Prof. Otieno Joseph A. M.

The objective of this research is to consider varying unemployment duration in the pricing of unemployment insurance with application to USA data. The study assumes that unemployment duration follows Burr XII mixture distribution while the discount rate to use in the pricing of the scheme will bedetermined by fitting market data into the capital asset pricing model. The Burr XII mixture distribution has been used to model unemployment duration in order to allow for heterogeniety in the unemployment duration of the insured employees. The results yield a mean unemployment duration of approximately 16 weeks and premium contribution rate of 5.10% of the taxable wage base per month for a benefit of 45% of the taxable wage base per month payable on weekly basis during spells of unemployment.

Keywords
Burr XII Mixture Distribution, Unemployment Insurace, Capital Asset Pricing Model, Taxable Wage Base, Discounted Cash Flow, Mean Present Value, Premium Rate

The hazard function of an exponential mixture characterizes an in- nitely divisible mixed Poisson distribution which is also a compound Poisson distribution. Given the hazard function, the probability generating functions (pgf) of the compound Poisson distribution and its independent and identically distributed (iid) random variables are derived.The recursive forms of the distributions are also given. Hofmann hazard function has been discussed and re-parameterized. The recursive form of the distribution of the iid random variables for the Hofmann distribution follows Panjer's model. Key words: Mixed Poisson distributions, Laplace transform, expo- nential mixtures, complete monotocity, in nite divisibility, compound Poisson distribution, Hofmann distributions, Panjer's model.

Mixed Poisson distributions can be expressed in explicit, recursive and expectation forms. It can also be expressed in terms of special functions. This paper expresses mixed Poisson distributions and their proba- bility generating functions in terms of Con uent Hypergeometric func- tions and modi ed Bessel function of the third kind. Keywords: Mixed Poisson; Con uent Hypergeometric; Bessel fun- tion; Probability generating function

In this paper, emphasis has been given to both the expected number of runs and the expected number of incorrect decisions and two stage group-screening designs have been obtained which minimise one fixing the other or minimise some sort of cost function which connects the two. Some group-screening plans have been given at the end as illustrations.
Keywords: group-screening plans, orthogonal main-effect plans, group-factor

Sometimes due to confidentiality purposes, the objective is to estimate the proportion P of individuals that posses a certain characteristic, such as a blood disease or an antibody, without necessarily identifying the individuals. In such a case, testing blood samples from k individuals may result in an estimator of P with substantial lower mean square error than traditional estimator. The mean square error varies with k and P , and we show a method for choosing the optimal value of k . Practical considerations as applied to determining proportions of people with a disease are discussed. Substantial cost savings may result when the prevalence of a disease is low.

Abstract

Mixed Poisson distributions are very significant in modeling non-homogeneous populations; for instance in Actuarial applications for modeling total claims in insurance. However, the setback is in their use since the probability mass functions are difficult to evaluate, except for a few mixing distributions. One way of dealing with this problem is to express Mixed Poisson distributions in terms of recursive relations.

In this paper, recursive relations of some mixed Poisson distributions are obtained by use of integration by parts technique.

Keywords: mixtures; recursive relation; generating functions; moments

This thesis is a study on the Dorfman-Sterrett group-screening designs assuming equal a-prior probabilities of items being defective. Based on a clear theoretical framework, we have studied both restricted and unrestricted Dorfman-Sterrett procedure without and without errors in decision, deriving expressions for expected number of tests (and cost functions), which are used to compare the performance of this procedures with the Dorfman procedure. \lVehave give an alternative approach to determining the expected number of tests in an unrestricted Dorfman-Sterrett design. The restricted Dorfman-Sterrett procedure without errors in decision has also be examined, giving conditions under which the restricted procedure converges to the unrestricted procedure. vVe have shown that for most prevalence rates, the two-stage Dorfman-Sterrett procedure performs just as well as the unrestricted procedure. We have in this thesis also examined the Restricted Dorfman-Sterrett procedure with error in decision. Expressions for the expected number of runs and cost functions have been derived. The results, based on the expected proportional red uct.ion in testing over individual testing, indicate that the single-step Dorfman-Sterrett procedure performs better than the Dorfman procedure for all the prevalence rates less than 30%. The result.s also indicate the t.here is little. if any, difference in performance between the single-step and the two-step Dorfman-Sterrett procedures. For the multi-step Modified Dorfman-Sterrett procedures without errors in decision we have derived the expected number. of runs and compared this results with the expected number of runs for the Dorfman procedure and the multi-step DorfmanSterrett procedure. In. addition, cost functions and expected number of tests for the multi-step Modified Dorfman-Sterrett procedures with errors in decision have also been derived. The results indicate that the modified procedure perform slightly better than Dorfman procedure fer most prevalence rates but is less efficient than the Dorfman-Sterrett procedure. The expressions for expected number of runs and the expected number of incorrect decisions in screening with errors are derived using vVatsons testing of hypothesis approach. Under the testing of hypothesis approach group factors are tested using orthogonal fractional factorial designs of the type giveil in Plackett and Burman (1946). We have also derived expressions for expected number of runs for both the restricted and the unrestricted Modified Dorfman-Sterrett procedures. The results indicate that the Modified Dorfman-Sterrett procedure performs better than the Dorfman procedure for prevalence rates less than 30%. There is also a greater saving for larger prevalence rates. There is a saving of 3~2%in using the Modified procedure instead of the Dorfman procedure when p = 0.29 compared with a s~,ving of 0.18% for p = 0.001. The results however indicate that Sterrett's procedure is more efficient than t.he Modified procedures for prevalence rates less than 30%. These figures seem to disprove the a.ssertion by Huang et al (1089) that a modified Dorfman-Sterrett procedure is more efficient than Sterrett's procedure when the a-priori probability of an item being defective, P. is somewha.t higher (though still quite low) .

Abstract

A Sum of hazard functions of exponential mixtures characterizes a convolution of in nitely divisible mixed Poisson distributions which is also a convolution of compound Poisson distri- butions. For each sum of two special cases of Hofmann hazard function, the following have been ob- tained: the probability generating function (pgf) of the convolution of the mixed Poisson distri- butions. the pgf of the independent and identically distributed (iid) random variables for the convolution of the compound Poisson distributions. the recursive form of the convolution of the compound Poisson distribution. We also wish to nd out whether Panjer's recursive model holds for all cases.

Key words: convolutions, exponential mixtures, mixed Poisson distribution, Hofmann hazard functions, characterization, compound Poisson distribution, Panjer's recursive model, Laplace transform

This paper aims at working out economic groupscreening plans to sort out defective items from a population which consists of tems with unequal a-priori probabilities of being defective. It is shown that in the case of group-screening from a population with unequal a-priori probabilities of factors being defective, the number of obseruations needed on the average is considerably smaller than that required in the case of a population with factors having the same a-priori probability of being defective. Tables at the end give some group-screening plans as illustrations.
Keywords: Group-factors, Group-Screening Plans

In this paper, we shall use clearly identified random variables required to study the Dorfman-Sterrett procedure first introduced by Sterrett. Based on the identified random variables, we shall derive probability distributions, conditional probability distributions and the expected number of runs (tests) in the Dorfman-Sterrett designs. Using combinatorial theory, we shall derive results obtained by earlier authors in a much simplified version.