A fourth order rotatable design in four dimensions

Patel and Arap Koske (1985) gave the necessary and sufficient conditions for a set of points to form a rotatable design of order four, A set of 256 points for four factors which satisfies these conditions is given in this paper.     

Extreme Values and Haar Series Estimates of Point Process Boundaries

ABSTRACT.  We present a new method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on Haar series and extreme values of the point process. We give conditions for various kind of convergence and we obtain remarkably different possible limit distributions. We propose a method of reducing the negative bias, illustrated by a simulation.     

Local transformation of facila features

This paper describes methodology for carrying out local transformations of a set of landmarkpoints describingfacialfeatures, so that they provide a good match to a given set of facial landmarkpoints: the landmark points are divided into groups of 'similar 'points that take facial symmetry into account, and the coordinate values of the points within each group are differentially stretched and contracted. The methodology is illustrated by application to sets of 195 facial landmark points.     

Fifth order rotability

In this paper, conditions for a set of points to form a rotatable design of order five in three dimensions have been worked out. These conditions form a basis for obtaining a set of rotatable designs of order five. An attempt has been made to generalise these results with partial success.     

Conditions for fourth order rotatability in K dimensions

In this paper, conditions for a set of points to form a rotatable design of order four have been worked out. These condi- tions form the basis for obtaining a set of rotatable designs of order four.     

Smoothed Extreme Value Estimators of Non-Uniform Point Processes Boundaries with Application to Star-Shaped Supports Estimation

We address the problem of estimating the edge of a bounded set in ? ^d given a random set of points drawn from the interior. Our method is based on a transformation of estimators dedicated to uniform point processes and obtained by smoothing some of its bias corrected extreme points. An application to the estimation of star-shaped supports is presented.     

An economical acceptance–rejection algorithm for uniform random variate generation over constrained simplexes

This paper develops an algorithm for uniform random generation over a constrained simplex, which is the intersection of a standard simplex and a given set. Uniform sampling from constrained simplexes has numerous applications in different fields, such as portfolio optimization, stochastic multi-criteria decision analysis, experimental design with mixtures and decision problems involving discrete joint distributions with imprecise probabilities. The proposed algorithm is developed by combining the acceptance–rejection and conditional methods along with the use of optimization tools. The acceptance rate of the algorithm is analytically compared to that of a crude acceptance–rejection algorithm, which generates points over the simplex and then rejects any points falling outside the intersecting set. Finally, using convex optimization, the setup phase of the algorithm is detailed for the special cases where the intersecting set is a general convex set, a convex set defined by a finite number of convex constraints or a polyhedron.     

A characterization of the independence - distribution - preserving covariance structure for the multivariate maximum squared - radii statistic

Necessary and sufficient conditions on the observation covariance structure and on the set of linear transformations are given for which the distribution of the multivariate maximum squared - radii statistic for detecting a single multivariate outlier is invariant from the distribution assuming the usual independence covariance structure. Thus, we extend the work of Baksalary and Puntanen (1990), who have given necessary and sufficient conditions for an independence-distribution-preserving covariance structure for Grubbs' statistic for detecting a univariate outlier. We also extend the work of Marco, Young, and Turner (1987) and Pavur and Young (1991), who have given sufficient conditions for an independence-distribution-preserving dependency structure for the multivariate squared - radii statistic.     

Analysis op failure time data by burr distribution

Sometimes it is appropriate to model the survival and failure time data by a non-monotonic failure rate distribution. This may be desirable when the course of disease is such that mortality reaches a peak after some finite period and then slowly declines.In this paper we study Burr, type XII model whose failure rate exhibits the above behavior. The location of the critical points (at which the monotonicity changes) for both the failure rate and the mean residual life function (MRLF) are studied. A procedure is described for estimating these critical points. Necessary and sufficient conditions for the existence and uniqueness of the maximum likelihood estimates are provided and it is shown that the conditions provided by Wingo (1993) are not sufficient. A data set pertaining to fibre failure strengths is analyzed and the maximum likelihood estimates of the critical points are obtained.     

Uniqueness of estimation and identifiability in mixture models

Further properties of the nonparametric maximum-likelihood estimator of a mixing distribution are obtained by exploiting the properties of totally positive kernels. Sufficient conditions for uniqueness of the estimator are given. This result is more general, and the proof is substantially simpler, than given previously. When the component density has support on N points, it is shown that all identifiable mixing distributions have support on no more than N/2 points. Identifiable mixtures are shown to lie on the boundary of the mixture model space. The maximum-likelihood estimate is shown to be unique if the vector of observations lies outside this space.     

Bayesian Estimation of the Number of Change Points in Simple Linear Regression Models

A Bayesian approach is considered to detect the number of change points in simple linear regression models. A normal-gamma empirical prior for the regression parameters based on maximum likelihood estimator (MLE) is employed in the analysis. Under mild conditions, consistency for the number of change points and boundedness between the estimated location and the true location of the change points are established. The Bayesian approach to the detection of the number of change points is suitable whether the switching simple regression is continuous or discontinuous. Some simulation results are given to confirm the accuracy of the proposed estimator.     

Extensions of Samuelson's Inequality

In this note we obtain upper and lower bounds for the kth largest number in a set of real numbers in terms of their mean and standard deviation. For each inequality necessary and sufficient conditions for equality are given.     

Useful generalized properties of L1 estimators

Recent results by G. Appa and C. Smith, as well as I. Barrodale and F. D. K. Roberts, underscore several properties exhibited for fitting a linear model to a set of observation points under the criterion of least sum of absolute deviations(commonly denoted as the L₁ criterion). This paper will generalize these properties to the non-full rank case and relax in a natural way some assumptions given by Appa and Smith.     

Extension Of Dale's Moment Conditions With Application To The Wright–fisher Model

Dale's necessary and sufficient conditions for an array to contain the joint moments for some probability distribution on the unit simplex in R² are extended to the unit simplex in R ^d . These conditions are then used in a computational method, based on linear programming, to evaluate the stationary distribution for the diffusion approximation of the Wright–Fisher model in population genetics. The computational method uses a characterization of the diffusion through an adjoint relation between the diffusion operator and its stationary distribution. Application of this adjoint relation to a set of functions in the domain of the generator leads to one set of constraints for the linear program involving the moments of the stationary distribution. The extension of Dale's conditions on the moments add another set of linear conditions and the linear program is solved to obtain bounds on numerical quantities of interest. Numerical illustrations are given to illustrate the accuracy of the method.

     

The Multiplicative Definition of Interaction

The theory of the multiplicative definition of second order interaction is considered. Necessary and sufficient conditions in terms of the correlations in the marginal two-dimensional distributions are found for the existence of a “perfect” set of probabilities. Some progress is reported on the proof of the uniqueness of the multiplicative set of probabilities for a given set of two-dimensional distributions. The multiplicative property of such sets is preserved under pooling of marginal sets only in trivial cases, and so the multiplicative definition cannot be said to be a straightforward generalization from the definition of no interaction in two-dimensional distributions.     

Detection of outliers in the exponential distribution based on prediction

In this paper, we present a test procedure to detect outliers in the one-parameter exponential distribution based on prediction. The distribution of the test statistic is obtained. The proposed test can be used to detect more than one outlier and the required percentage points can be easily determined. Furthermore, the test provides a simple procedure to detect whether a given set of data is free from outliers or spurious observations.     

Sums of weighted spacings and a test for the logistic distribution

A simple proof for a theorem of Csörgö and Révész (1981b and 1984) concerning sums of weighted spacings is given, The conditions of the theorem are relaxed. As an application, a goodness-of-fit test for the logistic distribution is proposed. The percentage points of the proposed test statistic are obtained by a simulation experiment.     

The rainbow test for lack of fit in regression

A test for lack of fit in regression is presented. Unlike other methods, this one doesn't require replicates or a prior estimate of variance. It can be used for linear or multiple regression, and would be easy to add to existing computer packages. It is based on comparing a fit over low leverage points with a fit over the entire set of data. Distribution theory results are pre¬sented, with examples of power. A discussion of its use for de¬tecting violations of other regression assumptions is also given.     

A Note on Collapsibility in DAG Models of Contingency Tables

Abstract.  Necessary and sufficient conditions for collapsibility of a directed acyclic graph (DAG) model for a contingency table are derived. By applying the conditions, we can easily check collapsibility over any variable in a given model either by using the joint probability distribution or by using the graph of the model structure. It is shown that collapsibility over a set of variables can be checked in a sequential manner. Furthermore, a DAG is compared with its moral graph in the context of collapsibility.     

The limiting bound of Efron's W-formula for hypothesis testing when a nuisance parameter is present only under the alternative

When testing a hypothesis with a nuisance parameter present only under the alternative, the maximum of a test statistic over the nuisance parameter space has been proposed. Different upper bounds for the one-sided tail probabilities of the maximum tests were provided. Davies (1977. Biometrika 64, 247–254) studied the problem when the parameter space is an interval, while Efron (1997. Biometrika 84, 143–157) considered the problem with some finite points of the parameter space and obtained a W-formula. We study the limiting bound of Efron's W-formula when the number of points in the parameter space goes to infinity. The conditions under which the limiting bound of the W-formula is identical to that of Davies are given. The results are also extended to two-sided tests. Examples are used to illustrate the conditions, including case-control genetic association studies. Efficient calculations of upper bounds for the tail probability with finite points in the parameter space are described.