Confidence regions for bivariate probability density functions using polygonal areas

Yıl 2020, Cilt: 69 Sayı: 1, 276 - 306, 30.06.2020

Orhan Kesemen , Eda Özkul , Ülkü Ünsal

https://doi.org/10.31801/cfsuasmas.542499

Öz

In this study, a polygonal approach is suggested to generalize the

notion of the condence region of the univariate probability density function

for the bivariate probability density function. The equal density approach is

used to demonstrate that condence regions can be polygonal shapes. The bisection

method is the preferred method in nding the equal density value that

reveals the desired condence coecient. Condence regions estimate not only

bivariate unimodal probability functions but also bivariate multimodal probability

functions. An approach is enhanced to estimate these condence regions

for probability density functions which are dened as rectangular, polygonal

and innite expanse areas.

Anahtar Kelimeler

bivariate probability density functions, Bivariate confidence region, multimodal confidence region, polygonal confidence region

Kaynakça

• Neyman, J., Outline of a theory of statistical estimation based on the classical theory of probability, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 236(767), (1937), 333--380.
• Tate, R. F., and Klett, G. W., Optimal confidence intervals for the variance of a normal distribution, Journal of the American statistical Association 54(287), (1959), 674--682.
• Dunn, O., Multiple comparisons among means, Journal of the American Statistical Association 56(293), (1961), 52--64.
• Dunn, O., Multiple comparisons using rank sums, Technometrics 6(3), (1964), 241--252.
• Chew, V., Confidence, prediction, and tolerance regions for the multivariate normal distribution, Journal of the American Statistical Association 61(315), (1966), 605--617.
• Sidak, Z., Rectangular confidence regions for the means of multivariate normal distributions, Journal of the American Statistical Association 62(318), (1967), 626--633.
• Hu, Z., and Yang, R.-C., A new distribution-free approach to constructing the confidence region for multiple parameters, PloS one 8(12), (2013), e81179.
• Mammen, E., and Polonik, W., Confidence regions for level sets, Journal of Multivariate Analysis 122, (2013), 202--214.
• Martin, R., Random sets and exact confidence regions, Sankhya A 76(2), (2014), 288--304.
• Rambaud-Althaus, C., Althaus, F., Genton, B., and D'Acremont, V., Clinical features for diagnosis of pneumonia in children younger than 5 years: a systematic review and meta-analysis, The Lancet Infectious Diseases 15(4), (2015), 439--450.
• Harrar, S., and Xu, J., Confidence regions for level differences in growth curve models, Journal of Statistical Planning and Inference 175, (2016), 11--24.
• Öztürk, F., and Karabulut, İ.,Interval Estimators for the Parameters of the Normal Distribution, Commun.Fac.Sci.Univ.Ank.Series A1 55(1), (2006), 23--32.
• Ünsal, Ü., Determining Polygonal Confidence Zone with Artificial Bee Colony Algorithm and an Application in Mines, MSc Thesis, Karadeniz Technical University, Turkey, 2014.
• Kesemen O., Tiryaki, B. K., Özkul, E. and Tezel, Ö., Determination of the Confidence Intervals for Multimodal Probability Density Functions, Gazi University Journal of Science 31(1), (2018) 310--326.
• Kesemen O., Tiryaki, B. K. and Uluyurt, T., A Novel Approximation for Computation Bivariate Distribution Functions in Polygonal Area, Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi 9(1), (2019) 88--98.
• Douglas, D., and Peucker, T., Algorithms for the reduction of the number of points required to represent a digitized line or its caricature, Cartographica: The International Journal for Geographic Information and Geovisualization 10(2), (1973), 112--122.
• Kesemen O. and Tiryaki, B. K., Non-Uniform Random Number Generation from Arbitrary Bivariate Distribution in Polygonal Area, Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22(2), (2018), 443--457.
• Chew, L., Constrained delaunay triangulations, Algorithmica 4(1-4), (1989), 97--108.
• Thacker, W. C., A brief review of techniques for generating irregular computational grids, International Journal for Numerical Methods in Engineering 15(9), (1980), 1335-1341.
• Fulton, S. R., Ciesielski, P. E., and Schubert, W. H., Multigrid methods for elliptic problems: A review. Monthly Weather Review 114 (5), (1986), 943-959.
• Sulman, M., Williams, J. F., and Russell, R. D., Optimal mass transport for higher dimensional adaptive grid generation, Journal of computational physics 230(9), (2011), 3302-3330.
• Stilitz, I., and Yitzhaky, J., The effect of grid size on street location time in maps, Applied ergonomics 10(4), (1979), 235-239. Özmen, B., Nurlu, M., and Güler, H., Coğrafi Bilgi Sistemi ile Deprem Bölgelerinin İncelenmesi, Bayındırlık ve İskan Bakanlığı, Afet İşleri Genel Müdürlüğü, Ankara, 1997.