##### Author

Bryant, John L.

##### Other Contributors

Paulson, A. S.; Sullo, Pasquale; Voytuk, James A.; Wilkinson, John W.;

##### Date Issued

1977-05

##### Subject

Operations research and statistics

##### Degree

PhD;

##### Terms of Use

This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.;

##### Abstract

The estimation of the parameters of a certain modified compound Poisson distribution with normal summands is investigated. This problem is of importance in a variety of applications and in particular arises from a security price fluctuation model advocated by Press. A new estimation procedure is proposed which essentially consists of performing an orthogonalized least squares regression on the empirical characteristic function, or more correctly on a real function uniquely associated with it. A fixed-point maximum likelihood algorithm is also developed. The performance of the estimators derived by these two methods is compared with that of the cumulant matching estimator employed by Press.; The application of the integral measure of the deviation of empirical and population characteristic functions to the estimation of parameters is also considered. Some results due to Thornton and Paulson concerning the consistency and asymptotic normality of the estimators obtained by minimizing this integral are extended. Finite sample results are derived in the case where the underlying population is a mixture of completely specified components. More precisely, estimators based on the empirical characteristic function are found in this case to be unbiased, and a finite sample expression for their covariance matrix is given. Such estimators are compared with others which have been proposed in the literature in the specific situation where the parent population is a mixture of two normal components.; One problem considered in this research is the assessment of the feasibility of a goodness-of-fit test based on an integral of the weighted squared modulus of the difference of the empirical' and population characteristic functions. The statistic of such a test is therefore analogous to that of the Cramer-von Mises test, with the exception that now the distance between populations is measured in the transform space rather than directly in the space of distribution functions. A number of properties of the test have been derived, including the asymptotic null distribution of its statistic. It is shown that under mild regularity conditions the test is consistent. Some relations to the Cramer-von Mises procedure are noted. A number of approximations to the null distribution of the test statistic are considered, and are found to be successful in simplifying its application without undue loss of accuracy. The asymptotic power of the goodness-of-fit test against certain simple alternates is calculated under a null hypothesis of normality, and is compared to those of other tests. Preliminary results indicate that the goodness-of-fit test based on the empirical characteristic function may be of use in the analysis of multivariate data.; Application of the theory of characteristic functions to the analysis of data has recently begun to receive considerable attention, motivated primarily by the problem of estimating the parameters of the stable laws. Define the empirical characteristic function associated with a sample drawn from a given population to be the transform of the sample distribution function. Then a measure of the deviation between the population and empirical characteristic functions may be introduced and used in some way to draw inferences concerning the population. For example, if the distribution function of the population is assumed to be a member of a given class of distributions indexed by a vector of unknown parameters, then estimates of the parameters may be generated by minimizing this measure of deviation over the parameter space. Such an estimation procedure is justified by the well-known one-to-one correspondence of distributions and their associated characteristic functions.;

##### Description

May 1977; School of Engineering

##### Department

Dept. of Decision Sciences and Engineering Systems.;

##### Publisher

Rensselaer Polytechnic Institute, Troy, NY

##### Relationships

Rensselaer Theses and Dissertations Online Collection;

##### Access

Restricted to current Rensselaer faculty, staff and students. Access inquiries may be directed to the Rensselaer Libraries.;