Acta mathematica scientia, Series B >
DISPERSION COMPARISONS OF TWO PROBABILITY VECTORS UNDER MULTINOMIAL SAMPLING
Received date: 2007-10-08
Revised date: 2008-05-12
Online published: 2010-05-20
Supported by
Sponsored by the National NSFC (10771126, 10801130)
We consider testing hypotheses concerning comparing dispersions between two parameter vectors of multinomial distributions in both one-sample and two-sample cases. The comparison criterion is the concept of Schur majorization. A new dispersion index is proposed for testing the hypotheses. The corresponding test for the one-sample problem is an exact test. For the two-sample problem, the bootstrap is used to approximate the null distribution of the test statistic and the p-value. We prove that the bootstrap test is asymptotically correct and
consistent. Simulation studies for the bootstrap test are reported and a real life example is presented.
Key words： Multinomial distribution; dispersion; diversity; bootstrap; hypothesis testing; majorization
XIONG Shi-Feng , LI Guo-Ying . DISPERSION COMPARISONS OF TWO PROBABILITY VECTORS UNDER MULTINOMIAL SAMPLING[J]. Acta mathematica scientia, Series B, 2010 , 30(3) : 907 -918 . DOI: 10.1016/S0252-9602(10)60088-4
[1] Dykstra R L, et al. Order restricted inference for hypotheses concerning qualitative dispersion. J Statist Plann Inference, 2002, 107: 249--265
[2] Patil G P, Taillie C. Diversity as a concept and its measurement. J Amer Statist Assoc, 1982, 77: 548--561
[3] Gilula Z, Haberman S J. Dispersion of categorical variables and penalty functions; derivation, estimation and comparability. J Amer Statist Assoc, 1995, 90: 1447--1452
[4] Gove J H, et al. Ecological diversity and forest management//Patil G P, Rao C R. Handbook of Statistics 12. Environmental statistics Amsterdam: North-Holland, 1994: 409--462
[5] Marshall A W, Oklin I. Inequalities: Theory of Majorization and Its Applications. New York: Academic Press, 1979
[6] Patil G P, Taillie C. An overview of diversity//Grassle J F, et al. Ecological Diversity in Theory and Practice. Fairland, MD: International Co-operative Publishing House, 1979
[7] Robertson T, Wright F T. Likelihood ratio tests for and against a stochastic ordering between multinomial populations. Ann Statist,
1981, 9: 1248--1257
[8] Silvapulle M J, Sen P K. Constrained Statistical Inference: Inequality, Order, and Shape Restrictions. New York: John Wiley & Sons, 2004
[9] Cohen A, Kolassa J, Sackrowitz H B. A new test for stochastic order of k≥ 3 ordered multinomial populations. Statist Probab Lett, 2006, 76: 1017--1024
[10] Feng Y, Wang J. Likelihood ratio test against simple stochastic ordering among several multinomial populations. J Statist Plann
Inference, 2007, 137: 1362--1374
[11] Xiong S, Li G. Testing for the maximum cell probabilities in multinomial distributions. Sci China Ser A, 2005, 48: 972--985
[12] Marcheselli M. A generalized delta method with applications to intrinsic diversity profiles. J Appl Prob, 2000, 37: 504--510
[13] Read T R C, Cressie N. Goodness-of-Fit Statistics for Discrete Multivariate Data. New York: Springer, 1988
[14] Xiong S, Li G. Inference for ordered parameters in multinomial distributions. Sci China Ser A, 2009, 52: 526--538
[15] Bichel P J, G\"{o}tze F, Zwet W R. Resampling fewer than n observations: gains, losses, and remedies for losses. Statist Sinica, 1997, 7: 1--37
[16] Shao J, Tu D. The Jackknife and Bootstrap. New York: Springer, 1995
[17] Barabesi L, Fattorini L. The use of replicated plot, line and point sampling for estimating species abundance and ecological
diversity. Environ Ecol Statist, 1998, 5: 353--370
[18] Fattorini L, Marcheselli M. Inference on intrinsic diversity profiles of Biological populations. Environmetrics, 1999, 10: 589--599
[19] Marcheselli M. Asymptotic results in jackknifing nonsmooth functions of the sample mean vector. Ann Statist, 2003, 31: 1885--1904
[20] Xiong S, Li G. Some results on the convergence of conditional distributions. Statist Probab Lett, 2008, 78: 3249--3253
/
| 〈 |
|
〉 |