Properties of estimators for the gamma distribution:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York u.a.
Dekker
1988
|
| Ausgabe: | 1. print. |
| Schriftenreihe: | Statistics
89. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 250 - 260 |
| Beschreibung: | XVI, 268 S. |
| ISBN: | 0824775562 |
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| adam_text | Titel: Properties of estimators for the gamma distribution
Autor: Bowman, Kimiko O
Jahr: 1988
CONTENTS Preface .......................................................................................................... iii 1. BASIC FORMULAS I. DEFINITIONS OF THE GAMMA DENSITY .................................... 1 A. Statistical ...................................................................................... 1 B. Particular Case of the Normal .................................................... 2 C. Relation to Goodness of Fit ........................................................ 3 D. Application ................................................................................... 3 II. MOMENTS .......................................................................................... 4 A. Moment Generating Function ..................................................... 4 B. Recurrence for Moments .............................................................. 4 C. Approach to Normality ............................................................... 5 D. A Cube-Root Transformation
..................................................... 6 III. THE INCOMPLETE GAMMA FUNCTION .................................... 7 A. Definition ...................................................................................... 7 B. Series for y(a,x) and T(a, x~) .................................................. 8 C. Link with Stieltjes Transform ................................................... 8 IV. THE PROBABILITY INTEGRALS .................................................. 9 A. Stieltjes Continued Fractions ..................................................... 9 B. Continued Fraction of Schlomilch .............................................. 10 C. Gauss Continued Fraction ........................................................... 11 D. Remarks on Computation ........................................................... 12 E. Computation of the Lower Tail ................................................. 13 F. Computation of the Upper Tail ..................................................
16 V. THE INVERSE PROBABILITY INTEGRAL .................................... 17 A. Lower Tail Probability Deviate ................................................. 18 ix
X Contents B. Upper Tall Probability Deviate .................................................. 19 1. Approximate Formula Derived from a Cornish-Fisher Expansion ................................................................................ 19 2. Upper Tail Deviate Using c.f. at (1.20b) ............................ 20 3. Approximate Percentage Points for The Pearson Family Using The Skewness and Kurtosis Parameters ................... 22 VI. THE STATISTIC Arithmetic Mean Geometric Mean 24 VII. FORMULA FOR THE POLYGAMMA FUNCTIONS ................... 26 A. Convergence .................................................................................. 27 B. Polygamma Functions .................................................................. 29 C. Inequalities .................................................................................... 31 D. Computing the Polygamma Functions ...................................... 31 VIII. FURTHER EXAMPLES .................................................................”
33 2. MAXIMUM LIKELIHOOD ESTIMATORS FOR THE TWO PARAMETER GAMMA DISTRIBUTION I. INTRODUCTION ................................................................................. 45 II. EXPANSION FOR p .......................................................................... 46 III. OBJECTIVES ..................................................................................... 47 IV. OPTIONS ........................................................................................... 47 V. A SIMPLE APPROXIMATION TO THE DISTRIBUTION OF p .................................................................................................... 48 VI. MOMENTS OF y .............................................................................. 49 A. Illustration ................................................................................... 50 B. Pearson-Tukey Approximations to Mean and Variance .......... 52 VII. SERIES IN DESCENDING POWERS OF n FOR THE MOMENTS OF p , THE SHAPE PARAMETER ............................
53 A. The Bivariate Moments fi r s ....................................................... 54 B. The Bivariate Derivatives ............................................................ 55 C. Order of Magnitude ..................................................................... 57 D. An Illustration ............................................................................. 57
Contents xi E. Terms in the First Four Moments of p ..................................... 59 F. Illustrations and Further Comments ......................................... 61 G. General Comments on the p Series in Descending Powers of n .............. 63 H. Usage ............................................................................................. 63 VIII. SERIES IN DESCENDING POWERS OF n FOR THE MOMENTS OF â ............................................................................ 69 IX. RANGE OF THE COEFFICIENTS IN THE MOMENT SERIES FOR p,a ............................................................................................ 76 X. MOMENTS OF p AND â USING THE APPROXIMATE DENSITY OF y AND GRAM-CHARLIER SERIES ......................................... 77 A. The Orthogonal Polynomial Coefficients {k s ) .......................... 78 B. Moments of p and â ................................................................... 79 XI. MOMENTS OF p AND â USING EXPANSIONS IN
DESCENDING POWERS OF p ........................................................ 80 A. Recurrence for E (y -s ) ............................................................... 81 B. Expressions for E (y s ), s = 1,2, • • •....................................... 82 C. First Four Moments of p the Shape Parameter Estimator in Descending Powers of p .............................................................. 84 D. First Four Moments of the Scale Parameter Estimator â in Descending Powers of p ............................................................. 85 XII. COMPARISONS OF ASSESSMENTS FOR THE MOMENTS OF p AND â BY DESCENDING POWERS OF n AND p ................ 86 XIII. TABULATIONS OF THE FIRST FOUR MOMENT PARAMETERS OF â AND p ...................................................... 87 A. The p Series .................................................................................. 87 B. The â Series ..................................................................................
89 XIV. APPLICATIONS ............................................................................. 89 A. Storm Data ................................................................................... 89 B. Asian Monsoon Data .................................................................... 90 C. Another Application .................................................................... 92 XV. THE PARAMETRIC FORM (p, c ) WHERE c = 1/a ................. 93 XVI. UNBIASED ESTIMATORS ............................................................ 93 XVII. FIRST AND SECOND ORDER COVARIANCES FOR LARGE SAMPLES ........................................................................ 95
xü Contents XVIII. JOINT ACCEPTANCE REGION FOR a , p USING JOHNSON’S TRANSLATION SYSTEM OF DISTRIBUTIONS ...................... 95 3. APPROXIMATE DISTRIBUTION OF PERCENTAGE POINTS I. INTRODUCTION ................................................................................. 100 II. THE W-H APPROXIMATION .......................................................... 100 III. MOMENTS OF THE MEAN mi ........................................................ 101 IV. MOMENTS OF T a (p ) ........................................................................ 102 A. Acceptance Parameter Space ......................................................... 104 V. ILLUSTRATION .................................................................................. 107 A. The W-H Method and Five Probability Levels ........................ 108 B. A Precipitation Data Example .................................................... 109 VI. AN EXPANSION FOR X a ...............................................................
Ill A. Linked Expressions for F s .......................................................... 112 B. Third and Fourth Derivatives ..................................................... 113 C. A General Approach ............ 114 VII. THE INTEGRAL METHOD AND THE W-H METHOD .............. 116 VIII. FURTHER COMMENTS .................................................................. 117 IX. FURTHER EXAMPLES ...................................................................... 121 4. SIMULATION AND ESTIMATION PROBLEMS ASSOCIATED WITH THE 3-PARAMETER GAMMA DENSITY I. INTRODUCTION ................................................................................. 127 II. BASIC MAXIMUM LIKELIHOOD Cmle, mmle) ........................... 127 III. BASIC MOMENT ESTIMATORS (moe) ......................................... 129 IV. MAXIMUM LIKELIHOOD SOLUTIONS ....................................... 130 A. Equations ...................................................................................... 130 B.
Algorithm Choice ......................................................................... 131 C. When Do Solutions Exist? ........................................................... 132 D. Lam’s Program ............................................................................. 133
Contents xlii V. CYCLE LENGTH FOR SIMULATIONS .......................................... 135 VI. THE TRANSFORMATION g ip) = Inip) - i/f (p) FOR THE ml AND mml ESTIMATORS OF (p) ................................................... 137 A. Simulation Stability .................................................................... 137 B. Stability of Percentage Points of p Using gip), a, and s ..... 138 VH. PERCENTAGE POINTS OF p AND p ......................................... 139 VIII. COHEN AND WHITTEN’S MODIFIED MAXIMUM LIKELIHOOD ESTIMATORS ....................................................... 141 IX. TABULATIONS OF MOMENTS RELATING TO s ,a, AND p .. 143 X. CONCLUDING REMARKS ............................................ 153 XI. APPENDIX 4A.1 EXISTENCE OF MOMENT ESTIMATOR MOMENTS ....... 154 4A.2 MAXIMUM LIKELIHOOD ESTIMATION ........................... 155 5. THE DISTRIBUTIONS OF THE STANDARD DEVIATION, SKEWNESS AND KURTOSIS IN RANDOM SAMPLES FROM A GAMMA DENSITY I. INTRODUCTION .................................................................................
160 H. HISTORICAL ...................................................................................... 160 A. The Semi-Invariant Approach .................................................... 161 III. NUMERICAL STUDIES OF THE STANDARD DEVIATION ...... 163 A. Egon Pearson’s Contribution ...................................................... 163 B. The Question of Divergency ........................................................ 163 C. Early Views on Divergency ........................................................ 163 D. Terms in the First and Third Moments of s* in ) .................... 166 E. Computer Oriented Extended Taylor Series for Moments 167 F. Algebraic Expansions from the Extended Series ....................... 168 G. Characteristics of Moment Series for the Standard Deviation s* in ) ........................................................................... 171 H. A Check for Terms in the Moment Series for the Variance and Fourth Moment of s* in ) ....................................................
172 I. Simple Approximations to the Four Moments of s* in ) .......... 173 J. A Brief Description of Summation Algorithms Used ............... 174 K. Further Comparisons for the Four Moments of s*in ) ........... 176
xiv Contents IV. THE DISTRIBUTION OF THE SKEWNESS .................................. 177 A. Moment Series for Vô 2 ............................................................... 177 B. Effect of Increasing Population Skewness on Terms in the Moment Series for the Sample Skewness ................................. 177 C. Algebraic Expressions for Terms in Ei Jbi) ............................ 179 D. Problems with the Summation Algorithms ..... 180 E. Comparisons with Normal Sampling (p oo) ........................ 182 F. Sample Size and Summation Algorithms .................................. 184 V. DISTRIBUTION OF THE KURTOSIS .............................................. 186 A. The Problem with its Distributional Approximation ............. 186 B. A Transformation to Asymptotic Zero Skewness .................... 188 C. Several Transforms and Robustness .......................................... 190 D. Further Remarks on the Transforms 1/ b 2 and lnb 2 — »Kb 2 ) l I. 11 III. IV. tbe
Normal Case ............................................. 191 E. The Kurtosis under Uniform Sampling: (a) Using 1/ b 2 _ (b) Using Asymptotic Series, A Remarkable Verification ........ 193 F. Transformations for the Standard Deviation ............................ 193 G. Pearson - Tukey Type Transformation and Approximate Zero Skewness ........................................................................................ 194 1. Background ............................................................................. 194 2. Applications ............................................................................ 196 3. The Kurtosis and Normality ................................................ 196 4. Solutions ....... 197 5. Application to b 2 for Gamma Sampling ..................... 197 6. A Guess at the Value of the Kurtosis When the Skewness Is Made Negligible ............. 200 7. Uses of the fc-Percentile Transformations .......................... 200 APPENDIX A SERIES INTO SUMS ASSOCIATED
WITH THE EXPONENTIAL I. INTRODUCTION ................................................................................. 201 II. COMMONLY ACCEPTED NOTIONS ARE THE MOST DIFFICULT TO CHANGE OR DIFFUSE ......................................... 201 III. THE LINK BETWEEN SERIES AND STIELTJES INTEGRALS .. 202 A. The Problem ................................................................................. 202 B. The Rational Fraction Approximants ........................................ 203 C. Important Features ...................................................................... 205 IV. THE CONTINUED FRACTION FORM .......................................... 206 A. The Orthogonal Set {P s (f )} ....................................................... 206
Contents xv B. A Remainder Formula ................................................................. 207 C. Setting up the Continued Fraction from the Series .................. 208 D. Examples .......... 209 E. The Ladder Form .......................................................................... 210 F. Even and Odd Parts of R(z) ........................................................ 211 G. Equivalence Transformations ..................................................... 211 V. STIELTJES AND THE MOMENT PROBLEM ................................. 212 VI. APPROXIMATE RATE OF CONVERGENCE OF CONTINUED FRACTIONS ...................................................................................... 214 VII. CARLEMAN’S CRITERION FOR THE STIELTJES MOMENT PROBLEM ......................................................................................... 219 VIII. PRACTICAL DIFFICULTIES ........................................................ 220 APPENDIX B ALTERNATIVE SUMMATION ALGORITHMS: LEVIN
AND MODIFIED BOREL I. SHANKS’NONLINEAR TRANSFORMATION .................................. 221 II. LEVIN S ALGORITHM ....................................................................... 222 III. MODIFIED BOREL ALGORITHMS .................................................. 229 APPENDIX C MISCELLANEOUS FORMULAS I. BERNOULLI NUMBERS ....................................................................... 230 A. Numbers ......................................................................................... 230 B. Bernoulli Number Bounds ............................................................. 230 C. List of Bernoulli Numbers ........................................................... 231 II. PEARSON DISTRIBUTION ................................................................. 231 A. Moments ......................................................................................... 231 B. Asymptotic Covariances of the Skewness and Kurtosis in — oo) ..........................................................................................
232 III. EXAMPLES ......................................................................................... 232 IV. EQUALITIES ....................................................................................... 237 A. Equalities Relating to V Cm 2 /M 2 )................................................... 237 B. Inequalities for £v Cm 2 /M 2 ) ......................................................... 237 C. Bounds for the Mean of the Standard Deviation ....................... 238 D. Bounds for the s.d. Under Normality ......................................... 238
xvi Contents V. FORMULAS FOR THE SAMPLING MOMENTS .............................. 239 VI. COMPUTERPROGRAMS .................................................................. 247 A. Polygamma Functions .................................................................... 247 B. Computation of M.L. Estimator of p .......................................... 249 BIBLIOGRAPHY ......................................................................................... 250 NOMENCLATURE ..................................................................................... 261 INDEX ......................................................................................................... 265
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| indexdate | 2024-07-09T17:23:31Z |
| institution | BVB |
| isbn | 0824775562 |
| language | English |
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| spelling | Bowman, Kimiko O. Verfasser aut Properties of estimators for the gamma distribution K. O. Bowman ; L. R. Shenton 1. print. New York u.a. Dekker 1988 XVI, 268 S. txt rdacontent n rdamedia nc rdacarrier Statistics 89. Literaturverz. S. 250 - 260 Distribution (Théorie des probabilités) Distribution (Théorie des probabilités) ram Estimation, Théorie de l' Pearson type 3, Loi de Taylor, séries de ram Distribution (Probability theory) Estimation theory Series, Taylor's Gammaverteilung (DE-588)4155928-9 gnd rswk-swf Schätztheorie (DE-588)4121608-8 gnd rswk-swf Gammaverteilung (DE-588)4155928-9 s Schätztheorie (DE-588)4121608-8 s DE-604 Shenton, L. R. Verfasser aut Statistics 89. (DE-604)BV000003265 89 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005741818&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bowman, Kimiko O. Shenton, L. R. Properties of estimators for the gamma distribution Statistics Distribution (Théorie des probabilités) Distribution (Théorie des probabilités) ram Estimation, Théorie de l' Pearson type 3, Loi de Taylor, séries de ram Distribution (Probability theory) Estimation theory Series, Taylor's Gammaverteilung (DE-588)4155928-9 gnd Schätztheorie (DE-588)4121608-8 gnd |
| subject_GND | (DE-588)4155928-9 (DE-588)4121608-8 |
| title | Properties of estimators for the gamma distribution |
| title_auth | Properties of estimators for the gamma distribution |
| title_exact_search | Properties of estimators for the gamma distribution |
| title_full | Properties of estimators for the gamma distribution K. O. Bowman ; L. R. Shenton |
| title_fullStr | Properties of estimators for the gamma distribution K. O. Bowman ; L. R. Shenton |
| title_full_unstemmed | Properties of estimators for the gamma distribution K. O. Bowman ; L. R. Shenton |
| title_short | Properties of estimators for the gamma distribution |
| title_sort | properties of estimators for the gamma distribution |
| topic | Distribution (Théorie des probabilités) Distribution (Théorie des probabilités) ram Estimation, Théorie de l' Pearson type 3, Loi de Taylor, séries de ram Distribution (Probability theory) Estimation theory Series, Taylor's Gammaverteilung (DE-588)4155928-9 gnd Schätztheorie (DE-588)4121608-8 gnd |
| topic_facet | Distribution (Théorie des probabilités) Estimation, Théorie de l' Pearson type 3, Loi de Taylor, séries de Distribution (Probability theory) Estimation theory Series, Taylor's Gammaverteilung Schätztheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005741818&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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