Advances in Stochastic Simulation Methods pp 207-244 | Cite as
Higher Order Moments of Order Statistics from the Pareto Distribution and Edgeworth Approximate Inference
Abstract
In this paper, we first derive exact explicit expressions for the triple and quadruple moments of order statistics from the Pareto distribution. Also, we establish recurrence relations for single, double, triple and quadruple moments of order statistics from the Pareto distribution. These relations will enable one to find all moments (of order up to four) of order statistics for all sample sizes in a simple recursive manner. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. These are then utilized to develop approximate confidence intervals for the Pareto parameters using the Edgeworth approximation. Finally, we extend the recurrence relations to the case of the doubly truncated Pareto distribution.
Keywords and phrases
Order statistics exact moments single moments double moments triple moments quadruple moments Pareto distribution doubly truncated distribution recurrence relations Edgeworth approximation coefficients of skewness and kurtosis approximate confidence interval pivotal quantityPreview
Unable to display preview. Download preview PDF.
References
- 1.Arnold, B. C. (1983). Pareto Distributions, International Cooperative Publishing House, Fairland, Maryland.Google Scholar
- 2.Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics, New York: John Wiley & Sons.MATHGoogle Scholar
- 3.Balakrishnan, N. and Cohen, A. C. (1991). Order Statistics and Inference: Estimation Methods, San Diego: Academic Press.MATHGoogle Scholar
- 4.Balakrishnan, N. and Gupta, S. S. (1996). Higher order moments of order statistics from exponential and right-truncated exponential distributions and applications to life-testing problems, In Handbook of Statistics-16: Order Statistics and Their Applications (Eds., C.R. Rao and N. Balakr-ishnan), Amsterdam: North-Holland (to appear).Google Scholar
- 5.Balakrishnan, N. and Joshi, P. C. (1982). Moments of order statistics from doubly truncated Pareto distribution, Journal of Indian Statistical Association, 20 109–117.MathSciNetGoogle Scholar
- 6.. Balakrishnan, N., Childs, A., Govindarajulu, Z. and Chandramouleeswaran, M. P. (1996). Inference on parameters of the Laplace distribution based on Type-II censored samples using Edgeworth approximation, submitted for publication. Google Scholar
- 7.Barton, D. E. and Dennis, K. E. R. (1952). The conditions under which Gram-Charlier and Edgeworth curves are positive definite and unimodal, Biometrika, 39 425–427.MathSciNetMATHGoogle Scholar
- 8.Childs, A. and Balakrishnan, N.(1996). Generalized recurrence relations for momeents of order statistics from non-identical Pareto and truncated Pareto random variables with applications to robustness, In Handbook of Statistics-16: Order Statistics and Their Applications (Eds., C. R. Rao and N. Balakrishnan), Amsterdam: North-Holland (to appear).Google Scholar
- 9.David, H. A. (1981). Order Statistics, Second edition, New York: John Wiley & Sons.MATHGoogle Scholar
- 10.Huang, J. S. (1975). A note on order statistics from the Pareto distribution, Scandinavian Actuarial Journal, 2 187–190.CrossRefGoogle Scholar
- 11.Johnson, N. L. Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. 1, Second edition, New York: John Wiley & Sons.MATHGoogle Scholar
- 12.Kulldorff, G. and Vännman, K. (1973). Estimation of the location and scale parameters of a Pareto distribution by linear functions of order statistics, Journal of the American Statistical Association, 68 218–227.MATHCrossRefGoogle Scholar
- 13.Malik, H. J. (1966). Exact moments of order statistics from the Pareto distribution, Skandinavisk Aktuarietidskrift, 1966 144–157.MathSciNetGoogle Scholar