Chung k l 1974 a course in probability theory second

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Chung, K. L. (1974). A Course in Probability Theory , second edition. Academic Press, New York. Ferguson, T. S. (1967). Mathematical Statistics . Academic Press, New York. Lehmann, E. L. (1986). Testing Statistical Hypotheses , second edition. Springer-Verlag, New York. Lehmann, E. L. and Casella, G. (1998). Theory of Point Estimation , second edition. Springer-Verlag, New York. Rao, C. R. (1973). Linear Statistical Inference and Its Applications , sec- ond edition. Wiley, New York. Rohatgi, V. K. (1976). An Introduction to Probability Theory and Math- ematical Statistics . Wiley, New York. 351
352 References Royden, H. L. (1968). Real Analysis , second edition. Macmillan, New York. Searle, S. R. (1971). Linear Models . Wiley, New York. Serfling, R. J. (1980). Approximation Theorems of Mathematical Statis- tics . Wiley, New York. Shao, J. (2003). Mathematical Statistics , second edition. Springer-Verlag, New York.
Index 0-1 loss, 73, 83 χ 2 goodness-of-fit test, 301-302 σ -field, xv, 1 A Absolute error loss, 80, 155, 161 Admissibility, xv, 73-75, 155-160, 169-174, 187-188 Ancillary statistic, xv, 68-70 Approximate unbiasedness, 80 Asymptotic bias, xv, 88 Asymptotic correctness of confidence sets, xvi, 333, 335-343 Asymptotic distribution, see limiting distribution Asymptotic efficiency, 193-194, 238, 240 Asymptotic level, xv, 93-94, 306-308 Asymptotic mean squared error, xv, 88, 91, 134 Asymptotic pivotal quantity, 333-335, 339 Asymptotic relative efficiency, xv, 89-91, 132, 135-136, 140, 187-189, 196- 197, 236-238, 240-242 B Bayes action, xvi, 145-150, 152 Bayes estimator, xvi, 153-154, 156-160, 168-170 353
354 Index Bayes factor, 304 Bayes risk, xvi, 81, 154, 156 Bayes rule, xvi, 80-84 Bayes test, 304 Best linear unbiased estimator (BLUE), 127-129, 132 Bootstrap, 247-249 Borel function, xvi, 2-4 Bounded completeness, xvi, 65-67 Bounded in probability, 39-40 C Characteristic function, xvi, 19, 21, 23-25 Cochran’s theorem, 16 Completeness, xvi, 64-70 Conditional distribution, 32 Conditional expectation, xvi, 26-34 Confidence interval, xvi, 85 Confidence region or set, xvi, 85-86 Conjugate prior, 141-142 Consistency of estimator, xvii, 86-88, 158-160, 189-191, 222, 344-345 Consistency of test, xvii, 92-93, 306-308 Contingency table, 299-301 Convergence almost surely, 35-36, 49 Convergence in distribution, 36-37, 42-45, 47-48 Convergence in moments, 35-37, 49 Convergence in probability, 36-37, 40-41, 46-47, 49 Correlation, 18
Index 355 Correlation coefficient, 56-58, 93-94, 279-280, 292 Cram´ er-Rao low bound, 116 Credible set, 320-321 D Density estimation, 216-221 Dunnett’s interval, 348-349 E Empirical Bayes, xvii, 150-151, 153 Empirical distribution, xvii, 212, 216, 222, 228, 232-234, 243, 245, 305-306, 321, 343 Estimability, xvii, 120-123, 132-133 Expectation, 17 Expected length, 85, 324 Exponential family, xvii, 51-53, 59-60 F Fieller’s interval, 309 Fisher information, 112-115, 124 Fisher-scoring, 180-181, 206-208 G ateaux differentiability, 222-223, 232 Generalized Bayes, xvii, 147-153, 157-158, 169 Goodness of fit test, see χ 2 goodness-of-fit test Good sets principle, 1 H Highest posterior density credible set, 320-321 Hodges-Lehmann estimator, 236
356 Index Huber’s estimator, 241-242, 245 I Independence, xvii, 14-16, 18, 53, 68-69 Influence function, 223-227, 230-232, 243 Integration, xviii, 5-8 Inverting acceptance regions, 317-319, 327, 334-338, 340-343 J Jackknife, 246-247 K Kaplan-Meier estimator, 214 L L-functional, 224-225, 227-228, 243 Least squares estimator (LSE), 119-120, 122, 125-132 Lebesgue density, 13-15, 19-20, 24 Length of a confidence interval, 322-324, 334 Liapounov’s condition, 50

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