Applied Asymptotics: Case Studies in Small-Sample Statistics
In fields such as biology, medical sciences, sociology, and economics researchers often face the situation where the number of available observations, or the amount of available information, is sufficiently small that approximations based on the normal distribution may be unreliable. Theoretical work over the last quarter-century has led to new likelihood-based methods that lead to very accurate approximations in finite samples, but this work has had limited impact on statistical practice. This book illustrates by means of realistic examples and case studies how to use the new theory, and investigates how and when it makes a difference to the resulting inference. The treatment is oriented towards practice and comes with code in the R language (available from the web) which enables the methods to be applied in a range of situations of interest to practitioners. The analysis includes some comparisons of higher order likelihood inference with bootstrap or Bayesian methods.
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̂ ̂ adjusted profile analysis ancillary statistic Bartlett correction Bellio bibliographic notes binomial bootstrap calculate canonical parameter censoring Chapter Code components computed confidence intervals corresponding covariates Cox and Snell data set derivatives difference discussed dose exact example exponential family exponential family model expression Fisher information fitted function(th given higher order approximations higher order asymptotics higher order inference illustrate Laplace approximation likelihood function likelihood ratio statistic linear model linear regression log likelihood function logistic regression make.V marginal marginal likelihood matrix maximum likelihood estimate modified likelihood root nlogL nlreg package nonlinear regression normal approximation nuisance parameters obtained P-value panel of Figure parameter of interest plots posterior profile log likelihood quantities regression model residuals response sample space Section shows significance level simulation standard normal studentized residuals sufficient statistic Table tail area approximation tangent exponential model testing Total number values variables variance function variance parameters vector Wald pivot
Page 11 - Examples are presented to show how ambiguities may arise from attempts to define and apply analogues of sufficiency and ancillarity in the presence of nuisance parameters. It is necessary to tread with caution if one wishes to avoid inconsistencies or unexpected consequences inherent in principles which, on first acquaintance, appear unexceptionable.