Estimation of the probability parameters in a contingency table with linear and/or log-linear constraints on the parameters is the principal concern of this thesis. Sequential unbiased estimates of the cell probabilities as well as some Bayes posterior mean type estimates are considered. / Chapter I is a review of some earlier work on the sequential unbiased estimation of the probability parameter in a Bernoulli process. The review begins with the classical work of Girshick, Mosteller and Savage (1946) and some follow-up studies like Wolfowitz (1946), Savage (1947), Blackwell (1947), Lehmann and Stein (1950), Degroot (1959) and Kagan, Linnik and Rao (1973). In several cases the original proofs have been simplified and the arguments streamlined. / Chapter II deals with the problem of sequential unbiased estimation of the parameters in a contingency table with linear and/or log-linear constraints. Multinomial Girschick, Mosteller and Savage (GMS) type stopping rules are discussed and the corresponding unbiased estimates based on the minimal sufficient statistic described. Consistency, in the sence of Wolfowitz (1947), of such estimates is demonstrated. Unbiased estimates of parametric functions like log-contrasts are derived. Sufficient conditions for the completeness of the GMS-type stopping rules are given. / In Chapter III, the problem of sequential unbiased estimation of the probability parameters in the Bradley-Terry (1952) model of paired comparisons is studied.g The Bradley-Terry model can be summarized as follows. Suppose that there are t treatments T(,1), ..., T(,t) that can be pairwise compared. The Bradley-Terry model postulates that associated with treatement T(,i) is a :strenth" parameter (PI)(,i) > 0, i = 1, ..., t, such that if treatments T(,i) and T(,j) are compared, the probability that T(,i) is preferred to T(,j) is (theta)(,ij) = (PI)(,i)/((PI)(,i) + (PI)(,j)). The model imposes log-linear constraints on he (theta)(,ij)'s so that techniques similar to those in Chapter II may be used to obtain unbiased estimates, based on a sufficient statistic. / In Chapter IV, two Bayes-type procedures for estimating the multnomial cell probability vector p, in he presence of linear constraints on the parameters, are proposed and illustrated with examples. A general prior is used with the restriction that the moment generating function of the prior exists in a closed form. The estimators are shown to be strongly consistent. Estimation under log-linear constraints is also considered. Finally, Bayes-type estimators for the covariance matrix of the cell frequencies are presented for some special cases of linearly and log-linearly constrained problems. / Chapter V is concerned with a Bayesian approach to the estimation of parameters in the Bradley-Terry model of paired comparisons. It is assumed that the sum of the treatment parameters (PI)(,i) is 1, and a Dirichlet prior for (PI) = ((PI)(,1), ..., (PI)(,t)) is used. Using the induced prior of (theta)(,ij) and Z(,ij) = (PI)(,i) + (PI)(,j), an estimate (PI)(,ij) of (PI)(,i), based on the data arising from the comparisons of(' ) treatments T(,i) and T(,j), is obtained. An estimate of (PI)(,i) based on all the data is a weighted combination of the (PI)(,ij)'s that minimizes a(' ) risk function. Similarly, estimates for log-contrasts of the (PI)(,i)'s areobtained. This technique of estimation is extended to the Luce-model of multiple comparisons.(,) / Source: Dissertation Abstracts International, Volume: 42-06, Section: B, page: 2436. / Thesis (Ph.D.)--The Florida State University, 1981.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_74559 |
| Contributors | CHEN, CHENG-CHUNG., Florida State University |
| Source Sets | Florida State University |
| Detected Language | English |
| Type | Text |
| Format | 107 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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