PubMed · 9497267
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P M Karunaratne, R C Elston. 1998. Likelihood calculation conditional on observed pedigree structure.. https://doi.org/10.1086/301770
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The maximum-likelihood (ML) solution to a simple phylogenetic estimation problem is obtained analytically The problem is estimation of the rooted tree for three species using binary characters with a symmetrical rate of substitution under the molecular clock. ML estimates of branch lengths and log-likelihood scores are obtained analytically for each of the three rooted binary trees. Estimation of the tree topology is equivalent to partitioning the sample space (space of possible data outcomes) into subspaces, within each of which one of the three binary trees is the ML tree. Distance-based least squares and parsimony-like methods produce essentially the same estimate of the tree topology, although differences exist among methods even under this simple model. This seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogeny estimation. The solution to this real phylogeny estimation problem will be useful for studying the problem of significance evaluation.
Inferences on mixtures of probability distributions, in general, and of life distributions, in particular, are receiving considerable importance in recent years. The likelihood ratio procedure of testing for the null hypothesis of no contamination is often very cumbersome and lacks its usual asymptotic properties. Recently, SenGupta (1991) has introduced the notion of an 'L-optimal' test for such testing problems. The idea is to recast the original several parametric hypotheses representation of the null hypothesis in terms of only a single hypothesis involving an appropriately chosen parametric function. This approach is shown to be both mathematically elegant and operationally simple for a quite general class of mixture distributions which contains, in particular, all mixtures of the one-parameter exponential family and also a very rich subclass of mixtures useful in life-testing and reliability analysis. It is also illustrated through two examples--one based on real-life data and the other on a simulated sample.