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Kolassa J.E. Series Approximation Methods in Statistics
Kolassa J.E.  Series Approximation Methods in Statistics







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: Series Approximation Methods in Statistics

: Kolassa J.E.

:

This book is intended primarily for advanced graduate students and researchers in the field needing a collection of core results in a uniform notation, with bibliographical references to further examples and applications. It assumes familiarity with real and univariate complex analysis, and vector calculus.


: en

: /

:

ed2k: ed2k stats

: Third Edition

: 2006

: 222

: 13.05.2008

: | | ID
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$p^*$ formula      175
$r^*$ approximation      103 150 187
$\chi^2_n$ distribution      48
Absolute error      37 63 67 81
Adjusted profile log likelihood      145
Albers, Bickel, and van Zwet, 1976      54 197
Albert and Anderson, 1984      158
Analytic statistical model      1
Ancillary      178
Andrews and Stafford, 1993      199
Asymptotic expansion      6
Asymptotic methods      1
Bahadur and Ranga Rao, 1960      9192 95
Bak and Newman, 1996      vi 11 71
Barndorff Nielsens formula      1 171 175 182188 190 192
BarndorffNielsen and Cox, 1979      140 169 171
BarndorffNielsen and Cox, 1984      171
BarndorffNielsen and Cox, 1989      1 4748
BarndorffNielsen and Kluppelberg, 1999      82
BarndorffNielsen, 1978      70 157 187
BarndorffNielsen, 1980      175 179 182 188
BarndorffNielsen, 1983      175 187188
BarndorffNielsen, 1984      185
BarndorffNielsen, 1986      101 187
BarndorffNielsen, 1990a      101
BarndorffNielsen, 1990b      187
Bartlett, 1953      167168
Bartlett, 1955      167
Bartletts correction      165 167168 171 184 193
Bayesian paradigm      191
Beek, 1972      26
Berman, et. al., 1990      163
Bernoulli distribution      12
Bernoulli number      30 55 61
Bernoulli polynomials      51
Berry Esseen theorem      9 25
Bhattacharya and Denker, 1990      1
Bhattacharya and Rao, 1976      1 45 53 125 166167
Billingsley, 1986      13 16 19 111
binomial      108
Binomial distribution      12
Blackwell and Hodges, 1959      67 7778 127
Blackwell Hodges formula      161
Blaesild and Jensen, 1985      81
Bleistein, 1966      104
Bochner and Martin, 1948      126
Bootstrap      49
Cauchy distribution      12 29
Central limit theorem      31 40 166
Chambers, 1967      44 59 116
Characteristic function      v 716 21 25 2930 3739 41 5556 111 124
Clarkson and Jennrich, 1991      159160
Conditional distributions      139 145
Conditional inference      139
Conditional profile log likelihood      142
Contingency tables      154
Converge with probability 1      2
Convergence almost surely      2
Convergence in distribution      2
Convergence in mean      2
Convergence in probability      2
Convolution      8 18
Cordeiro and Paula, 1989      171
Cornish and Fisher, 1937      47
Cornish Fisher expansion      31 4748 54 62
Cox and Reid, 1987      142
Cox and Snell, 1981      198
Cox, 1980      180
Cramer, 1925      4
Cramer, 1946      14
Cramers condition      40 46 124 166
Cumulant generating function      9 64 6871 7778 8285 106 113 149 193195
Cumulants      910
Cumulative distribution functions      28
Curved exponential family      179 184185
Daniels, 1954      63 6769 7172 75 8081
Daniels, 1958      175
Daniels, 1983      189
Daniels, 1987      89 94 135
Davis, 1976      32
Davison and Hinkley, 1988      193194
Davison, 1986      192
Davison, 1988      150 157
DiCiccio and Martin, 1991      129
DiCiccio, Hall, and Romano, 1991      175
DiCiccio, Martin, and Young, 1993      141
Directed log likelihood ratio statistic      185
digamma function      176
Double exponential distribution      29
Double saddlepoint density approximation      140 154
Double saddlepoint distribution function approximation      147150 156
Durbin, 1980      83 175 177
Edgeworth series      28 3132 36 5155 5860 6267 7273 75 77 80 89 91 94 166167
Efron, 1975      179 196197
Efron, 1982      193
Empirical likelihood      175
Esseen, 1945      51 166
Esseen, 1956      26
Esseens series      51 60 125 167
Estimating equations      189
Euler Maclaurin summation formula      5758
Expected information      168
Exponential distribution      29 128
Exponential empirical likelihood      193
Exponential family      6364 117 139 148149 168 175 178 184187 193
Exponential random variables      142
Feller, 1971      v 19 21 38 51
Field and Ronchetti, 1990      1 190
Fisher and Cornish, 1960      48
Fisher information      175 179 196
Fisher, 1925      195196
Fisher, 1934      178 187
Fourier inversion      19 32 71 115
Fourier transform      7
Fraser, 1968      188
Fraser, 1979      188
Fraser, Reid, and Wong, 1991      141142
Frydenberg and Jensen, 1989      173
Fubinis Theorem      1718 90
Gamma distribution      68 8183 88 172 175 203
Geman and Geman, 1984      162
Generalized lattice      114 147 149
Gibbs Skovgaard algorithm      162
Gram Charlier series      34
Hall, 1983      48
Hall, 1992      1
Haynsworth and Goldberg, 1965      30 51 57
Hermite polynomial      35 37 67 116118
Hettmansperger, 1984      54 62
High contact      56
Hocking, 1985      143 153
Hodges and Lehmann, 1970      195 197
Hypergeometric distribution      154155
Invariant      10 48 54 91
Inverse gaussian distribution      30 8182 88 203
Inverting characteristic functions      16
Jacobsen, 1989      157158
Jeffreys, 1962      75
Jensen, 1988      8081 92
Jensen, 1992      101
Jensen, 1995      1
Jing and Robinson, 1994      68 119 127129 141 150
Jing and Wood, 1996      193
Kass, 1988      123
Kass, Tierney, and Kadane, 1990      191
Kendall, Stuart, and Ord, 1987      910 12
Kolassa and McCullagh, 1990      58 166
Kolassa and Tanner, 1994      157 159160 162
Kolassa, 1989      125 134 167
Kolassa, 1991      67
Kolassa, 1992      62 107
Kolassa, 1994      162
Kolassa, 1995      162
Kolassa, 1996a      127
Kolassa, 1996b      97 107 150
Kolassa, 1996c      195 198
Kolassa, 1997      150 157
Kolassa, 2001      162
Kolassa, 2003a      94 107 114 116 131
Kolassa, 2003b      166
Kolassa, 2004      145 162
Kolassa, 2005      97
Kong and Levin, 1996      55
Koning and Does, 1988      48
Konishi, Niki, and Gupta, 1988      49
Laplace transform      8 33
Laplaces method      123 191
Lattice      127
Lattice distribution      12 2728 40 4849 5153 5558 106 166167 172 195
Lawley, 1956      171
Le Cam, 1969      1
Legendre transform      66 83 119 128
Levin and Kong, 1990      167
Levin, 1990      142
Likelihood ratio statistic      101 126 167 182
Logistic distribution      29 7879 85 96
Logistic regression      49 55 163 172 188
Lognormal distribution      30
Lugannani and Rice approximation      9597 194
Lugannani and Rice, 1980      95 135 146
M-estimate      189 194
Mardia, 1970      119
Mathews and Walker, 1964      72
Maximum likelihood estimator      63 167 175 189190 196
McCullagh and Cox, 1986      171
McCullagh and Nelder, 1989      162
McCullagh and Tibshirani, 1990      145
McCullagh, 1984      182
McCullagh, 1986      119
McCullagh, 1987      v 910 3233 35 118 171 182
Moment generating function      89 29 32 38 68 109110
Monotone Convergence Theorem      15
Multivariate BlackwellHodges approximations      140
Multivariate cumulant generating function      110 117
Multivariate Edgeworth series      165166
Multivariate moment generating function      109
Multivariate saddlepoint density      125 154
Multivariate saddlepoints      146
Newton Raphson method      85
Non-independently distributed variables      4446 195
Normal correlation      182
Normal distribution      11 68
Normalize      64 83
numeric integration      89
Observed information      168 175 179
Path of steepest descent      72
Permutation tests      194
Petrov, 1975      100
Pfaff and Pfanzagl, 1985      197
Pfanzagl, 1980      197
Phillips, 1983      127128
Pitman efficiency      197
Poisson regression      188
Poisson variables      154
Posterior moments      191
PraskovaVizkova, 1976      55
Press, et. al., 1986      161
Prior distribution      191
Pseudo-cumulant      3335 37
Pseudo-moments      3335 37 59 115 118
Rao, 1960      166167
Rao, 1961      166167
Rao, 1962      196
Rao, 1963      196
Regularity conditions      40
Reid, 1988      v 165 171 175 187
Reid, 1996      1 90
Relative error      42 44 63 67 8182
Riemann Lebesgue theorem      41 72
Robinson, 1982      55 92 95 107 135 195
Routledge and Tsao, 1995      135
Rudin, 1973      18
Rudin, 1976      vi 29
Saddlepoint      63 65 73
Saddlepoint cumulative distribution function approximations      90 106
Saddlepoint density approximation      6372 81 147
Score statistic      165 167
Secant approximation      104
Secant method      86
Semi-invariant      910
Sequential saddlepoint approximation      141 157
Serfling, 1980      12 167 189
Series Theorem      23 3839 124
Sheppards correction      55 5859 61 167
Sign test      12
simple linear regression      44 46
Singular distribution      40 195
Skates, 1993      104
Skovgaard, 1987      135 145 148 150151 155 157 163
Skovgaard, 1989      1
Skovgaard, 1990      1
Smoothing Theorem      2022 25 3840 124125
Spearmans Rank Correlation      48 54
Stafford and Andrews, 1993      199
Statistical curvature      196
Steepness      70
Strawderman, Casella, and Wells, 1996      163
Sufficiency      178
Taniguchi, 1991      198
Tanner, 1996      162
Temme, 1982      99
Temmes Theorem      150 153
Thisted, 1988      89
Tierney and Kadane, 1986      191192
Tierney, Kass, and Kadane, 1989      127128
Transformation family      188
Turnbull, Brown, and Hu, 1974      192
Uniform distribution      12 68
Uniform error      94
Uniform relative error      63 78 92
Wald statistic      165
Wallace, 1958      3
Wang, 1990a      194
Wang, 1990b      133 135
Waterman and Lindsay, 1996      162
Watsons lemma      75 86 9192 96 107 119 147
Weak convergence      19
Widder, 1946      8
Wilcoxon signed rank statistic      54
Withers, 1988      49
Wong and Li, 1992      192
Wong, 1992      196
Yang and Kolassa, 2002      96 148
Yarnold, 1972      167
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