility of joints such as the wrists and elbows may be an asset, whereas hypermobility of less frequently moved joints such as the knees and spine may be a liability.

Generalized weibull family: a structural analysis

A two shape parameter generalization of the well known family of the Weibull distributions is presented and its properties are studied. The properties examined include the skewness and kurtosis, density shapes and tail character, and relation of the members of the family to those of the Pear-sonian system. The members of the family are grouped in four classes in terms of these properties. Also studied are the extreme value distributions and the limiting distributions of the extreme spacings for the members of the family. It is seen that the generalized Weibull family contains distributions with a variety of density and tail shapes, and distributions which in terms of skewness and kurtosis approximate the main types of curves of the Pearson system. Furthermore, as shown by the extreme value and extreme spacings distributions the family contains short, medium and long tailed distributions. The quantile and density quantile functions are the principle tools used for the structural analysis of the family.

An entropy characterization of the inverse Gaussian distribution and related goodness-of-fit test

The inverse Gaussian family is widely used for modeling positively skewed measurements. In this note, we present an entropy characterization of the inverse Gaussian family and use it to construct a goodness-of-fit test for ascertaining appropriateness of such models. This test is consistent against all alternatives without a singular component and has good power properties.

Generalized Multivariate Estimator for the Mean of Finite Populations

Abstract Just as ratio estimators are often indicated when the variable y is positively correlated with an auxiliary variable x so product estimators are indicated when y is negatively correlated with x. The method of estimation suggested by Olkin [4] is extended to the case where some or all of the auxiliary variables are positively correlated and some or all are negatively correlated with y.