Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics
We find the distribution that has maximum entropy conditional on having specified values of its first r L -moments. This condition is equivalent to specifying the expected values of the order statistics of a sample of size r. The maximum-entropy distribution has a density-quantile function, the reci...
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| Published in | Journal of statistical planning and inference Vol. 137; no. 9; pp. 2870 - 2891 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Lausanne
Elsevier B.V
01.09.2007
New York,NY Elsevier Science Amsterdam |
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| Abstract | We find the distribution that has maximum entropy conditional on having specified values of its first
r
L
-moments. This condition is equivalent to specifying the expected values of the order statistics of a sample of size
r. The maximum-entropy distribution has a density-quantile function, the reciprocal of the derivative of the quantile function, that is a polynomial of degree
r; the quantile function of the distribution can then be found by integration. This class of maximum-entropy distributions includes the uniform, exponential and logistic, and two new generalizations of the logistic distribution. It provides a new method of nonparametric fitting of a distribution to a data sample. We also derive maximum-entropy distributions subject to constraints on expected values of linear combinations of order statistics. |
|---|---|
| AbstractList | We find the distribution that has maximum entropy conditional on having specified values of its first
r
L
-moments. This condition is equivalent to specifying the expected values of the order statistics of a sample of size
r. The maximum-entropy distribution has a density-quantile function, the reciprocal of the derivative of the quantile function, that is a polynomial of degree
r; the quantile function of the distribution can then be found by integration. This class of maximum-entropy distributions includes the uniform, exponential and logistic, and two new generalizations of the logistic distribution. It provides a new method of nonparametric fitting of a distribution to a data sample. We also derive maximum-entropy distributions subject to constraints on expected values of linear combinations of order statistics. |
| Author | Hosking, J.R.M. |
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| Cites_doi | 10.1111/j.2517-6161.1990.tb01775.x 10.1623/hysj.50.3.427.65021 10.1016/S0022-1694(01)00520-0 10.1023/A:1016367418778 10.1061/(ASCE)1084-0699(2002)7:2(137) 10.1016/S0167-4730(99)00041-7 10.2307/2286734 10.1007/s004770050004 10.1016/S0378-3758(03)00213-1 10.1016/S0378-3758(01)00232-4 10.1016/0304-4076(93)90031-Y 10.2307/2290988 10.1103/PhysRev.106.620 10.1016/S0022-1694(03)00089-1 10.1016/j.jweia.2004.02.003 |
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| Keywords | Density estimation Density-quantile function Nonparametric Logistic distribution Polynomial Conditional distribution Exponential distribution Sample size Optimization method L statistics Non parametric method Non parametric estimation Entropy Linear constraint Uniform distribution Linear combination Distribution function Quantile Integration Fitting Statistical moment Order statistic Method of maximum entropy Statistical decision Constrained optimization Polynomial function Statistical method Function derivative Density function |
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| References | Hosking, Wallis (bib11) 1997 Asadi, Ebrahimi, Hamedani, Soofi (bib1) 2004; 50 Kjeldsen, Smithers, Schulze (bib17) 2002; 255 Rényi (bib21) 1961 moments, version 3. Research Report RC20525, IBM Research Division, Yorktown Heights, NY. Karvanen, Eriksson, Koivunen (bib16) 2002; 32 Troutman (bib25) 1983 Hosking (bib6) 1990; 52 Silverman (bib23) 1986 Hosking (bib8) 1998; update vol. 2 Soofi (bib24) 1994; 89 Yue, Pilon (bib28) 2005; 50 Hosking, J.R.M., 1996. Fortran routines for use with the method of Goel, Burn, Pandey, An (bib4) 2004; 92 Pandey (bib19) 2000; 22 Elamir, Seheult (bib3) 2004; 124 Johnson, N.L., Kotz, S., Balakrishnan, N., 1995. Continuous Univariate Distributions, second ed., vol. 2, Wiley, New York. Parzen (bib20) 1979; 74 Hosking, J.R.M., 2007. Supplement to “Distributions with maximum entropy subject to constraints on their L-moments”. Research Report RC24177, IBM Research Division, Yorktown Heights, NY. Ulrych, Velis, Woodbury, Sacchi (bib26) 2000; 14 Jones, Balakrishnan (bib14) 2002; 103 Wand, Jones (bib27) 1995 Zaidman, Keller, Young, Cadman (bib29) 2003; 277 moments. Research Report RC21691, IBM Research Division, Yorktown Heights, NY. Jaynes (bib12) 1957; 106 Holm (bib5) 1993; 59 Hosking, J.R.M., 2000. Maximum-entropy characterization of the logistic distribution using Sansone (bib22) 1959 Kapur (bib15) 1994 Ebrahimi (bib2) 2000; 62 Kroll, Vogel (bib18) 2002; 7 Ebrahimi (10.1016/j.jspi.2006.10.010_bib2) 2000; 62 Hosking (10.1016/j.jspi.2006.10.010_bib8) 1998; update vol. 2 Zaidman (10.1016/j.jspi.2006.10.010_bib29) 2003; 277 Jones (10.1016/j.jspi.2006.10.010_bib14) 2002; 103 Sansone (10.1016/j.jspi.2006.10.010_bib22) 1959 Kjeldsen (10.1016/j.jspi.2006.10.010_bib17) 2002; 255 10.1016/j.jspi.2006.10.010_bib9 Troutman (10.1016/j.jspi.2006.10.010_bib25) 1983 Asadi (10.1016/j.jspi.2006.10.010_bib1) 2004; 50 10.1016/j.jspi.2006.10.010_bib7 Rényi (10.1016/j.jspi.2006.10.010_bib21) 1961 Jaynes (10.1016/j.jspi.2006.10.010_bib12) 1957; 106 Parzen (10.1016/j.jspi.2006.10.010_bib20) 1979; 74 Soofi (10.1016/j.jspi.2006.10.010_bib24) 1994; 89 Wand (10.1016/j.jspi.2006.10.010_bib27) 1995 Silverman (10.1016/j.jspi.2006.10.010_bib23) 1986 Hosking (10.1016/j.jspi.2006.10.010_bib11) 1997 Yue (10.1016/j.jspi.2006.10.010_bib28) 2005; 50 Holm (10.1016/j.jspi.2006.10.010_bib5) 1993; 59 Hosking (10.1016/j.jspi.2006.10.010_bib6) 1990; 52 10.1016/j.jspi.2006.10.010_bib10 Kroll (10.1016/j.jspi.2006.10.010_bib18) 2002; 7 10.1016/j.jspi.2006.10.010_bib13 Kapur (10.1016/j.jspi.2006.10.010_bib15) 1994 Goel (10.1016/j.jspi.2006.10.010_bib4) 2004; 92 Karvanen (10.1016/j.jspi.2006.10.010_bib16) 2002; 32 Elamir (10.1016/j.jspi.2006.10.010_bib3) 2004; 124 Pandey (10.1016/j.jspi.2006.10.010_bib19) 2000; 22 Ulrych (10.1016/j.jspi.2006.10.010_bib26) 2000; 14 |
| References_xml | – year: 1959 ident: bib22 article-title: Orthogonal Functions – reference: -moments, version 3. Research Report RC20525, IBM Research Division, Yorktown Heights, NY. – year: 1995 ident: bib27 article-title: Kernel Smoothing – reference: Hosking, J.R.M., 2000. Maximum-entropy characterization of the logistic distribution using – volume: 32 start-page: 83 year: 2002 end-page: 92 ident: bib16 article-title: Adaptive score functions for maximum likelihood ICA publication-title: J. VLSI Sig. Proc. – volume: 22 start-page: 61 year: 2000 end-page: 79 ident: bib19 article-title: Direct estimation of quantile functions using the maximum entropy principle publication-title: Struct. Safety – volume: 50 start-page: 427 year: 2005 end-page: 438 ident: bib28 article-title: Probability distribution type of Canadian annual minimum streamflow publication-title: Hydrol. Sci. J. – volume: 277 start-page: 195 year: 2003 end-page: 213 ident: bib29 article-title: Flow-duration-frequency behaviour of British rivers based on annual minima data publication-title: J. Hydrol. – volume: 62 start-page: 236 year: 2000 end-page: 243 ident: bib2 article-title: The maximum entropy method for lifetime distributions publication-title: Sankhyā A – volume: 124 start-page: 337 year: 2004 end-page: 359 ident: bib3 article-title: Exact variance structure of sample L-moments publication-title: J. Statist. Plann. Inference – volume: 50 start-page: 177 year: 2004 end-page: 183 ident: bib1 article-title: Maximum dynamic entropy models publication-title: IEEE Trans. Inform. Theory – volume: 52 start-page: 105 year: 1990 end-page: 124 ident: bib6 article-title: L-moments: analysis and estimation of distributions using linear combinations of order statistics publication-title: J. Roy. Statist. Soc. B – year: 1994 ident: bib15 article-title: Measures of Information and their Applications – start-page: 547 year: 1961 end-page: 561 ident: bib21 article-title: On measures of entropy and information publication-title: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability – volume: 89 start-page: 1243 year: 1994 end-page: 1254 ident: bib24 article-title: Capturing the intangible concept of information publication-title: J. Amer. Statist. Assoc. – reference: Hosking, J.R.M., 1996. Fortran routines for use with the method of – reference: -moments. Research Report RC21691, IBM Research Division, Yorktown Heights, NY. – volume: 14 start-page: 50 year: 2000 end-page: 68 ident: bib26 article-title: L-moments and C-moments publication-title: Stoch. Envir. Res. Risk Assess. – volume: update vol. 2 start-page: 357 year: 1998 end-page: 362 ident: bib8 article-title: L-moments publication-title: Encyclopedia of Statistical Sciences – volume: 74 start-page: 105 year: 1979 end-page: 121 ident: bib20 article-title: Nonparametric statistical data modeling publication-title: J. Amer. Statist. Assoc. – volume: 106 start-page: 620 year: 1957 end-page: 630 ident: bib12 article-title: Information theory and statistical mechanics publication-title: Phys. Rev. – volume: 92 start-page: 509 year: 2004 end-page: 528 ident: bib4 article-title: Wind quantile estimation using a pooled frequency analysis approach publication-title: J. Wind Eng. Indust. Aerodynam. – volume: 7 start-page: 137 year: 2002 end-page: 146 ident: bib18 article-title: Probability distribution of low streamflow series in the United States publication-title: J. Hydrol. Eng. – year: 1983 ident: bib25 article-title: Variational Calculus with Elementary Convexity – reference: Johnson, N.L., Kotz, S., Balakrishnan, N., 1995. Continuous Univariate Distributions, second ed., vol. 2, Wiley, New York. – volume: 103 start-page: 377 year: 2002 end-page: 390 ident: bib14 article-title: How are moments and moments of spacings related to distribution functions? publication-title: J. Statist. Plann. Inference – year: 1997 ident: bib11 article-title: Regional Frequency Analysis: An Approach Based on – volume: 255 start-page: 194 year: 2002 end-page: 211 ident: bib17 article-title: Regional flood frequency analysis in the KwaZulu-Natal province, South Africa, using the index-flood method publication-title: J. Hydrol. – year: 1986 ident: bib23 article-title: Density Estimation for Statistics and Data Analysis – reference: Hosking, J.R.M., 2007. Supplement to “Distributions with maximum entropy subject to constraints on their L-moments”. Research Report RC24177, IBM Research Division, Yorktown Heights, NY. – volume: 59 start-page: 377 year: 1993 end-page: 389 ident: bib5 article-title: Maximum entropy Lorenz curves publication-title: J. Econometrics – volume: 52 start-page: 105 year: 1990 ident: 10.1016/j.jspi.2006.10.010_bib6 article-title: L-moments: analysis and estimation of distributions using linear combinations of order statistics publication-title: J. Roy. Statist. Soc. B doi: 10.1111/j.2517-6161.1990.tb01775.x – start-page: 547 year: 1961 ident: 10.1016/j.jspi.2006.10.010_bib21 article-title: On measures of entropy and information – ident: 10.1016/j.jspi.2006.10.010_bib13 – year: 1983 ident: 10.1016/j.jspi.2006.10.010_bib25 – volume: 62 start-page: 236 year: 2000 ident: 10.1016/j.jspi.2006.10.010_bib2 article-title: The maximum entropy method for lifetime distributions publication-title: Sankhyā A – year: 1986 ident: 10.1016/j.jspi.2006.10.010_bib23 – volume: 50 start-page: 427 year: 2005 ident: 10.1016/j.jspi.2006.10.010_bib28 article-title: Probability distribution type of Canadian annual minimum streamflow publication-title: Hydrol. Sci. J. doi: 10.1623/hysj.50.3.427.65021 – volume: 255 start-page: 194 year: 2002 ident: 10.1016/j.jspi.2006.10.010_bib17 article-title: Regional flood frequency analysis in the KwaZulu-Natal province, South Africa, using the index-flood method publication-title: J. Hydrol. doi: 10.1016/S0022-1694(01)00520-0 – ident: 10.1016/j.jspi.2006.10.010_bib7 – ident: 10.1016/j.jspi.2006.10.010_bib9 – volume: 32 start-page: 83 year: 2002 ident: 10.1016/j.jspi.2006.10.010_bib16 article-title: Adaptive score functions for maximum likelihood ICA publication-title: J. VLSI Sig. Proc. doi: 10.1023/A:1016367418778 – volume: 7 start-page: 137 year: 2002 ident: 10.1016/j.jspi.2006.10.010_bib18 article-title: Probability distribution of low streamflow series in the United States publication-title: J. Hydrol. Eng. doi: 10.1061/(ASCE)1084-0699(2002)7:2(137) – year: 1995 ident: 10.1016/j.jspi.2006.10.010_bib27 – volume: 22 start-page: 61 year: 2000 ident: 10.1016/j.jspi.2006.10.010_bib19 article-title: Direct estimation of quantile functions using the maximum entropy principle publication-title: Struct. Safety doi: 10.1016/S0167-4730(99)00041-7 – volume: 74 start-page: 105 year: 1979 ident: 10.1016/j.jspi.2006.10.010_bib20 article-title: Nonparametric statistical data modeling publication-title: J. Amer. Statist. Assoc. doi: 10.2307/2286734 – year: 1997 ident: 10.1016/j.jspi.2006.10.010_bib11 – ident: 10.1016/j.jspi.2006.10.010_bib10 – volume: 14 start-page: 50 year: 2000 ident: 10.1016/j.jspi.2006.10.010_bib26 article-title: L-moments and C-moments publication-title: Stoch. Envir. Res. Risk Assess. doi: 10.1007/s004770050004 – volume: 124 start-page: 337 year: 2004 ident: 10.1016/j.jspi.2006.10.010_bib3 article-title: Exact variance structure of sample L-moments publication-title: J. Statist. Plann. Inference doi: 10.1016/S0378-3758(03)00213-1 – volume: update vol. 2 start-page: 357 year: 1998 ident: 10.1016/j.jspi.2006.10.010_bib8 article-title: L-moments – volume: 103 start-page: 377 year: 2002 ident: 10.1016/j.jspi.2006.10.010_bib14 article-title: How are moments and moments of spacings related to distribution functions? publication-title: J. Statist. Plann. Inference doi: 10.1016/S0378-3758(01)00232-4 – volume: 59 start-page: 377 year: 1993 ident: 10.1016/j.jspi.2006.10.010_bib5 article-title: Maximum entropy Lorenz curves publication-title: J. Econometrics doi: 10.1016/0304-4076(93)90031-Y – volume: 89 start-page: 1243 year: 1994 ident: 10.1016/j.jspi.2006.10.010_bib24 article-title: Capturing the intangible concept of information publication-title: J. Amer. Statist. Assoc. doi: 10.2307/2290988 – volume: 106 start-page: 620 year: 1957 ident: 10.1016/j.jspi.2006.10.010_bib12 article-title: Information theory and statistical mechanics publication-title: Phys. Rev. doi: 10.1103/PhysRev.106.620 – volume: 50 start-page: 177 year: 2004 ident: 10.1016/j.jspi.2006.10.010_bib1 article-title: Maximum dynamic entropy models publication-title: IEEE Trans. Inform. Theory – year: 1959 ident: 10.1016/j.jspi.2006.10.010_bib22 – volume: 277 start-page: 195 year: 2003 ident: 10.1016/j.jspi.2006.10.010_bib29 article-title: Flow-duration-frequency behaviour of British rivers based on annual minima data publication-title: J. Hydrol. doi: 10.1016/S0022-1694(03)00089-1 – year: 1994 ident: 10.1016/j.jspi.2006.10.010_bib15 – volume: 92 start-page: 509 year: 2004 ident: 10.1016/j.jspi.2006.10.010_bib4 article-title: Wind quantile estimation using a pooled frequency analysis approach publication-title: J. Wind Eng. Indust. Aerodynam. doi: 10.1016/j.jweia.2004.02.003 |
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r
L
-moments. This condition is equivalent to specifying... |
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| SubjectTerms | Density estimation Density-quantile function Distribution theory Exact sciences and technology General topics Logistic distribution Mathematics Nonparametric Nonparametric inference Probability and statistics Probability theory and stochastic processes Sciences and techniques of general use Statistics |
| Title | Distributions with maximum entropy subject to constraints on their L-moments or expected order statistics |
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