Determination of the Confidence Intervals for Multimodal Probability Density Functions

Yıl 2018, Cilt: 31 Sayı: 1, 310 - 326, 01.03.2018

ORHAN Kesemen BUĞRA Tiryaki EDA Özkul ÖZGE Tezel

Öz

The shortest
interval approach can be solved as an optimization problem, while the equally
tailed approach is determined by using the distribution function. The equal
density approach is proposed instead of the optimization problem for
determining the shortest confidence interval. It is applied to multimodal
probability density functions to determine the shortest confidence interval.
Furthermore, the equal density and optimization approach for the shortest
confidence interval and the equally tailed approach were applied to numerical
examples and their results were compared. Nevertheless, the main subject of
this study is the calculation of the shortest confidence intervals for any
multimodal distribution.

Anahtar Kelimeler

Multimodal probability density function, Confidence interval estimators, The shortest confidence interval, The equally tailed confidence interval, The equal density confidence interval

Kaynakça

• J. Neyman, "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, vol. 236, no. 767, pp. 333-380, (1937).
• A. Wald, "Asymptotically shortest confidence intervals," The Annals of Mathematical Statistics, vol. 13, no. 2, pp. 127-137, (1942).
• C. R. Blyth and D. W. Hutchinson, "Table of Neyman-shortest unbiased confidence intervals for the binomial parameter," Biometrika, vol. 47, no. 3/4, pp. 381-391, (1960).
• Z. Sidak, "Rectangular confidence regions for the means of multivariate normal distributions," Journal of the American Statistical Association, vol. 62, no. 318, pp. 626-633, (1967).
• K. Levy and S. Narula, "Shortest confidence intervals for the ratio of two normal variances," Canadian Journal of Statistics, vol. 2, no. 1-2, pp. 83-87, (1974).
• T. J. DiCiccio and J. P. Romano, "A review of bootstrap confidence intervals," Journal of the Royal Statistical Society. Series B (Methodological), pp. 338-354, (1988).
• A. B. Owen, "Empirical likelihood ratio confidence intervals for a single functional," Biometrika, vol. 75, no. 2, pp. 237-249, (1988).
• K. K. Ferentinos, "Shortest confidence intervals for families of distributions involving truncation parameters," The American Statistician, vol. 44, no. 2, pp. 167-168, (1990).
• K. Ferentinos and S. Kourouklis, "Shortest confidence interval estimation for families of distributions involving two truncation parameters," Metrika, vol. 37, no. 1, pp. 353-363, (1990).
• R. Juola, "More on shortest confidence intervals," The American Statistician, vol. 47, no. 2, pp. 117-119, (1993).
• S. Weerahandi, "Generalized confidence intervals," in Exact Statistical Methods for Data Analysis, Springer New York, pp. 143-168, (1995).
• R. G. Newcombe, "Two-sided confidence intervals for the single proportion: comparison of seven methods," Statistics in medicine, vol. 17, no. 8, pp. 857-872, (1998).
• R. Willink, "A confidence interval and test for the mean of an asymmetric distribution," Communications in Statistics—Theory and Methods, vol. 34, no. 4, pp. 753-766, (2005).
• X. H. Zhou and P. Dinh, "Nonparametric confidence intervals for the one-and two-sample problems," Biostatistics, vol. 6, no. 2, pp. 187-200, (2005).
• G. B. Kibria, "Modified confidence intervals for the mean of the asymmetric distribution," Pak. J. Statist, vol. 22, no. 2, pp. 109-120, (2006).
• B. D. Burch, "Comparing equal-tail probability and unbiased confidence intervals for the intraclass correlation coefficient," Communications in Statistics—Theory and Methods, vol. 37, no. 20, pp. 3264-3275, (2008).
• M. Evans and M. Shakhatreh, "Optimal properties of some Bayesian inferences," Electronic Journal of Statistics, vol. 2, pp. 1268-1280, (2008).
• A. Baklizi and B. Golam Kibria, "One and two sample confidence intervals for estimating the mean of skewed populations: an empirical comparative study," Journal of Applied Statistics, vol. 36, no. 6, pp. 601-609, (2009).
• S. Banik and B. G. Kibria, "Comparison of some parametric and nonparametric type one sample confidence intervals for estimating the mean of a positively skewed distribution," Communications in Statistics—Simulation and Computation®, vol. 39, no. 2, pp. 361-389, (2010).
• S. Banik and B. G. Kibria, "Estimating the population coefficient of variation by confidence intervals," Communications in Statistics-Simulation and Computation, vol. 40, no. 8, pp. 1236-1261, (2011).
• M. Gulhar, G. K. Kibria, A. N. Albatineh and N. U. Ahmed, "A comparison of some confidence intervals for estimating the population coefficient of variation: a simulation study," SORT: statistics and operations research transactions, vol. 36, no. 1, pp. 45-68, (2012).
• M. Alizadeh, A. Parchami and M. Mashinchi, "Unbiased confidence intervals for distributions involving truncation parameter," in ProbStat Forum, (2013).
• E. Mammen and W. Polonik, "Confidence regions for level sets," Journal of Multivariate Analysis, vol. 122, pp. 202-214, (2013).
• M. W. Fagerland, S. Lydersen and P. Laake, "Recommended confidence intervals for two independent binomial proportions," Statistical methods in medical research, vol. 24, no. 2, pp. 224-254, (2015).
• J. W. Pratt, "Length of confidence intervals," Journal of the American Statistical Association, vol. 56, no. 295, pp. 549-567, (1961).
• G. Casella and R. L. Berger, Statistical inference, vol. 2, Duxbury Pacific Grove, CA, (2002).
• M. Smithson, Confidence intervals, vol. 140, Sage Publications, (2002).
• W. C. Guenther, "Unbiased confidence intervals," The American Statistician, vol. 25, no. 1, pp. 51-53, (1971).
• J. Stoer and R. Bulirsch, Introduction to numerical analysis, vol. 12, Springer Science & Business Media, (2013).
• R. F. Tate and G. W. Klett, "Optimal confidence intervals for the variance of a normal distribution," Journal of the American statistical Association, vol. 54, no. 287, pp. 674-682, (1959).
• W. C. Guenther, "Shortest confidence intervals," The American Statistician, vol. 23, no. 1, pp. 22-25, (1969).
• S. Gao, Z. Zhang and C. Cao, "Particle swarm optimization algorithm for the shortest confidence interval problem," Journal of Computers, vol. 7, no. 8, pp. 1809-1816, (2012).
• G. G. Roussas, A course in mathematical statistics, Academic Press, (1997).