《MATHEMATICAL STATISTICS》

CHAPTER Ⅰ．INTRODUCTION1

CHAPTER Ⅱ．DISTRIBUTION FUNCTIONS5

2．1Cumulative Distribution Functions5

2．11 Univariate Case5

2．12 Bivariate Case8

2．13 k-Variate Case11

2．2 Marginal Distributions12

2．3 Statistical Independence13

2．4 Conditional Probability15

2．5The Stieltjes Integral17

2．51 Univariate Case17

2．52 Bivariate Case20

2．53 k-Variate Case21

2．6Transformetion of Variables23

2．61 Univariate Case24

2．62 Bivariate Case24

2．63 k-Variate Case28

2．7Mean Value29

2．71 Univariate Case;Tohebycheff＇s Inequality30

2．72 Bivariate Case31

2．73 k-Variate Case32

2．74 Mean and Variance of a Linear Combination of Random Variables33

2．75 Covariance and Correlation between two Linear Combinations of Random Variables34

2．76 The Moment Problem35

2．8Moment Generating Functions36

2．81 Univariate Case36

2．82 Multivariate Case39

2．9Regression40

2．91 Regression Functions40

2．92 Variance about Regression Functions41

2．93 Partial Correlation42

2．94 Multiple Correlation42

CHAPTER Ⅲ．SOME SPECIAL DISTRIBUTIONS47

3．1Discrete Distributions47

3．11 Binomial Distribution47

3．12 Multinomial Distribution50

3．13 The Poisson Distribution52

3．14 The Negative Binomial Distribution54

3．2The Normal Distribution56

3．21 The Univariate Case56

3．22 The Normal Bivarlate Distribution59

3．23 The Normal Multivariate Distribution63

3．5 Pearson System of Distribution Functions72

3．4 The Gram-Charlier Series76

CHAPTER Ⅳ．SAMPLING THEORY79

4．1General Remarks79

4．2Application of Theorems on Mean Values to Sampling Theory80

4．21 Distribution of Sample Mean81

4．22 Expected Value of Sample Variance83

4．3 Sampling from a Finite Population83

4．4Representative Sampling86

4．41 Sampling when the Pi are known87

4．42 Sampling when the σi are also known88

4．5Sampling Theory of Order Statistics89

4．51 Simultaneous Distribution of any k Order Statistics89

4．52 Distribution of Largest(or Smallest)Variate91

4．53 Distribution of Median91

4．54 Distribution of Sample Range92

4．55 Tolerance Limits93

4．6 Mean Values of Sample Moments when Sample Values are Grouped;Sheppard Corrections94

4．7 Appendix on Lagrange＇s Multipliers97

CHAPTER Ⅴ．SAMPLING FROM A NORMAL POPULATION98

5．1Distribution of Sample Mean98

5．11 Distribution of Difference between Two Sample Means100

5．12 Joint Distribution of Means in Samples from a Normal Bivariate Distribution100

5．2The X2-distribution102

5．21 Distribution of Sum of Squares of Normally and Independently Distributed Variables102

5．22 Distribution of the Exponent in a Multivariate Normal Distribution103

5．23 Reproductive Property of X2-Distribution105

5．24 Cochran＇s Theorem105

5．25 Independence of Mean and Sum of Squared Deviations from Mean in Samples from a Normal Population108

5．3 The＂Student＂t-Distribution110

5．4 Snedecor＇s F-Distribution113

5．5 Distribution of Second Order Sample Moments in Samples from a Bivariate Normal Distribution116

5．6 Independence of Second Order Moments and Means in Samples from a Normal Multivariate Distribution120

CHAPTER Ⅵ．ON THE THEORY OF STATISTICAL ESTIMATION122

6．1Confidence Intervals and Confidence Regions122

6．11 Case in which the Distribution Depends on only one Parameter122

6．12 Confidence Limits from Large Samples127

6．13 Confidence Intervals in the Case where the Distribution Depends on Several Parameters130

6．14 Confidence Regions132

6．2Point Estimetion:Maximum Likelihood Statistics133

6．21 Consistency133

6．22 Efficiency134

6．23 Sufficiency135

6．24 Maximum Likelihood Estimates136

6．3 Tolerance Interval Estimation142

6．4 The Fitting of Distribution Functions145

CHAPTER Ⅶ．TESTS OF STATISTICAL HYPPTHESES147

7．1Statistical Tests Related to Confidence Intervals147

7．2 Likelihood Ratio Tests150

7．3 The Neyman-Pearson Theory of Testing Hypotheses152

CHAPTER Ⅷ．NORMAL REGRESSION THEORY157

8．1Case of One Fixed Variate157

8．2 The Case of k Fixed Variates160

8．3 A General Normal Regression Significance Test166

8．4Remarks on the Generality of Theorem(A),8．3171

8．41 Case 1171

8．42 Case 2172

8．43 Case 3173

8．5 The Minimum of a sum of Squares of Deviations with Respect to Regression Coefficients which are Subject to Linear Restrictions174

CHAPTER Ⅸ．APPLICATIONS OF NORMAL REGRESSION THEORY TO ANALYSIS OF VARIANCE PROBLEMS176

9．1Testing for the Equality of Means of Normal Populations with the Same Variance176

9．2 Randomized Blocks or Two-way Layouts177

9．3 Three-way and Higher Order Layouts;Interaction181

9．4 Latin Squares186

9．5 Graeco-Latin Squares190

9．6 Analysis of Variance in Incomplete Layouts192

9．7 Analysis of Covariance195

CHAPTER Ⅹ．ON COMBINATORIAL STATISTICAL THEORY200

10．1On the Theory of Runs200

10．11 Case of Two Kinds of Elements200

10．12 Case of k Kinds of Elements205

10．2 Application of Run Theory to Ordering Within Samples206

10．3Matching Theory208

10．31 Case of Two Decks of Cards208

10．32 Case of Three or More Decks of Cards212

10．4Independence in Contingency Tables213

10．41 The Fartitional Approach213

10．42 Karl Pearson＇s Original Chi-Square Problems and its Application to Contingency Tables217

10．5Sampling Inspection220

10．51 Single Sampling Inspection221

10．52 Double Sampling Inspection224

CHAPTER Ⅺ．AN INTRODUCTION TO MULTIVARIATE STATISTICAI ANALYSIS226

11．1The Wishart Distribution226

11．2 Reproductive Property of the Wiahart Distribution232

11．3 The Independence of Means and Second Order Moments in Samples from a Normal Multivariate Population233

11．4 Hotelling＇s Generelized＂Student＂Test234

11．5 The Hypothesis of Equality of Means in Multivariate Normal Populations238

11．6 The Hypothesis of Independence of Sets of Variables in a Normal Multivariate Population242

11．7 Linear Regression Theory in Normal Multivariate Populations245

11．8 Remarks on Multivariate Analysis of Variance Theory250

11．9 Principal Components of a Total Variance252

11．10 Canonical Correlation Theory257

11．11The Sampling Theory of the Roots of Certain Determinantal Equations260

11．111 Characteristic Roots of One Sample Variance-covariance Matrix261

11．112 Characteristic Roots of the Difference of Two Sample Variance-covariance Matrices265

11．113 Distribution of the Sample Canonical Correlations268

LITERATURE FOR SUPPLEMENTARY READING271

INDEX279