# Inference on a genetic model

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This Dissertation deals with statistical inference on the mutation rates α₁ and α₂ of a population genetic model introduced by Moran [Proc. Camb. Phil. Soc. 54 (1958), pp. 60-71]. The deductive theory by approximate methods of such models has reached an advanced stage but little has been done along the line of statistical inference. Moran's model is a model of the Markov chain type. It was selected for investigation because it is the only finite population genetic model for which the cteductive theory by exact methods is well enough established to stimulate an investigation of statistical inference.

The first broad area of discussion of this dissertation deals with the simultaneous consideration of the mutation rates α₁ and α₂. Maximum likelihood estimates for α₁ and α₂ are obtained iteratively from the Newton-Raphson scheme for simultaneous solution of two equations in two unknowns. Several theorems are given which ensure that the log likelihood function involving α₁ and α₂ has a unique maximum in the parameter space of useful values.

The transition matrix consists of conditional probability elements involving the unknown parameters α₁ and α₂. These elements are the probability of a transition from one state to another in at most unit steps. The eigenvalue expression along with the corresponding pre- and post-eigenvector matrices are given. The post-eigenvector matrix has elements consisting of Hahn polynomials. The pre-eigenvector matrix is obtained by inverting the post-eigenvector matrix for which an expression is given. The Hahn polynomials form a family of orthogonal polynomials. They were introduced by Hahn [Math. Nach. 2 (1949), pp. 4-34], and further discussed by Karlin and McGregor [Scripta Math, 26 (1961), pp. 33-46]. These polynomials form the foundation and are basic to many of the results of the dissertation. The expression for the expected value of the number of transitions from one state to another is given and this expression is also in terms of Hahn polynomials.

Finally for this positively regular transition matrix involving both of the mutation rates α₁ and α₂, asymptotic multivariate normality of the maximum likelihood estimates α₁, α₂ is discussed along with hypothesis testing. Also discussed are large sample approximations., methods of designing and conducting experiments and replicated experiments.

The second broad area of this dissertation deals with an absorbing Markov chain. That is, α₂ is set equal to zero and investigation on α₁ only is carried out. For this case the above transition matrix becomes an absorbing one (regular) and inferences are obtained from realizations on this absorbing chain whose peculiarities provide some unique difficulties. The eigenvalue expression with the corresponding post-eigenvector matrix whose elements are also Hahn polynomials and the expression (in terms of Hahn polynomials) for the expected number of transitions from one state to another are all given.

Of particular interest are several postulated theorems on the maximum likelihood estimate α₁ of the mutation rate α₁ of the absorbing Markov chain in which an attempt is made at establishing the properties and normality of α₁. The estimate is again obtained iteratively. An outline of the proofs of the postulated theorems is presented. Gaps in the proof are a result of unresolved questions in positive regular Markov chain theory.

In connection with the above theory and postulated theorems a simulation study on the IBM 650 was undertaken. This study substantiated many of the assumptions of the postulated theorems. The study, however, was not extensive enough to be conclusive. A further study is proposed.

Replicated experiments are also discussed. Of particular interest here is a geometric type stopping rule in which the negative binomial is employed. Methods of conducting and designing experiments are discussed.

An appendix discusses the Hahn polynomial system along with many of its important properties.