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olive d statistical theory inference 2014

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Textbook in PDF format Many math and some statistics departments offer a one-semester graduate course in statistical theory using texts such as Casella and Berger (2002), Bickel and Doksum (2007) or Mukhopadhyay (2000, 2006). The course typically covers minimal and complete sufficient statistics, maximum likelihood estimators (MLEs), method of moments, bias and mean square error, uniform minimum variance unbiased estimators (UMVUEs) and the Fr´echet–Cram´er–Rao lower bound (FCRLB), an introduction to large sample theory, likelihood ratio tests, and uniformly most powerful (UMP) tests and the Neyman–Pearson Lemma. A major goal of this text is to make these topics much more accessible to students by using the theory of exponential families. One of the most important uses of exponential families is that the theory often provides two methods for doing inference. For example, minimal sufficient statistics can be found with either the Lehmann Scheff´e theorem or by finding T from the exponential family parameterization. Similarly, if Y1, . . . ,Yn are iid from a one parameter regular exponential family with complete sufficient statistic T(Y), then one-sided UMP tests can be found by using the Neyman–Pearson lemma or by using exponential family theory. The prerequisite for this text is a calculus-based course in statistics at the level of Hogg and Tanis (2005), Larsen and Marx (2011), Wackerly et al. (2008), or Walpole et al. (2006). Also see Arnold (1990), Gathwaite et al. (2002), Spanos (1999), Wasserman (2004), and Welsh (1996). The following intermediate texts are especially recommended: DeGroot and Schervish (2012), Hogg et al. (2012), Rice (2006), and Rohatgi (1984). A less satisfactory alternative prerequisite is a calculus-based course in probability at the level of Hoel et al. (1971), Parzen (1960), or Ross (2009). A course in Real Analysis at the level of Bartle (1964), Gaughan (2009), Rosenlicht (1985), Ross (1980), or Rudin (1964) would be useful for the large sample theory chapter. Probability and Expectations Probability Counting Conditional Probability and Independence The Expected Value and Variance The Kernel Method Mixture Distributions Complements Problems Multivariate Distributions and Transformations Joint, Marginal, and Conditional Distributions Expectation, Covariance, and Independence Conditional Expectation and Variance Location–Scale Families Transformations Sums of Random Variables Random Vectors The Multinomial Distribution The Multivariate Normal Distribution Elliptically Contoured Distributions Complements Problems Exponential Families Regular Exponential Families Properties of (t1(Y),...,tk(Y)) Complements Problems Sufficient Statistics Statistics and Sampling Distributions Minimal Sufficient Statistics Complements Problems Point Estimation I Maximum Likelihood Estimators Method of Moments Estimators Complements Problems Point Estimation II MSE and Bias Exponential Families, UMVUEs, and the FCRLB Complements Problems Testing Statistical Hypotheses Hypothesis Tests and Power Exponential Families, the Neyman–Pearson Lemma, and UMP Tests Likelihood Ratio Tests Complements Problems Large Sample Theory The CLT, Delta Method, and an Exponential Family Limit Theorem Asymptotically Efficient Estimators Modes of Convergence and Consistency Slutsky’s Theorem and Related Results Order Relations and Convergence Rates Multivariate Limit Theorems More Multivariate Results Complements Problems Confidence Intervals Some Examples Bootstrap and Randomization Tests Complements Problems Some Useful Distributions The Beta Distribution The Beta-Binomial Distribution The Binomial and Bernoulli Distributions The Birnbaum Saunders Distribution The Burr Type III Distribution The Burr Type X Distribution The Burr Type XII Distribution The Cauchy Distribution The Chi Distribution The Chi-Square Distribution The Discrete Uniform Distribution The Double Exponential Distribution The Exponential Distribution The Two-Parameter Exponential Distribution The F Distribution The Gamma Distribution The Generalized Gamma Distribution The Generalized Negative Binomial Distribution The Geometric Distribution The Gompertz Distribution The Half Cauchy Distribution The Half Logistic Distribution The Half Normal Distribution The Hypergeometric Distribution The Inverse Exponential Distribution The Inverse Gaussian Distribution The Inverse Weibull Distribution The Inverted Gamma Distribution The Largest Extreme Value Distribution The Logarithmic Distribution The Logistic Distribution The Log-Cauchy Distribution The Log-Gamma Distribution The Log-Logistic Distribution The Lognormal Distribution The Maxwell–Boltzmann Distribution The Modified DeMoivre’s Law Distribution The Negative Binomial Distribution The Normal Distribution The One-Sided Stable Distribution The Pareto Distribution The Poisson Distribution The Power Distribution The Two-Parameter Power Distribution The Rayleigh Distribution The Smallest Extreme Value Distribution The Student’s t Distribution The Topp–Leone Distribution The Truncated Extreme Value Distribution The Uniform Distribution The Weibull Distribution The Zero Truncated Poisson Distribution The Zeta Distribution The Zipf Distribution Complements Bayesian Methods Bayes’ Theorem Bayesian Point Estimation Bayesian Inference Bayesian Computation Complements Problems Stuff for Students R Statistical Software Hints and Solutions to Selected Problems Tables

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olive d statistical theory inference 2014

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