Invariant co-ordinate selection

Summary.  A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based on the eigenvalue–eigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant co-ordinate system for the multivariate data. Consequently, we view this method as a method for invariant co-ordinate selection . By plotting the data with respect to this new invariant co-ordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant co- ordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant co-ordinates corresponds to Fisher's linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given.     

Cumulative correspondence analysis of ordered categorical data from industrial experiments

Most studies of quality improvement deal with ordered categorical data from industrial experiments. Accounting for the ordering of such data plays an important role in effectively determining the optimal factor level of combination. This paper utilizes the correspondence analysis to develop a procedure to improve the ordered categorical response in a multifactor state system based on Taguchi's statistic. Users may find the proposed procedure in this paper to be attractive because we suggest a simple and also popular statistical tool for graphically identifying the really important factors and determining the levels to improve process quality. A case study for optimizing the polysilicon deposition process in a very large-scale integrated circuit is provided to demonstrate the effectiveness of the proposed procedure.     

Forecast mean squared error reductionin the VAR(1) process

When VAR models are used to predict future outcomes, the forecast error can be substantial. Through imposition of restrictions on the off-diagonal elements of the parameter matrix, however, the information in the process may be condensed to the marginal processes. In particular, if the cross-autocorrelations in the system are small and only a small sample is available, then such a restriction may reduce the forecast mean squared error considerably.

In this paper, we propose three different techniques to decide whether to use the restricted or unrestricted model, i.e. the full VAR(1) model or only marginal AR(1) models. In a Monte Carlo simulation study, all three proposed tests have been found to behave quite differently depending on the parameter setting. One of the proposed tests stands out, however, as the preferred one and is shown to outperform other estimators for a wide range of parameter settings.     

Aggregation of Variables in Least-Squares Regression

I consider the properties of the estimator when the true model is y = β₁ x ₁ + β₂ x ₂ + u, but the restriction β₁ = β₂ = β is incorrectly imposed. I show that the probability limit of is a weighted sum of β₁ and β₂; the weights sum to 1 but do not necessarily lie in the unit interval, so plim need not be bounded by β₁ and β₂. Sufficient conditions for such bounding are derived. Certain changes in the moments of x ₁ and x ₂ have “perverse” effects on the weights. I illustrate the consequences of inappropriate aggregation of variables with an empirical example of the effect of research and development investment on productivity.     

The asymptotic distribution of the partial attributable risk in cross-sectional studies

The partial attributable risk (PAR) has been introduced as a tool for partitioning the responsibility for causing an adverse event between various risk factors. It has arisen from epidemiology, but it is also a valid general risk allocation concept, which can, for example, be applied to data from customer satisfaction surveys. So far, a variance formula for the PAR has been missing so that the confidence intervals were not directly available. This paper provides the asymptotic normal distribution for the PAR determined from a cross-sectional study.     

Bivariate exponential-type distributions with linear regression

Let Xbe a bivariate exponential-type random vector (BIDLIKAR, PATIL (1968)), than it is proved:

1. If P(X ≥0) = 1 is valid, then Xhas linear regression to both directions if and only if Xpossesses a symmetric Γ-distribution.

2. Xpossesses linear regression to both directions with constant regression coefficients (independent of the parameter vector ? of the exponential-type distribution (BIDLIKAR, PATIL (1968)) if and only if Xis normal distributed.     

The existence of waiting time moments for single server queues with mixed renewal and poisson inputs

The following queuing system is considered: Two independent recurrent input streams (streams 1 and 2) arrive at a server. It is assumed that stream 1 is of Poisson type. Three priority disciplines are studied in case that customers of type 1 have priority: head-of-the-line, preemptive-resume, and preemptive-repeat discipline. For all three cases, the limiting distribution function of actual waiting times of low-priority customers is considered, and conditions are given for the existence of moments related to these limiting distributions.     

On the construction of optimal experimental designs: a penalty approach

Tn optimal experimental design theory there are well-known situations, in which additional constraints are implied to the design set. These constraints destroy in general the simplex structure of the set of feasible points of the design set. Thus the available iteration procedures for the unrestricted case are no longer applicable.

In this paper a penalty approach is suggested which transforms the restricted problem to the unrestricted case and allows the application of well-known algorithms such as the Fedorov-Wynn-type or the projected gradient procedure.     

R–estimation of normed bivariate density functions

Normed bivariate density funtions were introduced by HOEFFDING (1940/41). In the present paper estimators for normed bivariate ranks and on a FOURIER series expansion in LEGENDRE polynomials. The estimation of normed bivarate density functions under positive dependence is also described