Density estimation and some topics in multivariate analysis

Part I, entitled "A test of goodness-of-fit for multivariate distributions with special emphasis on multinornality", investigates a modification of the Chi-squared goodness-of-fit statistic which eliminates certain objectionable properties of other multivariate goodness-of-fit tests. Special emphasis is given to the multinormal distribution, and computer simulation is used to generate an empirical distribution for this goodness-of-fit statistic for the standardized bivariate normal density. Attempts to fit a four-parameter generalized gamma density function to this empirical distribution were only partially successful.

Part II, entitled "The centroid method of numerical integration", begins with a discussion of the often slighted midpoint method of numerical integration, then, using Taylor's theorem, generalized formulae for the centroid method of numerical integration of a function of several variables over a closed bounded region are developed. These formulae are in terms of the derivatives of the integrand and the moments of the region of integration with respect to its centroid. Since most nonpathological bounded regions can be well approximated by a finite set of simplexes, formulae are developed for the moments of general as well as special simplexes. Several numerical examples are given and a comparison is made between the midpoint and Gaussian quadrature methods. FORTRAN programs are included.

Part III - entitled "Non-parametric density estimation," begins with an extensive literature review of non-parametric methods for estimating probability densities based on a sample of N observations and goes on to suggest a new method which is to subtract a penalty for roughness fron the log-likelihood before maximizing. The roughness penalty is a functional of the assumed density function and the recommendation is to use a linear combination of the squares of the first and second derivatives of the square root of the density function. Many numerical examples and graphs are given and show that the estimated density function, for selected values of the coefficients in the linear expression, turns out to be very smooth even for very small sample sizes. Computer programs are not included but are available upon request.

Part IV, entitles "On separation of product and error variability," surveys standard techniques of partitioning the total variance into product (or item) variance and error (or testing) variance when destructive testing makes replication over the same item impossible. The problem of negative variance estimates is also investigated. The factor-analysis model and related iterative techniques are suggested as an alternative method for dealing with this separation when three or more independent measurements per item are available. The problem of dependent measurements is discussed. Numerical examples are included.