This dissertation is a contribution to the theory of Bayesian nonparametrics. A construction of the Dirichlet process (Ferguson {1973}) on a finite set (chi) is introduced in such a way that it leads to the Blackwell's (1973) constructive definition of a Dirichlet process on a Borel space ((chi),A). If ((chi),A) is a Borel space and P is a random probability measure on ((chi),A) with a Dirichlet process prior D('(alpha)), then under the condition that the (alpha)-measure of every open subset of (chi) is positive, for almost every realization P of P the set of discrete mass points of P is dense in (chi). / A more general constructive definition introduced by Sethuraman (1978) is used to derive several new properties of the Dirichlet process and to present in a unified way some of the known properties of the process. An alternative construction of Dalal's (1975) G-invariant Dirichlet process (G being a finite group of transformations) is presented. / The Bayes estimates of an estimable parameter of degree k(k (GREATERTHEQ) 1), namely / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / where h is a symmetric kernel, are derived for the no sample size and for a sample of size n from P under the squared error loss function and a Dirichlet process prior. Using the result of the Bayes estimate of (psi)(,k)(P) for the no sample size the (marginal) distribution of a sample from P (when the prior for P is the Dirichlet process) is obtained. The extension to the case when the prior for P is G-invariant Dirichlet process is also obtained.(,) / Let ((chi),A) be the one-dimensional Euclidean space (R(,1),B(,1)). Consider a sequence {D('(alpha)(,N)+(gamma))} of Dirichlet processes such that (alpha)(,N)((chi)) converges to zero as N tends to infinity, where (gamma) and (alpha)(,N)'s are finite measures on A. It is shown that D('(alpha)(,N)+(gamma)) converges weakly to D('(gamma)) in the topology of weak / convergence on P, the class of all probability measures on ((chi),A). As a corollary, it follows that D('(alpha)(,N)+nF(,n)) converges weakly to D('nF(,n)), where F(,n) is the empirical distribution of the sample. Suppose (alpha)(,N)((chi)) converges to zero and (alpha)(,N)/(alpha)(,N)((chi)) converges uniformly to (alpha)/(alpha)((chi)) as N tends to infinity. If / {D('(alpha)(,N))} is a sequence of Dirichlet process priors for a random probability measure P on ((chi),A), then P, in the limit, is a random probability measure concentrated on the set of degenerate probability measures on ((chi),A) and the point of degeneracy is distributed as (alpha)/(alpha)((chi)) on ((chi),A). To the sequence of priors (D('(alpha)(,N))) for P, there corresponds a sequence of the Bayes estimates of (psi)(,k)(P). The limit of this sequence of the Bayes estimates when (alpha)(,N)((chi)) converges to zero as N tends to infinity, called the limiting Bayes estimate of (psi)(,k)(P), is obtained. / When P is a random probability measure on {0, 1}, Sethuraman (1978) proposed a more general class of conjugate priors for P which contains both the family of Dirichlet processes and the family of priors introduced by Dubins and Freedman (1966). As an illustration, a numerical example is considered and the Bayes estimates of the mean and the variance of P are computed under three distinct priors chosen from Sethuraman's class of priors. The computer algorithm for this calculation is presented. / Source: Dissertation Abstracts International, Volume: 41-10, Section: B, page: 3829. / Thesis (Ph.D.)--The Florida State University, 1981.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_74329 |
| Contributors | TIWARI, RAM CHANDRA., Florida State University |
| Source Sets | Florida State University |
| Detected Language | English |
| Type | Text |
| Format | 160 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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