Probability and Statistical Inference

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CRC Press, Mar 22, 2000 - Mathematics - 665 pages
Priced very competitively compared with other textbooks at this level!
This gracefully organized textbook reveals the rigorous theory of probability and statistical inference in the style of a tutorial, using worked examples, exercises, numerous figures and tables, and computer simulations to develop and illustrate concepts.

Beginning with an introduction to the basic ideas and techniques in probability theory and progressing to more rigorous topics, Probability and Statistical Inference
  • studies the Helmert transformation for normal distributions and the waiting time between failures for exponential distributions
  • develops notions of convergence in probability and distribution
  • spotlights the central limit theorem (CLT) for the sample variance
  • introduces sampling distributions and the Cornish-Fisher expansions
  • concentrates on the fundamentals of sufficiency, information, completeness, and ancillarity
  • explains Basu's Theorem as well as location, scale, and location-scale families of distributions
  • covers moment estimators, maximum likelihood estimators (MLE), Rao-Blackwellization, and the Cramér-Rao inequality
  • discusses uniformly minimum variance unbiased estimators (UMVUE) and Lehmann-Scheffé Theorems
  • focuses on the Neyman-Pearson theory of most powerful (MP) and uniformly most powerful (UMP) tests of hypotheses, as well as confidence intervals
  • includes the likelihood ratio (LR) tests for the mean, variance, and correlation coefficient
  • summarizes Bayesian methods
  • describes the monotone likelihood ratio (MLR) property
  • handles variance stabilizing transformations
  • provides a historical context for statistics and statistical discoveries
  • showcases great statisticians through biographical notes

    Employing over 1400 equations to reinforce its subject matter, Probability and Statistical Inference is a groundbreaking text for first-year graduate and upper-level undergraduate courses in probability and statistical inference who have completed a calculus prerequisite, as well as a supplemental text for classes in Advanced Statistical Inference or Decision Theory.
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    Selected pages

    Contents

    12 About Sets
    		3
    13 Axiomatic Development of Probability
    		6
    14 The Conditional Probability and Independent Events
    		9
    141 Calculus of Probability
    		12
    142 Bayess Theorem
    		14
    143 Selected Counting Rules
    		16
    15 Discrete Random Variables
    		18
    151 Probability Mass and Distribution Functions
    		19
    661 Complete Sufficient Statistics
    		320
    662 Basus Theorem
    		324
    67 Exercises and Complements
    		327
    7 Point Estimation
    		341
    72 Finding Estimators
    		342
    722 The Method of Maximum Likelihood
    		344
    73 Criteria to Compare Estimators
    		351
    732 Best Unbiased and Linear Unbiased Estimators
    		354

    16 Continuous Random Variables
    		23
    162 The Median of a Distribution
    		28
    17 Some Standard Probability Distributions
    		32
    171 Discrete Distributions
    		33
    172 Continuous Distributions
    		37
    18 Exercises and Complements
    		50
    2 Expectations of Functions of Random Variables
    		65
    221 The Bernoulli Distribution
    		71
    222 The Binomial Distribution
    		72
    223 The Poisson Distribution
    		73
    226 The Laplace Distribution
    		76
    23 The Moments and Moment Generating Function
    		77
    231 The Binomial Distribution
    		80
    23 2 The Poisson Distribution
    		81
    233 The Normal Distribution
    		82
    234 The Gamma Distribution
    		84
    24 Determination of a Distribution via MGF
    		86
    25 The Probability Generating Function
    		88
    26 Exercises and Complements
    		89
    Multivariate Random Variables
    		99
    32 Discrete Distributions
    		100
    321 The Joint Marginal and Conditional Distributions
    		101
    322 The Multinomial Distribution
    		103
    33 Continuous Distributions
    		107
    332 Three and Higher Dimensions
    		115
    34 Covariances and Correlation Coefficients
    		119
    341 The Multinomial Case
    		124
    35 Independence of Random Variables
    		125
    36 The Bivariate Normal Distribution
    		131
    37 Correlation Coefficient and Independence
    		139
    38 The Exponential Family of Distributions
    		141
    382 Multiparameter Situation
    		144
    39 Some Standard Probability Inequalities
    		145
    392 Tchebysheffs Inequality
    		148
    393 CauchySchwarz and Covariance Inequalities
    		149
    394 Jensens and Lyapunovs Inequalities
    		152
    395 Holders Inequality
    		156
    396 Bonferroni Inequality
    		157
    397 Central Absolute Moment Inequality
    		158
    310 EXercises and Complements
    		159
    Functions of Random Variables and Sampling Distribution
    		177
    42 Using Distribution Functions
    		179
    422 Continuous Cases
    		181
    423 The Order Statistics
    		182
    424 The C0nvolution
    		185
    425 The Sampling Distribution
    		187
    43 Using the Moment Generating Function
    		190
    44 A General Approach with Transformations
    		192
    441 Several Variable Situations
    		195
    45 Special Sampling Distributions
    		206
    451 The Students t Distribution
    		207
    452 The F Distribution
    		209
    453 The Beta Distribution
    		211
    46 Special Continuous Multivariate Distributions
    		212
    462 The t Distribution
    		218
    463 The F Distribution
    		219
    47 Importance of Independence in Sampling Distributions
    		220
    472 Reproductivity of Chisquare Distributions
    		221
    473 The Students t Distribution
    		223
    48 Selected Review in Matrices and Vectors
    		224
    49 Exercises and Complements
    		227
    Concepts of Stochastic Convergence
    		241
    52 Convergence in Probability
    		242
    53 Convergence in Distribution
    		253
    531 Combination of the Modes of Convergence
    		256
    532 The Central Limit Theorems
    		257
    54 Convergence of Chisquare t and F Distributions
    		264
    543 The F Distribution
    		265
    55 Exercises and Complements
    		270
    6 Sufficiency Completeness and Ancillarity
    		281
    62 Sufficiency
    		282
    621 The Conditional Distribution Approach
    		284
    622 The Neyman Factorization Theorem
    		288
    63 Minimal Sufficiency
    		294
    631 The LehmannScheffi Approach
    		295
    64 Information
    		300
    641 Oneparameter Situation
    		301
    642 Multiparameter Situation
    		304
    65 Ancillarity
    		309
    651 The Location Scale and LocationScale Families
    		314
    652 Its Role in the Recovery of Information
    		316
    66 Completeness
    		318
    74 Improved Unbiased Estimators via Sufficiency
    		358
    75 Uniformly Minimum Variance Unbiased Estimator
    		365
    751 The CramerRao Inequality and UMVUE
    		366
    752 The LehmannScheffe Theorems and UMVUE
    		371
    753 A Generalization of the CramerRao Inequality
    		374
    754 Evaluation of Conditional Expectations
    		375
    76 Unbiased Estimation Under Incompleteness
    		377
    77 Consistent Estimators
    		380
    78 Exercises and Complements
    		382
    8 Tests of Hypotheses
    		395
    82 Error Probabilities and the Power Function
    		396
    821 The Concept of a Best Test
    		399
    83 Simple Null Versus Simple Alternative Hypotheses
    		401
    No Parameters Are Involved
    		413
    Observations Are NonIID
    		416
    84 OneSided Composite Alternative Hypothesis
    		417
    842 Monotone Likelihood Ratio Property
    		420
    843 UMP Test via MLR Property
    		422
    85 Simple Null Versus TwoSided Alternative Hypotheses
    		425
    852 An Example Where UMP Test Exists
    		426
    853 Unbiased and UMP Unbiased Tests
    		428
    86 Exercises and Complements
    		429
    9 Confidence Interval Estimation
    		441
    92 OneSample Problems
    		443
    921 Inversion of a Test Procedure
    		444
    922 The Pivotal Approach
    		446
    923 The Interpretation of a Confidence Coefficient
    		451
    924 Ideas of Accuracy Measures
    		452
    925 Using Confidence Intervals in the Tests of Hypothesis
    		455
    93 TwoSample Problems
    		456
    932 Comparing the Scale Parameters
    		460
    94 Multiple Comparisons
    		463
    942 Comparing the Means
    		465
    943 Comparing the Variances
    		467
    95 Exercises and Complements
    		469
    10 Bayesian Methods
    		477
    102 Prior and Posterior Distributions
    		479
    103 The Conjugate Priors
    		481
    104 Point Estimation
    		485
    105 Credible Intervals
    		488
    1051 Highest Posterior Density
    		489
    1052 Contrasting with the Confidence Intervals
    		492
    106 Tests of Hypotheses
    		493
    107 Examples with NonConjugate Priors
    		494
    108 Exercises and Complements
    		497
    11 Likelihood Ratio and Other Tests
    		507
    112 OneSample Problems
    		508
    1121 LR Test for the Mean
    		509
    1122 LR Test for the Variance
    		512
    113 TwoSample Problems
    		515
    1132 Comparing the Variances
    		519
    114 Bivariate Normal Observations
    		522
    1142 LR Test for the Correlation Coefficient
    		525
    11 43 Test for the Variances
    		528
    115 Exercises and Complements
    		529
    121 Introduction
    		539
    123 Confidence Intervals and Tests of Hypothesis
    		542
    1231 The Distribution Free Population Mean
    		543
    1232 The Binomial Proportion
    		548
    1233 The Poisson Mean
    		553
    124 The Variance Stabilizing Transformations
    		555
    1241 The Binomial Proportion
    		556
    1242 The Poisson Mean
    		559
    1243 The Correlation Coefficient
    		560
    125 Exercises and Complements
    		563
    Sample Size Determination TwoStage Procedures
    		569
    132 The Fixed Width Confidence Interval
    		573
    1322 Some Interesting Properties
    		574
    133 The Bounded Risk Point Estimation
    		579
    1331 The Sampling Methodology
    		581
    1332 Some Interesting Properties
    		582
    134 Exercises and Complements
    		584
    14 Appendix
    		591
    Selected Biographical Notes
    		593
    143 Selected Statistical Tables
    		621
    1432 Percentage Points of the Chi Square Distribution
    		626
    1433 Percentage Points of the Students t Distribution
    		628
    1434 Percentage Points of the F Distribution
    		630
    References
    		633
    Index
    		649
    Copyright

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    Nitis Mukhopadhyay
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    Common terms and phrases

    alternative hypothesis assumed known Bayes estimate bivariate normal Chi-square common pdf confidence interval Consider continuous random variable credible interval defined derive evaluate Example Exercise exponential family expression finite Fisher fixed H₁ Hence Hint independent inequality joint pdf Lehmann-Scheffé Theorems Let us denote Let X1 level a test likelihood function LR test minimal sufficient statistic Neyman Neyman-Pearson Lemma normal distribution null hypothesis observed order statistic pdf f(x pdf given pdf's pmf or pdf population positive numbers posterior preassigned probability random variables X1 Rao-Blackwellized real number real valued random Reject sample mean Section Show standard normal Student's t distribution sufficient statistic Suppose that X1 T₁ Theorem UMP level UMVUE unbiased estimator unknown parameter valued random variables verify versus H1 X₁ and X2 Xn are iid Xn be iid Xn:n Y₁ θο μ₁ μ₂ μο σ²

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    Bibliographic information

    TitleProbability and Statistical Inference
    Statistics: A Series of Textbooks and Monographs
    AuthorNitis Mukhopadhyay
    Editionillustrated
    PublisherCRC Press, 2000
    ISBN0824703790, 9780824703790
    Length665 pages
    Subjects ›  › 

    Mathematics / Probability & Statistics / Bayesian Analysis
    Mathematics / Probability & Statistics / General
      
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