This thesis is concerned with models and inference for single ion channels. Molecular modelling studies are used as the basis for biologically realistic, large state-space gating models of the nicotinic acetylcholine receptor which enable single-channel kinetic behaviour to be characterized in terms of a small number of free parameters. A model is formulated which incorporates known structural information concerning pentameric subunit composition, interactions between neighbouring subunits and knowledge of the behaviour of agonist binding sites within the receptor-channel proteins. Expressions are derived for various channel properties and results are illustrated using numerical examples. The model is adapted and extended to demonstrate how properties of the calcium ion-activated potassium ion channel may be modelled. A two-state stochastic model for ion channels which incorporates time interval omission is examined. Two new methods for overcoming a non-identifiability problem induced by time interval omission are introduced and simulation studies are presented in support of these methods. A framework is presented for analysing the asymptotic behaviour of the method-of-moments estimators of the mean lengths of open and closed sojourns. This framework is used to clarify the origin of the non-identifiability and to construct confidence sets for the mean sojourn lengths. A conjecture concerning the number of solutions of the moment estimating equations is proved.

Boundary hitting times for one-dimensional diffusion processes have applications in a variety of areas of mathematics. Unfortunately, for most choices of diffusions and boundaries, the exact exit distribution is unknown, and an approximation has to be made. The primary requirements of an approximation, from a practical viewpoint, is that it is both accurate and easily computable. The main, currently used approximations are discussed, and a new method is developed for two-sided boundaries, where current methodology provides very few techniques. In order to produce new approximations, we will make use of results about the ordering of stochastic processes, and conditioning processes not to have hit a boundary. These topics are introduced in full detail, and a number of results are proved. The ability to order conditioned processes is exploited to provide exact, analytic bounds on the exit distribution. This technique also produces a new approximation, which, for Brownian motion exiting concave or convex boundaries, is shown to be a superior approximation to the standard tangent approximation. To illustrate the uses of these approximations, and general boundary hitting time results, we investigate a class of optimal stopping problems, motivated by a sequential analysis problem. Properties of the optimal stopping boundary are found using analytic techniques for a wide class of cost functions, and both one- and two-sided boundaries. A number of results are proved concerning the expected stopping cost in cases of "near optimality". Numerical examples are used, throughout this thesis, to illustrate the principal results and exit distribution approximations.

Chapter I of this thesis considers the frequency theory of probability and in particular its treatmant of individual events. Attempts by frequentists to give an account of ordinary statements of probability which are generally about individual events are criticized at length: Reichenbach's attempts to deal with the problem of individual events is found to be unsatisfactory in virtue of his introduction of such dubious entities a 'fictitious probabilities'; Salmon's related suggestion to treat tha problem of the single case in terms of 'weights' determined by an 'application' of his theory of probability is also found to be unsatisfactory, for the concept of probability defined by his theory can not, on his own admission, be applied to the single case. Proposals by Keynes and Popper to treat the single case along frequentist lines, while apparently more promising, turn out in the end to be too rudimentary and sketchy to fulfill even the minimal condition for an adequate semantic definition of probability - the specification of a probability function which provides an interpretation of the axioms of probability. The basic difficulty for a frequentist in assigning probabilities to individual events is (not surprisingly) found in Chapter I to be that the probability of an individual event will vary depending on what reference class is chosen to determine its probability. In Chapter II however, the dependence of the probability of individual events on the reference class chosen is seen not to present a genuine difficulty in formulating an adequate semantic definition of probability. The variability of the probabilities assigned to individual events indicates that they are relational probabilities crucially involving a reference class in one term of the relation; once recognized the relational character of the probability of individual events provides the basis of an adequate semantic interpretation of the standard axioms of probability. It is shown that the resulting theory of probability can meet the objections raised by Salmon, Von Mises and Reichenbach against the assignment of probabilities to individual events and, indeed, this theory of probability resolves contradictions found to be inherent in Reichenbach and Salmon's treatment of the single case. Chapter III begins with a discussion of a methodological principle by which the relational theory given in Chapter II can be applied to actual situations to determine tbe unique probability values often needed for the purposes of action. This is the well-known principle of choosing the narrowest reference class for which statistics are available. This principle, usually referred to in the thesis as that of choosing the narrowest available reference class, is given a preliminary (and traditional) formulation at the start of Chapter III. Von Mises' requirement of randomness is then analyzed in light of this principle and it is found that if we employ this principle to determine unique probabilities, only random classes of events can be used to assign unique probabilities to every individual member of the class. This is found to account for the view - expressed in Van Mises' requirement of randomness - that only random classes are truly probabilistic. In Chapter IV various problems concerning the principle of choosing the narrowest available reference class are considered. Some of these are familiar, e.g. Ayer's objection to frequentists' use of this principle, the difficulties caused by unlawlike predicates, and the problems raised by the quite common absence of complete statistical knowledge. Less familiar problems are also considered. In particular Chapter IV is concerned with the question of precisely what kinds of reference classes can be considered 'available' for determining the probability of individual events. A solution to this problem is devised, based, in the first instance, on the concept of an effectively calculable function introduced by Church into discussions of randomness. A reference class is said to be available for determining the probability of an individual event if and only if there exists a determinate procedure which, if carried out, would yield the result that the event belongs to that reference class. In the case of reference classes formed by mathematical roles of selection, the existence of a determinate procedure leading to a result is equated with the existence of a recursive function leading to that result. In the case of reference classes based on empirical predicates, the class of determinate procedures is the totality of experimental procedures extant in the scientific field in question. This line of argument is extended to explicate the concept of 'available evidence', which figures prominently in many theories of probability. Various objections to this explication of available evidence are considered and a more refined analysis eventually emerges. Chapter V considers Carnap's thesis that there exist two distinct concepts of probability and it is shown that this conclusion can be avoided by adopting the frequency theory for the singe case given in Chapter II in place of the standard frequency theory - the definition of probability expressed in our theory is shown to be an instance of the definition of probability as a quantitative relation between evidence and hypothesis and thus to have distinct affinities to Carnap's theory of probability. Chapter V concludes with a discussion of subjectivism in theories which make probability a relation between evidence and hypothesis and it is shown how both our theory of probability and the logical relation theory avoid such subjectivism. Chapter VI and VII are concerned with the principle of indifference. Chapter VI begins with a survey of a variety of opinions on the principle and then presents a new interpretation of it as a rudimentary form of semantic definition of probability. It is shown that the principle, in stating conditions under which alternatives are equally probable, actually fixes identity conditions for the concept of probability and, as widely understood since Frege, any statement which fixes identity conditions for a concept fixes a particular sense for that concept, a particular semantics. The definition of probability encapsulated in the principle is that of probability as a comparative relation of evidential support. It is further explained how such a comparative conception of probability will have led to the numerical assignments of probability traditionally arrived at by use of the principle of indifference - when, as was the case with the principle's usual employment, a set of mutually exclusive and exhaustive hypothses are (comparatively) judged to be equally likely, it follows directly from the axioms of probability that each hypothesis has the numerical probability ¹-_n. That the principle of indifference encapsulates the semantic definition of probability as the comparative concept of evidential support is found to accord with the classical theorists' definition of probability as the ratio of favourable cases to possible cases, where each case is equi-possible. If we take the expression 'equi-possible' to be an undefined primitive term, the classical definition of probability constitutes a natural (and fruitful) uninterpreted axiomatic definition of probability.

In growth curve studies, measurements are made on individuals over a moderately large period of time and the main problem is set as the best possible evaluation of the curve which is believed to underlie the phenomenon. One important meaning of the term 'best possible evaluation' is the efficient estimation of the coefficients which, postulated by the model, characterize this curve. The important key for doing this, is the 'best' (in a certain way) estimation of the covariance matrix of observations which is supposed here to be the same for all individuals. The purpose of this thesis is to investigate certain methods which have been suggested to be appropriate for this problem both theoretically and empirically as well as to prove certain results concerning the problem of efficiency itself when an estimate of the covariance matrix of observations, irrespective of the method which calculated it, is at hand. More specifically, it is shown that REstricted Maximum Likelihood (REML) gives in general estimates of the regression coefficients whose estimated variance lies nearer to their true value quite independently from the form of the covariance matrix, than Maximum Likelihood (ML) and it is proved that the general estimated scatter of the estimated regression coefficients is greater for REML. When we measure more than one characteristics on time and for a special class of covariance models, the property that the REML estimates of the variance of estimated regression coefficients are larger than the corresponding ML, which holds for one characteristic, is lost. Asymptotically however, the two methods give identical results. Another method which is not based on the adoption of a certain parametric form for the covariance matrix is considered. A new method is suggested for its optimal choice and a comparison is made between this and three other already known methods. Empirical results suggest that it retains a very good balance between the variance of the regression coefficients and their real and optimal value for the majority of the covariance models which have been tested as possible population covariance matrices. Finally, upper and lower bounds of different types of efficiency are obtained by assuming that an estimate of the population covariance matrix has already been calculated and is not distant from the true covariance matrix more than a certain constant.

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Thesis (M.S.)--Ohio University, June, 1984. / Title from PDF t.p.

Thesis (LL. M.)--University of Melbourne, 1986. / Typescript (photocopy). Includes bibliographical references.

Thesis (M.A.) in Mathematics--University of Maine, 2008. / Includes vita. Includes bibliographical references (leaf 34).

Thesis (Ph. D.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Systems Science and Industrial Engineering, 2007. / Includes bibliographical references.