INSEAD Library

Includes bibliographical references and index

CD available inside back cover

Digitized

Handbook of Statistical Distributions with Applications Contents INTRODUCTION TO STATCALC 0.1 Introduction...............................................................................................1 0.2 Contents of StatCalc................................................................................... 4 1 PRELIMINARIES 1.1 Random Variables and Expectations......................................................... 9 1.2 Moments and Other Functions.................................................................. 12 1.2.1 Measures of Central Tendency....................................................... 12 1.2.2 Moments....................................................................................... 12 1.2.3 Measures of Variability.................................................................. 13 1.2.4 Measures of Relative Standing....................................................... 14 1.2.5 Other Measures ............................................................................ 14 1.2.6 Some Other Functions................................................................... 15 1.3 Some Functions Relevant to Reliability ..................................................... 15 1.4 Model Fitting ............................................................................................ 16 1.4.1 Q--Q Plot...................................................................................... 17 1.4.2 The Chi-Square Goodness-of-Fit Test.............................................17 1.5 Methods of Estimation.............................................................................. 18 1.5.1 Moment Estimation....................................................................... 18 1.5.2 Maximum Likelihood Estimation................................................... 19 1.6 Inference .................................................................................................. 19 1.6.1 Hypothesis Testing ....................................................................... 19 1.6.2 Interval Estimation........................................................................ 23 1.7 Random Number Generation .................................................................... 24 1.8 Some Special Functions............................................................................ 25 2 DISCRETE UNIFORM DISTRIBUTION 2.1 Description.............................................................................................. 29 2.2 Moments ................................................................................................ 30 3 BINOMIAL DISTRIBUTION 3.1 3.2 3.3 3.4 Description.............................................................................................. 31 Moments................................................................................................. 32 Computing Table Values.......................................................................... 34 Test for the Proportion............................................................................. 36 3.4.1 An Exact Test.............................................................................. 36 3.4.2 Power of the Exact Test................................................................ 36 3.5 Confidence Intervals for the Proportion.................................................... 38 3.5.1 An Exact Confidence Interval....................................................... 38 3.5.2 Computing Exact Limits and Sample Size Calculation ................. 39 3.6 A Test for the Difference between Two Proportions................................... 40 3.6.1 An Unconditional Test................................................................ 40 3.6.2 Power of the Unconditional Test.................................................. 41 3.7 Fisher's Exact Test.................................................................................. 42 3.7.1 Calculation of p-Values............................................................... 43 3.7.2 Exact Powers.............................................................................. 44 3.8 Properties and Results............................................................................. 45 3.8.1 Properties................................................................................... 45 3.8.2 Relation to Other Distributions................................................... 45 3.8.3 Approximations ......................................................................... 46 3.9 Random Number Generation .................................................................. 46 3.10 Computation of Probabilities ................................................................. 48 4 HYPERGEOMETRIC DISTRIBUTION 4.1 4.2 4.3 4.4 4.5 Description.............................................................................................. 51 Moments................................................................................................. 52 Computing Table Values.......................................................................... 54 Point Estimation...................................................................................... 56 Test for the Proportion............................................................................. 57 4.5.1 An Exact Test............................................................................. 57 4.5.2 Power of the Exact Test............................................................... 58 4.6 Confidence Intervals and Sample Size Calculation................................... 59 4.6.1 Confidence Intervals................................................................... 59 4.6.2 Sample Size for Precision............................................................ 60 4.7 A Test for the Difference between Two Proportions....................................62 4.7.1 The Test ..................................................................................... 62 4.7.2 Power Calculation....................................................................... 63 4.8 Properties and Results............................................................................. 64 4.8.1 Recurrence Relations.................................................................. 64 4.8.2 Relation to Other Distributions................................................... 64 4.8.3 Approximations........................................................................... 64 4.9 Random Number Generation.................................................................... 65 4.10 Computation of Probabilities ................................................................. 66 5 POISSON DISTRIBUTION 5.1 Description.............................................................................................. 71 5.2 Moments.................................................................................................. 72 5.3 Computing Table Values.......................................................................... 74 5.4 Point Estimation...................................................................................... 75 5.5 Test for the Mean..................................................................................... 75 5.5.1 An Exact Test..............................................................................75 5.5.2 Powers of the Exact Test ............................................................ 76 5.6 Confidence Intervals for the Mean............................................................ 77 5.6.1 An Exact Confidence Interval...................................................... 77 5.6.2 Sample Size Calculation for Precision.......................................... 78 5.7 Test for the Ratio of Two Means............................................................... 78 5.7.1 A Conditional Test.......................................................................78 5.7.2 Powers of the Conditional Test ................................................... 80 5.8 Confidence Intervals for the Ratio of Two Means....................................... 81 5.9 A Test for the Difference between Two Means........................................... 81 5.9.1 An Unconditional Test................................................................. 82 5.9.2 Powers of the Unconditional Test.................................................83 5.10 Model Fitting with Examples.................................................................. 84 5.11 Properties and Results........................................................................... 86 5.11.1 Properties................................................................................. 86 5.11.2 Relation to Other Distributions................................................. 86 5.11.3 Approximations ........................................................................ 87 5.12 Random Number Generation.................................................................. 87 5.13 Computation of Probabilities.................................................................. 88 6 GEOMETRIC DISTRIBUTION 6.1 6.2 6.3 6.4 6.5 Description.............................................................................................. 93 Moments................................................................................................. 94 Computing Table Values.......................................................................... 94 Properties and Results............................................................................. 95 Random Number Generation....................................................................96 7 NEGATIVE BINOMIAL DISTRIBUTION 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Description.............................................................................................. 97 Moments................................................................................................. 98 Computing Table Values.......................................................................... 100 Point Estimation...................................................................................... 101 A Test for the Proportion ......................................................................... 101 Confidence Intervals for the Proportion.................................................... 103 Properties and Results............................................................................. 103 7.7.1 Properties................................................................................... 103 7.7.2 Relation to Other Distributions................................................... 104 7.8 Random Number Generation....................................................................104 7.9 A Computational Method for Probabilities................................................ 106 8 LOGARITHMIC SERIES DISTRIBUTION 8.1 8.2 8.3 8.4 Description.............................................................................................. 107 Moments................................................................................................. 109 Computing Table Values.......................................................................... 109 Inferences ............................................................................................... 112 8.4.1 Point Estimation......................................................................... 112 8.4.2 Interval Estimation..................................................................... 112 8.5 Properties and Results............................................................................. 113 8.6 Random Number Generation....................................................................113 8.7 A Computational Algorithm for Probabilities ............................................ 114 9 UNIFORM DISTRIBUTION 9.1 Description.............................................................................................. 115 9.2 Moments................................................................................................. 116 9.3 Inferences ............................................................................................... 116 9.4 Properties and Results........................................................................... 117 9.5 Random Number Generation................................................................. 117 10 NORMAL DISTRIBUTION 10.1 10.2 10.3 10.4 Description.......................................................................................... 119 Moments............................................................................................. 123 Computing Table Values...................................................................... 123 One-Sample Inference.......................................................................... 127 10.4.1 Point Estimation.................................................................... 127 10.4.2 Test for the Mean and Power Computation.............................128 10.4.3 Interval Estimation for the Mean............................................130 10.4.4 Test and Interval Estimation for the Variance........................ 132 10.5 Two-Sample Inference.......................................................................... 134 10.5.1 Inference for the Ratio of Variances........................................135 10.5.2 Inference for the Difference between Two Means when the Variances Are Equal............................................. 136 10.5.3 Inference for the Difference between Two Means ................... 140 10.6 Tolerance Intervals.............................................................................. 142 10.6.1 Two-Sided Tolerance Intervals............................................... 142 10.6.2 One-Sided Tolerance Limits................................................... 143 10.6.3 Equal-Tail Tolerance Intervals .............................................. 145 10.6.4 Simultaneous Hypothesis Testing for Quantiles..................... 146 10.6.5 Tolerance Limits for One-Way Random Effects Model............. 147 10.7 Properties and Results ........................................................................ 149 10.8 Relation to Other Distributions............................................................ 150 10.9 Random Number Generation............................................................... 151 10.10 Computing the Distribution Function................................................ 152 11 CHI-SQUARE DISTRIBUTION 11.1 11.2 11.3 11.4 11.5 Description.......................................................................................... 155 Moments............................................................................................. 156 Computing Table Values ..................................................................... 157 Applications........................................................................................ 157 Properties and Results ........................................................................ 158 11.5.1 Properties.............................................................................. 158 11.5.2 Relation to Other Distributions.............................................. 159 11.5.3 Approximations..................................................................... 160 11.6 Random Number Generation............................................................... 161 11.7 Computing the Distribution Function.................................................. 161 12 F DISTRIBUTION 12.1 Description.......................................................................................... 163 12.2 Moments.............................................................................................. 165 12.3 Computing Table Values...................................................................... 165 12.4 Properties and Results......................................................................... 166 12.4.1 Identities............................................................................... 166 12.4.2 Relation to Other Distributions.............................................. 166 12.4.3 Series Expansions.................................................................. 167 12.4.4 Approximations .....................................................................168 12.5 Random Number Generation................................................................ 168 12.6 A Computational Method for Probabilities ............................................169 13 STUDENT'S t DISTRIBUTION 13.1 Description.......................................................................................... 171 13.2 Moments.............................................................................................. 172 13.3 Computing Table Values...................................................................... 173 13.4 Distribution of the Maximum of Several Variables ............................... 173 13.4.1 An Application....................................................................... 174 13.4.2 Computing Table Values........................................................ 175 13.4.3 An Example........................................................................... 175 13.5 Properties and Results ........................................................................ 176 13.5.1 Properties.............................................................................. 176 13.5.2 Relation to Other Distributions.............................................. 176 13.5.3 Series Expansions for Cumulative Probability.........................177 13.5.4 An Approximation.................................................................. 178 13.6 Random Number Generation................................................................ 178 13.7 A Computational Method for Probabilities ............................................178 14 EXPONENTIAL DISTRIBUTION 14.1 Description.......................................................................................... 179 14.2 Moments.............................................................................................. 180 14.3 Computing Table Values...................................................................... 180 14.4 Inferences............................................................................................ 181 14.5 Properties and Results ........................................................................ 182 14.5.1 Properties.............................................................................. 182 14.5.2 Relation to Other Distributions.............................................. 182 14.6 Random Number Generation................................................................ 183 15 GAMMA DISTRIBUTION 15.1 15.2 15.3 15.4 15.5 Description.......................................................................................... 185 Moments.............................................................................................. 186 Computing Table Values ......................................................................187 Applications with Some Examples........................................................ 188 Inferences............................................................................................ 189 15.5.1 Maximum Likelihood Estimators............................................ 189 15.5.2 Moment Estimators................................................................ 190 15.5.3 Interval Estimation................................................................ 190 15.6 Properties and Results ........................................................................ 191 15.7 Random Number Generation................................................................ 192 15.8 A Computational Method for Probabilities ............................................ 193 16 BETA DISTRIBUTION 16.1 16.2 16.3 16.4 16.5 16.6 Description.......................................................................................... 195 Moments.............................................................................................. 196 Computing Table Values.......................................................................197 Inferences............................................................................................ 198 Applications with an Example.............................................................. 198 Properties and Results ........................................................................ 201 16.6.1 An Identity and Recurrence Relations .................................... 201 16.6.2 Relation to Other Distributions.............................................. 202 16.7 Random Number Generation................................................................ 203 16.8 Evaluating the Distribution Function................................................... 205 17 NONCENTRAL CHI-SQUARE DISTRIBUTION 17.1 17.2 17.3 17.4 17.5 Description.......................................................................................... 207 Moments.............................................................................................. 209 Computing Table Values.......................................................................209 Applications......................................................................................... 210 Properties and Results ........................................................................ 211 17.5.1 Properties.............................................................................. 211 17.5.2 Approximations to Probabilities ............................................. 211 17.5.3 Approximations to Percentiles ............................................... 211 17.6 Random Number Generation................................................................ 212 17.7 Evaluating the Distribution Function................................................... 212 18 NONCENTRAL F DISTRIBUTION 18.1 Description.......................................................................................... 217 18.2 Moments.............................................................................................. 219 18.3 Computing Table Values...................................................................... 219 18.4 Applications......................................................................................... 219 18.5 Properties and Results ........................................................................ 220 18.5.1 Properties............................................................................... 220 18.5.2 Approximations ...................................................................... 221 18.6 Random Number Generation................................................................ 221 18.7 Evaluating the Distribution Function................................................... 222 19 NONCENTRAL t DISTRIBUTION 19.1 Description.......................................................................................... 225 19.2 Moments.............................................................................................. 226 19.3 Computing Table Values ..................................................................... 227 19.4 Applications......................................................................................... 227 19.5 Properties and Results ........................................................................ 228 19.5.1 Properties.............................................................................. 228 19.5.2 An Approximation.................................................................. 229 19.6 Random Number Generation................................................................ 229 19.7 Evaluating the Distribution Function................................................... 229 20 LAPLACE DISTRIBUTION 20.1 Description.......................................................................................... 233 20.2 Moments.............................................................................................. 234 20.3 Computing Table Values...................................................................... 235 20.4 Inferences............................................................................................ 235 20.4.1 Maximum Likelihood Estimators............................................ 235 20.4.2 Interval Estimation................................................................ 236 20.5 Applications......................................................................................... 236 20.6 Relation to Other Distributions............................................................ 238 20.7 Random Number Generation................................................................ 239 21 LOGISTIC DISTRIBUTION 21.1 Description.......................................................................................... 241 21.2 Moments.............................................................................................. 242 21.3 Computing Table Values...................................................................... 243 21.4 Maximum Likelihood Estimators.......................................................... 244 21.5 Applications......................................................................................... 244 21.6 Properties and Results ........................................................................ 245 21.7 Random Number Generation................................................................ 245 22 LOGNORMAL DISTRIBUTION 22.1 Description.......................................................................................... 247 22.2 Moments.............................................................................................. 248 22.3 Computing Table Values...................................................................... 249 22.4 22.5 22.6 22.7 22.8 Maximum Likelihood Estimators.......................................................... 250 Confidence Interval and Test for the Mean............................................ 250 Inferences for the Difference between Two Means................................. 251 Inferences for the Ratio of Two Means.................................................. 253 Applications ........................................................................................ 254 22.9 Properties and Results ........................................................................ 254 22.10 Random Number Generation.............................................................. 255 22.11 Computation of Probabilities and Percentiles ..................................... 255 23 PARETO DISTRIBUTION 23.1 Description.......................................................................................... 257 23.2 Moments.............................................................................................. 258 23.3 Computing Table Values ..................................................................... 259 23.4 Inferences............................................................................................ 259 23.4.1 Point Estimation.................................................................... 260 23.4.2 Interval Estimation................................................................ 260 23.5 Applications......................................................................................... 260 23.6 Properties and Results ........................................................................ 261 23.7 Random Number Generation................................................................ 261 23.8 Computation of Probabilities and Percentiles ....................................... 261 24 WEIBULL DISTRIBUTION 24.1 24.2 24.3 24.4 24.5 24.6 24.7 24.8 Description.......................................................................................... 263 Moments.............................................................................................. 264 Computing Table Values...................................................................... 265 Applications......................................................................................... 265 Point Estimation.................................................................................. 266 Properties and Results ........................................................................ 267 Random Number Generation................................................................ 267 Computation of Probabilities and Percentiles ....................................... 267 25 EXTREME VALUE DISTRIBUTION 25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8 Description.......................................................................................... 269 Moments.............................................................................................. 270 Computing Table Values...................................................................... 271 Maximum Likelihood Estimators.......................................................... 271 Applications......................................................................................... 272 Properties and Results ........................................................................ 273 Random Number Generation................................................................ 273 Computation of Probabilities and Percentiles........................................ 273 26 CAUCHY DISTRIBUTION 26.1 26.2 26.3 26.4 26.5 26.6 26.7 26.8 Description.......................................................................................... 275 Moments.............................................................................................. 276 Computing Table Values...................................................................... 276 Inference.............................................................................................. 277 26.4.1 Estimation Based on Sample Quantiles ................................. 277 26.4.2 Maximum Likelihood Estimators ........................................... 278 Applications......................................................................................... 278 Properties and Results......................................................................... 278 Random Number Generation................................................................ 279 Computation of Probabilities and Percentiles........................................ 279 27 INVERSE GAUSSIAN DISTRIBUTION 27.1 27.2 27.3 27.4 Description.......................................................................................... 281 Moments............................................................................................. 282 Computing Table Values...................................................................... 283 One-Sample Inference.......................................................................... 283 27.4.1 A Test for the Mean............................................................... 284 27.4.2 Confidence Interval for the Mean........................................... 284 27.5 Two-Sample Inference.......................................................................... 285 27.5.1 Inferences for the Difference between Two Means...................285 27.5.2 Inferences for the Ratio of Two Means.................................... 287 27.6 Random Number Generation............................................................... 287 27.7 Computational Methods for Probabilities and Percentiles .................... 288 28 RAYLEIGH DISTRIBUTION 28.1- Description......................................................................................... 289 28.2 Moments............................................................................................. 290 28.3 Computing Table Values...................................................................... 290 28.4 Maximum Likelihood Estimator........................................................... 291 28.5 Relation to Other Distributions............................................................ 291 28.6 Random Number Generation............................................................... 292 29 BIVARIATE NORMAL DISTRIBUTION 29.1 29.2 29.3 29.4 Description.......................................................................................... 293 Computing Table Values...................................................................... 294 An Example......................................................................................... 295 Inferences on Correlation Coefficients.................................................. 296 29.4.1 Point Estimation.................................................................. 297 29.4.2 Hypothesis Testing .............................................................. 297 29.4.3 Interval Estimation............................................................... 299 29.4.4 Inferences on the Difference between Two Correlation Coefficients..................................................... 301 29.5 Some Properties................................................................................... 303 29.6 Random Number Generation............................................................... 303 29.7 A Computational Algorithm for Probabilities.........................................305 30 DISTRIBUTION OF RUNS 30.1 Description.......................................................................................... 307 30.2 Computing Table Values...................................................................... 309 30.3 Examples............................................................................................. 309 31 SIGN TEST AND CONFIDENCE INTERVAL FOR THE MEDIAN 31.1 31.2 31.3 31.4 Hypothesis Test for the Median............................................................ 311 Confidence Interval for the Median....................................................... 312 Computing Table Values ..................................................................... 312 An Example......................................................................................... 313 32 WILCOXON SIGNED-RANK TEST 32.1 32.2 32.3 32.4 Description ......................................................................................... 315 Moments and an Approximation...........................................................316 Computing Table Values ..................................................................... 317 An Example......................................................................................... 317 33 WILCOXON RANK-SUM TEST 33.1 33.2 33.3 33.4 33.5 Description ......................................................................................... 319 Moments and an Approximation...........................................................320 Mann-Whitney U Statistic ................................................................... 320 Computing Table Values...................................................................... 321 An Example......................................................................................... 321 34 NONPARAMETRIC TOLERANCE INTERVAL 34.1 Description ......................................................................................... 323 34.2 Computing Table Values...................................................................... 324 34.3 An Example......................................................................................... 324 35 TOLERANCE FACTORS FOR A MULTIVARIATE NORMAL POPULATION 35.1 Description........................................................................................... 325 35.2 Computing Tolerance Factors............................................................... 326 35.3 Examples............................................................................................. 326 36 DISTRIBUTION OF THE SAMPLE MULTIPLE CORRELATION COEFFICIENT 36.1 Description .......................................................................................... 329 36.2 Moments.............................................................................................. 330 36.3 Inferences............................................................................................ 330 36.3.1 Point Estimation.....................................................................330 36.3.2 Interval Estimation................................................................. 331 36.3.3 Hypothesis Testing................................................................. 331 36.4 Some Results....................................................................................... 332 36.5 Random Number Generation................................................................ 332 36.6 A Computational Method for Probabilities ............................................ 332 36.7 Computing Table Values....................................................................... 334 REFERENCES................................................................................................... 335 INDEX............................................................................................................... 345