## Bayesian inference for linear and nonlinear functions of Poisson and binomial rates

##### Abstract

This thesis focuses on objective Bayesian statistics, by evaluating a number of noninformative priors. Choosing the prior distribution is the key to Bayesian inference. The probability matching prior for the product of different powers of k binomial parameters is derived in Chapter 2. In the case of two and three independently distributed binomial variables, the Jeffreys, uniform and probability matching priors for the product of the parameters are compared. This research is an extension of the work by Kim (2006), who derived the probability matching prior for the product of k independent Poisson rates. In Chapter 3 we derive the probability matching prior for a linear combination of binomial parameters. The construction of Bayesian credible intervals for the difference of two independent binomial parameters is discussed. The probability matching prior for the product of different powers of k Poisson rates is derived in Chapter 4. This is achieved by using the differential equation procedure of Datta & Ghosh (1995). The reference prior for the ratio of two Poisson rates is also obtained. Simulation studies are done to com- pare different methods for constructing Bayesian credible intervals. It seems that if one is interested in making Bayesian inference on the product of different powers of k Poisson rates, the probability matching prior is the best. On the other hand, if we want to obtain point estimates, credibility intervals or do hypothesis testing for the ratio of two Poisson rates, the uniform prior should be used. In Chapter 5 the probability matching prior for a linear contrast of Poisson parameters is derived, this prior is extended in such a way that it is also the probability matching prior for the average of Poisson parameters. This research is an extension of the work done by Stamey & Hamilton (2006). A comparison is made between the confidence intervals obtained by Stamey & Hamilton (2006) and the intervals derived by us when using the Jeffreys and probability matching priors. A weighted Monte Carlo method is used for the computation of the Bayesian credible intervals, in the case of the proba- bility matching prior. In the last section of this chapter hypothesis testing for two means is considered. The power and size of the test, using Bayesian methods, are compared to tests used by Krishnamoorthy & Thomson (2004). For the Bayesian methods the Jeffreys prior, probability matching prior and two other priors are used. Bayesian estimation for binomial rates from pooled samples are considered in Chapter 6, where the Jeffreys prior is used. Bayesian credibility intervals for a single proportion and the difference of two binomial proportions estimated from pooled samples are considered. The results are compared This thesis focuses on objective Bayesian statistics, by evaluating a number of noninformative priors. Choosing the prior distribution is the key to Bayesian inference. The probability matching prior for the product of different powers of k binomial parameters is derived in Chapter 2. In the case of two and three independently distributed binomial variables, the Jeffreys, uniform and probability matching priors for the product of the parameters are compared. This research is an extension of the work by Kim (2006), who derived the probability matching prior for the product of k independent Poisson rates. In Chapter 3 we derive the probability matching prior for a linear combination of binomial parameters. The construction of Bayesian credible intervals for the difference of two independent binomial parameters is discussed. The probability matching prior for the product of different powers of k Poisson rates is derived in Chapter 4. This is achieved by using the differential equation procedure of Datta & Ghosh (1995). The reference prior for the ratio of two Poisson rates is also obtained. Simulation studies are done to com- pare different methods for constructing Bayesian credible intervals. It seems that if one is interested in making Bayesian inference on the product of different powers of k Poisson rates, the probability matching prior is the best. On the other hand, if we want to obtain point estimates, credibility intervals or do hypothesis testing for the ratio of two Poisson rates, the uniform prior should be used. In Chapter 5 the probability matching prior for a linear contrast of Poisson parameters is derived, this prior is extended in such a way that it is also the probability matching prior for the average of Poisson parameters. This research is an extension of the work done by Stamey & Hamilton (2006). A comparison is made between the confidence intervals obtained by Stamey & Hamilton (2006) and the intervals derived by us when using the Jeffreys and probability matching priors. A weighted Monte Carlo method is used for the computation of the Bayesian credible intervals, in the case of the proba- bility matching prior. In the last section of this chapter hypothesis testing for two means is considered. The power and size of the test, using Bayesian methods, are compared to tests used by Krishnamoorthy & Thomson (2004). For the Bayesian methods the Jeffreys prior, probability matching prior and two other priors are used. Bayesian estimation for binomial rates from pooled samples are considered in Chapter 6, where the Jeffreys prior is used. Bayesian credibility intervals for a single proportion and the difference of two binomial proportions estimated from pooled samples are considered. The results are compared to those from other methods. In Chapters 7 and 8, Bayesian process control for the p - chart and the c - chart are considered. The Jeffreys prior is used for the Bayesian methods. Control chart limits, average run lengths and false alarm rates are determined. The results from the Bayesian method are compared to the results obtained from the classical (frequentist) method. Bayesian tolerance intervals for the binomial and Poisson distributions are studied in Chapter 9, where the Jeffreys prior is used.

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