1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

Paternity inference using highly polymorphic codominant markers is becoming common in the study of natural populations. However, multiple males are often found to be genetically compatible with each offspring tested, even when the probability of excluding an unrelated male is high. While various methods exist for evaluating the likelihood of paternity of each nonexcluded male, interpreting these likelihoods has hitherto been difficult, and no method takes account of the incomplete sampling and error-prone genetic data typical of large-scale studies of natural systems. We derive likelihood ratios for paternity inference with codominant markers taking account of typing error, and define a statistic delta for resolving paternity. Using allele frequencies from the study population in question, a simulation program generates criteria for delta that permit assignment of paternity to the most likely male with a known level of statistical confidence. The simulation takes account of the number of candidate males, the proportion of males that are sampled and gaps and errors in genetic data. We explore the potentially confounding effect of relatives and show that the method is robust to their presence under commonly encountered conditions. The method is demonstrated using genetic data from the intensively studied red deer (Cervus elaphus) population on the island of Rum, Scotland. The Windows-based computer program, CERVUS, described in this study is available from the authors. CERVUS can be used to calculate allele frequencies, run simulations and perform parentage analysis using data from all types of codominant markers.

Statistical confidence for likelihood-based paternity inference in natural populations

In many important statistical applications, the number of variables or parameters p is much larger than the number of observations n. Suppose then that we have observations y=Xβ+z, where β∈Rp is a parameter vector of interest, X is a data matrix with possibly far fewer rows than columns, n≪p, and the zi’s are i.i.d. N(0, σ^2). Is it possible to estimate β reliably based on the noisy data y?

https://projecteuclid.org/download/pdfview_1/euclid.aos/1201012958

The Dantzig selector: Statistical estimation when p is much larger than n

Statistical Analysis with Missing Data

Conventional analyses of single spell duration models control for unobservables using a random effect estimator with the distribution of unobservables selected by ad hoc criteria. Both theoretical and empirical examples indicate that estimates of structural parameters obtained from conventional procedures are very sensitive to the choice of mixing distribution. Conventional procedures overparameterize duration models. We develop a consistent nonparametric maximum likelihood estimator for the distribution of unobservables and a computational strategy for implementing it. For a sample of unemployed workers our estimator produces estimates in concordance with standard search theory while conventional estimators do not. ECONOMIC THEORIES of search unemployment (Lippman and McCall [34]; Flinn and Heckman [14]), job turnover (Jovanovic [25]), mortality (Harris [17]), labor supply (Heckman and Willis [23]) and marital instability (Becker [3]) produce structural distributions for durations of occupancy of states. These theories generate qualitative predictions about the effects of changes in parameters on these structural distributions, and occasionally predict their functional forms.2 In order to test economic theories about durations and recover structural parameters, it is necessary to account for population variation in observed and unobserved variables unless it is assumed a priori that individuals are homogeneous.3 In every microeconomic study in which the hypothesis of heterogeneity is subject to test, it is not rejected. Temporally persistent unobserved components are an empirically important fact of life in microeconomic data (Heckman [19]). Since the appearance of papers by Silcock [39] and Blumen, Kogan, and McCarthy [5], social scientists have been aware that failure to adequately control for population heterogeneity can produce severe bias in structural estimates of duration models. Serious empirical analysts attempt to control for these unob

https://www.jstor.org/stable/pdfplus/1911491.pdf

A method for minimizing the impact of distributional assumptions in econometric models for duration data

Mixture models : theory, geometry, and applications

Genomic comparisons provide evidence for ancient genome-wide duplications in a diverse array of animals and plants. We developed a birth–death model to identify evidence for genome duplication in EST data, and applied a mixture model to estimate the age distribution of paralogous pairs identified in EST sets for species representing the basal-most extant flowering plant lineages. We found evidence for episodes of ancient genome-wide duplications in the basal angiosperm lineages including Nuphar advena (yellow water lily: Nymphaeaceae) and the magnoliids Persea americana (avocado: Lauraceae), Liriodendron tulipifera (tulip poplar: Magnoliaceae), and Saruma henryi (Aristolochiaceae). In addition, we detected independent genome duplications in the basal eudicot Eschscholzia californica (California poppy: Papaveraceae) and the basal monocot Acorus americanus (Acoraceae), both of which were distinct from duplications documented for ancestral grass (Poaceae) and core eudicot lineages. Among gymnosperms, we found equivocal evidence for ancient polyploidy in Welwitschia mirabilis (Gnetales) and no evidence for polyploidy in pine, although gymnosperms generally have much larger genomes than the angiosperms investigated. Cross-species sequence divergence estimates suggest that synonymous substitution rates in the basal angiosperms are less than half those previously reported for core eudicots and members of Poaceae. These lower substitution rates permit inference of older duplication events. We hypothesize that evidence of an ancient duplication observed in the Nuphar data may represent a genome duplication in the common ancestor of all or most extant angiosperms, except Amborella.

/pdf/widespread-genome-duplications-throughout-the-history-of-2o2h7rwoo1.pdf

Widespread genome duplications throughout the history of flowering plants

In this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set. The existence, support size, likelihood equations, and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull. It is shown using geometric techniques that the estimator exists under quite general conditions, with a support size no larger than the number of distinct observations. Analysis of the convex dual of the likelihood set leads to a dual maximization problem. A convergent algorithm is described. The defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures. Sufficient conditions for uniqueness are given. Part II will deal with a special theory for exponential family mixtures.

/pdf/the-geometry-of-mixture-likelihoods-a-general-theory-fcznmj6q98.pdf

The Geometry of Mixture Likelihoods: A General Theory

It is shown how and why the influence curve poorly measures the robustness properties of minimum Hellinger distance estimation. Rather, for this and related forms of estimation, there is another function, the residual adjustment function, that carries the relevant information about the trade-off between efficiency and robustness. It is demonstrated that this function determines various second-order measures of efficiency and robustness through a scalar measure called the estimation curvature. The function is also shown to determine the breakdown properties of the estimators through its tail behavior. A 50% breakdown result is given. It is shown how to create flexible classes of estimation methods in the spirit of $M$-estimation, but with first-order efficiency (or even second-order efficiency) at the chosen model, 50% breakdown and a minimum distance interpretation.

/pdf/efficiency-versus-robustness-the-case-for-minimum-hellinger-1dtb6v0e8q.pdf

Efficiency versus robustness : the case for minimum Hellinger distance and related methods

SUMMARY Generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified. However, under such misspecification, the estimator of the regression parameter can be inefficient. In this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter, and that remains optimal even if the working correlation structure is misspecified. The idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices, a representation that is valid for the working correlations most commonly used. Both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations. This approach also provides a chi-squared inference function for testing nested models and a chi-squared regression misspecification test. Furthermore, the test statistic follows a chi-squared distribution asymptotically whether or not the working correlation structure is correctly specified.

Bruce G. Lindsay

Papers

Mixture models : theory, geometry, and applications

Widespread genome duplications throughout the history of flowering plants

The Geometry of Mixture Likelihoods: A General Theory

Efficiency versus robustness : the case for minimum Hellinger distance and related methods

Improving generalised estimating equations using quadratic inference functions