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<p><b>1 Preliminary Tools and Foundations 1</b></p> <p>1.1 Preliminary Notation and Definitions 1</p> <p>1.2 Modes of Convergence of a Sequence of Random Variables 6</p> <p>1.3 Relationships Among the Modes of Convergence 9</p> <p>1.4 Convergence of Moments; Uniform Integrability 13</p> <p>1.5 Further Discussion of Convergence in Distribution 16</p> <p>1.6 Operations on Sequences to Produce Specified Convergence Properties 22</p> <p>1.7 Convergence Properties of Transformed Sequences 24</p> <p>1.8 Basic Probability Limit Theorems: The WLLN and SLLN 26</p> <p>1.9 Basic Probability Limit Theorems: The CLT 28</p> <p>1.10 Basic Probability Limit Theorems: The LIL 35</p> <p>1.11 Stochastic Process Formulation of the CLT 37</p> <p>1.12 Taylor’s Theorem; Differentials 43</p> <p>1.13 Conditions for Determination of a Distribution by Its Moments 45</p> <p>1.14 Conditions for Existence of Moments of a Distribution 46</p> <p>1.15 Asymptotic Aspects of Statistical Inference Procedures 47</p> <p>1.P Problems 52</p> <p><b>2 The Basic Sample Statistics 55</b></p> <p>2.1 The Sample Distribution Function 56</p> <p>2.2 The Sample Moments 66</p> <p>2.3 The Sample Quantiles 74</p> <p>2.4 The Order Statistics 87</p> <p>2.5 Asymptotic Representation Theory for Sample Quantiles Order Statistics and Sample Distribution Functions 91</p> <p>2.6 Confidence Intervals for Quantiles 102</p> <p>2.7 Asymptotic Multivariate Normality of Cell Frequency Vectors 107</p> <p>2.8 Stochastic Processes Associated with a Sample 109</p> <p>2.P Problems 113</p> <p><b>3 Transformations of Given Statistics 117</b></p> <p>3.1 Functions of Asymptotically Normal Statistics: Univariate Case 118</p> <p>3.2 Examples and Applications 120</p> <p>3.3 Functions of Asymptotically Normal Vectors 122</p> <p>3.4 Further Examples and Applications 125</p> <p>3.5 Quadratic Forms in Asymptotically Multivariate Normal Vectors 128</p> <p>3.6 Functions of Order Statistics 134</p> <p>3.P Problems 136</p> <p><b>4 Asymptotic Theory in Parametric Inference 138</b></p> <p>4.1 Asymptotic Optimality in Estimation 138</p> <p>4.2 Estimation by the Method of Maximum Likelihood 143</p> <p>4.3 Other Approaches toward Estimation 150</p> <p>4.4 Hypothesis Testing by Likelihood Methods 151</p> <p>4.5 Estimation via Product-Multinomial Data 160</p> <p>4.6 Hypothesis Testing via Product-Multinomial Data 165</p> <p>4.P Problems 169</p> <p><b>5 <i>U</i>-Statistics 171</b></p> <p>5.1 Basic Description of <i>U</i>-Statistics 172</p> <p>5.2 The Variance and Other Moments of a <i>U</i>-Statistic 181</p> <p>5.3 The Projection of a <i>U</i>-Statistic on the Basie Observations 187</p> <p>5.4 Almost Sure Behavior of <i>U</i>-Statistics 190</p> <p>5.5 Asymptotic Distribution Theory of <i>U</i>-Statistics 192</p> <p>5.6 Probability Inequalities and Deviation Probabilities for <i>U</i>-Statistics 199</p> <p>5.7 Complements 203</p> <p>5.P Problems 207</p> <p><b>6 Von Mises Differentiable Statistical Functions 210</b></p> <p>6.1 Statistics Considered as Functions of the Sample Distribution Function 211</p> <p>6.2 Reduction to a Differential Approximation 214</p> <p>6.3 Methodology for Analysis of the Differential Approximation 221</p> <p>6.4 Asymptotic Properties of Differentiable Statistical Functions 225</p> <p>6.5 Examples 231</p> <p>6.6 Complements 238</p> <p>6.P Problems 241</p> <p><b>7 <i>M</i>-Estimates 243</b></p> <p>7.1 Basic Formulation and Examples 243</p> <p>7.2 Asymptotic Properties of <i>M</i>-Estimates 248</p> <p>7.3 Complements 257</p> <p>7.P Problems 260</p> <p><b>8 <i>L</i>-Estimates</b></p> <p>8.1 Basic Formulation and Examples 262</p> <p>8.2 Asymptotic Properties of <i>L</i>-Estimates 271</p> <p>8.P Problems 290</p> <p><b>9 <i>R</i>-Estimates</b></p> <p>9.1 Basic Formulation and Examples 292</p> <p>9.2 Asymptotic Normality of Simple Linear Rank Statistics 295</p> <p>9.3 Complements 311</p> <p>9.P Problems 312</p> <p><b>10 Asymptotic Relative Efficiency</b></p> <p>10.1 Approaches toward Comparison of Test Procedures 314</p> <p>10.2 The Pitman Approach 316</p> <p>10.3 The Chernoff Index 325</p> <p>10.4 Bahadur’s “Stochastic Comparison,” 332</p> <p>10.5 The Hodges-Lehmann Asymptotic Relative Efficiency 341</p> <p>10.6 Hoeffding’s Investigation (Multinomial Distributions) 342</p> <p>10.7 The Rubin‒Sethuraman “Bayes Risk” Efficiency 347</p> <p>I0.P Problems 348</p> <p>Appendix 351</p> <p>References 553</p> <p>Author Index 365</p> <p>Subject Index 369</p>

"...even today it still provides a really good introduction into asymptotic statistics..."(Zentralblatt Math, Vol. 1001, No.01, 2003)

<b>ROBERT J. SERFLING</b>, PhD, is a Professor at the Department of Mathematical Sciences at the University of Texas at Dallas.

<i><b>Approximation Theorems of Mathematical Statistics</b></i> <p>This convenient paperback edition makes a seminal text in statistics accessible to a new generation of students and practitioners. Approximation Theorems of Mathematical Statistics covers a broad range of limit theorems useful in mathematical statistics, along with methods of proof and techniques of application. The manipulation of "probability" theorems to obtain "statistical" theorems is emphasized. Besides a knowledge of these basic statistical theorems, this lucid introduction to the subject imparts an appreciation of the instrumental role of probability theory.</p> <p>The book makes accessible to students and practicing professionals in statistics, general mathematics, operations research, and engineering the essentials of:</p> <ul> <li>The tools and foundations that are basic to asymptotic theory in statistics</li> <li>The asymptotics of statistics computed from a sample, including transformations of vectors of more basic statistics, with emphasis on asymptotic distribution theory and strong convergence</li> <li>Important special classes of statistics, such as maximum likelihood estimates and other asymptotic efficient procedures; W. Hoeffding’s U-statistics and R. von Mises’s "differentiable statistical functions"</li> <li>Statistics obtained as solutions of equations ("M-estimates"), linear functions of order statistics ("L-statistics"), and rank statistics ("R-statistics")</li> <li>Use of influence curves</li> <li>Approaches toward asymptotic relative efficiency of statistical test procedures</li> </ul>

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