Mathematical Statistics With Applications

Students with a background that is less focused on rigorous mathematical theory but more applied in their chosen field often have a difficult time with the theoretical gravity of a course in mathematical statistics. Unfortunately for these students, most textbooks supply only a cursory, if any, review of the prerequisite topics for the course. Mathematical Statistics with Applications provides a thorough grounding in the prerequisite material along with an extensive link to the applications of the theory to the student's area of interest. This is a crucial resource for students who are not applied statisticians, but who wish to apply statistics to enrich their field of interest.

Mathematical statistics typically represents one of the most difficult challenges in statistics, particularly for those with more applied, rather than mathematical, interests and backgrounds. Most textbooks on the subject provide little or no review of the advanced calculus topics upon which much of mathematical statistics relies and furthermore contain material that is wholly theoretical, thus presenting even greater challenges to those interested in applying advanced statistics to a specific area.Mathematical Statistics with Applications presents the background concepts and builds the technical sophistication needed to move on to more advanced studies in multivariate analysis, decision theory, stochastic processes, or computational statistics. Applications embedded within theoretical discussions clearly demonstrate the utility of the theory in a useful and relevant field of application and allow readers to avoid sudden exposure to purely theoretical materials.With its clear explanations and more than usual emphasis on applications and computation, this text reaches out to the many students and professionals more interested in the practical use of statistics to enrich their work in areas such as communications, computer science, economics, astronomy, and public health.

INTRODUCTIONREVIEW OF MATHEMATICSIntroductionCombinatoricsPascal's TriangleNewton's Binomial FormulaExponential FunctionStirling's FormulaMultinomial TheoremMonotonic FunctionsConvergence and DivergenceTaylor's TheoremDifferentiation and SummationSome Properties of IntegrationIntegration by PartsRegion of FeasibilityMultiple IntegrationJacobianMaxima and MinimaLagrange MultiplierL'Hôpital's RulePartial Fraction ExpansionCauchy-Schwarz InequalityGenerating FunctionsDifference EquationsVectors, Matrices and DeterminantsReal NumbersPROBABILITY THEORYIntroductionSubjective Probability, Relative Frequency and Empirical ProbabilitySample SpaceDecomposition of a Union of Events: Disjoint EventsSigma Algebra and Probability SpaceRules and Axioms of Probability TheoryConditional ProbabilityLaw of Total ProbabilityBayes RuleSampling With and Without ReplacementProbability and SIMULATIONBorel SetsMeasure Theory in ProbabilityApplication of Probability Theory: Decision AnalysisRANDOM VARIABLESIntroductionDiscrete Random VariablesCumulative Distribution FunctionContinuous Random VariablesJoint DistributionsIndependent Random VariablesDistribution of the Sum of Two Independent Random VariablesMoments, Expected Values and VarianceCovariance and CorrelationDistribution of a Function of a Random VariableMultivariate Distributions and Marginal DensitiesConditional ExpectationsConditional Variance and CovarianceMoment Generating FunctionsCharacteristic FunctionsProbability Generating FunctionsDISCRETE DISTRIBUTIONSIntroductionBernoulli DistributionBinomial DistributionMultinomial DistributionHypergeometric Distributionk-Variate Hypergeometric DistributionGeometric DistributionNegative Binomial DistributionNegative Multinomial DistributionPoisson DistributionDiscrete Uniform DistributionLesser Known DistributionsJoint DistributionsConvolutionsCompound DistributionsBranching ProcessesHierarchical DistributionsCONTINUOUS RANDOM VARIABLESLocation and Scale ParametersDistribution of Functions of Random VariablesUniform DistributionNormal DistributionExponential DistributionPoisson ProcessGamma DistributionBeta DistributionChi-square DistributionStudent's t-DistributionF-DistributionCauchy DistributionExponential FamilyHierarchical Models-Mixture DistributionsOther DistributionsDistributional RelationshipsAdditional Distributional FindingsDISTRIBUTIONS OF ORDER STATISTICSIntroductionRank OrderingThe Probability Integral TransformationDistributions of Order Statistics in i.i.d. SamplesExpectations of Minimum and Maximum Order StatisticsDistributions of Single Order StatisticsJoint Distributions of Order StatisticsASYMPTOTIC DISTRIBUTION THEORYIntroductionIntroducing Probability to the Limit ProcessIntroduction to Convergence in DistributionNon-convergenceIntroduction to Convergence in ProbabilityConvergence Almost Surely (with Probability One)Convergence in rth MeanRelationships Between Convergence ModalitiesApplication of Convergence in DistributionProperties of Convergence in ProbabilityThe Law of Large Numbers and Chebyshev's InequalityThe Central Limit TheoremProof of the Central Limit TheoremThe Delta MethodConvergence Almost Surely (with probability one)POINT ESTIMATIONIntroductionMethod of Moments EstimatorsMaximum Likelihood EstimatorsBayes EstimatorsSufficient StatisticsExponential FamiliesOther Estimators*Criteria of a Good Point EstimatorHYPOTHESIS TESTINGStatistical Reasoning and Hypothesis TestingDiscovery, the Scientific Method, and Statistical Hypothesis TestingSimple Hypothesis TestingStatistical SignificanceThe Two Sample TestTwo Sided vs. One Sided TestingLikelihood Ratios and the Neyman Pearson LemmaOne SampleTesting and the Normal DistributionTwo Sample Testing for the Normal DistributionLikelihood Ratio Test and the Binomial DistributionLikelihood Ratio Test and the Poisson DistributionThe Multiple Testing IssueNonparametric TestingGoodness of Fit TestingFisher's Exact TestSample Size ComputationsINTERVAL ESTIMATIONIntroductionDefinitionConstructing Confidence IntervalsBayesian Credible IntervalsApproximate Confidence Intervals and MLE PivotThe Bootstrap Method*Criteria of a Good Interval EstimatorConfidence Intervals and Hypothesis TestsINTRODUCTION TO COMPUTATIONAL METHODSThe Newton-Raphson MethodThe EM AlgorithmSimulationMarkov ChainsMarkov Chain Monte Carlo MethodsINDEX

Asha Seth Kapadia, Wenyaw Chan, Lemuel A. Moyé