Fred M. Hoppe

Asymptotic rates of growth of the extinction probability of a mutant gene

We prove that a result of Haldane (1927) that relates the asymptotic behaviour of the extinction probability of a slightly supercritical Poisson branching process to the mean number of offspring is true for a general Bienaymé-Galton-Watson branching process, provided that the second derivatives of the probability-generating functions converge uniformly to a non-zero limit. We show also by examples that such a result is true more widely than our proof suggests and exhibit some counter-examples.

The survival probability of a mutant in a multidimensional population.

We prove a general result about the asymptotic behaviour of the survival probability of a slightly supercritical multitype Bienaymé-Galton-Watson branching process. This is the complete analogue of a result which Ewens (1968) obtained for a Poisson branching process.

Improving probability bounds by optimization over subsets

The simple device of maximization over subsets of events can provide substantial improvement over the Dawson-Sankoff degree two lower bound on the probability of a union of events and can also exceed a sharper bound that uses individual and pairwise joint event probabilities developed by Kuai, Alajaji, and Takahara. In each of their examples, the maximized bound achieves the exact probability of the union using a subset of events containing no redundant events of the original set of events.

Beyond Inclusion-and-Exclusion: Natural Identities for P(exactly t events) and P(at least t events) and Resulting Inequalities

Summary We present universal identities for the probability of occurrence of at least t, or exactly t, out of n arbitrary events. Well-known, as well as new, bounds can be derived simply from these identities. The approach is probabilistic and provides a natural and unified approach to these inequalities. It also has the advantage of showing how these relationships are consequences of two elementary results: P(AB) = P(A) - P(AB); and P(E) > P(F) if E D F.

Simultaneous Confidence Intervals

These provide a confidence region for a multivariate parameter, comprising individual intervals for the separate components of the parameter. There is a specified confidence level of the simultaneous correctness of all the statements. The approach is useful for multiple comparisons of treatment means. The methods of Bonferroni, Tukey, Scheffe and Duncan are described, with examples. Keywords: multiple comparisons; Bonferroni; Scheffe; Tukey; Duncan; analysis of variance; studentized range

The effect of redundancy on probability bounds

Lower bounds on the probability of a union obtained by applying optimal bounds to subsets of events can provide excellent bounds. Comparisons are made with bounds obtained by linear programming and in the cases considered, the best bound is obtained with a subset that contains no redundant events contributing to the union. It is shown that redundant events may increase or decrease the value of a lower bound but surprisingly even removal of a non-redundant event can increase the bound.

Faà di Bruno’s formula and the distributions of random partitions in population genetics and physics

We show that the formula of Faà di Bruno for the derivative of a composite function gives, in special cases, the sampling distributions in population genetics that are due to Ewens and to Pitman. The composite function is the same in each case. Other sampling distributions also arise in this way, such as those arising from Dirichlet, multivariate hypergeometric, and multinomial models, special cases of which correspond to Bose-Einstein, Fermi-Dirac, and Maxwell-Boltzmann distributions in physics. Connections are made to compound sampling models.

A Hybrid Logistic Model for Case-Control Studies

For logistic regression in case-control studies, when risk factors associated with the outcome are exceedingly rare in the control group, the estimation of parameters in the model becomes difficult. In this paper, we propose a two-stage hybrid method to achieve this. In the first stage, we model the risk due to the rare factor, and in the second stage we model the residual risk due to the other factors using standard logistic model.

Bonferroni-type inequalities and the methods of indicators and polynomials

A difficulty in the application of the method of polynomials for deducing Bonferroni-type inequalities as exposited by Galambos, is investigated. The method, recast as the method of indicators, in a form due originally to Renyi, is applied to the situation of non-constant coefficients

Benford's law and distractors in multiple choice exams

ABSTRACT Suppose that in a multiple choice examination the leading significant digit of the correct options follows Benfords Law, while the leading digit of the distractors is uniform. Consider a strategy for guessing at answers that selects the option with the lowest leading digit with ties broken at random. We provide an expression for both the probability that this strategy selects the correct option and also the generalization to the probability of selecting the option with the lowest r significant digit string.