Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T16:17:24.175Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME CONVEXITY PROPERTIES OF THE DISTRIBUTION OF LOWER k-RECORD VALUES WITH EXTENSIONS

Published online by Cambridge University Press:  17 March 2014

Mahdi Alimohammadi ,
Mohammad Hossein Alamatsaz  and
Erhard Cramer
Mahdi Alimohammadi
Affiliation:
Department of Statistics, University of Isfahan, Isfahan, 81746-73441, Iran
Mohammad Hossein Alamatsaz
Affiliation:
Department of Statistics, University of Isfahan, Isfahan, 81746-73441, Iran
Erhard Cramer
Affiliation:
Institute of Statistics, RWTH Aachen University, D-52056 Aachen, Germany. Email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Unimodality and strong unimodality of the distribution of ascendingly ordered random variables have been extensively studied in the literature, whereas these properties have not received much attention in the case of descendingly ordered random variates. In this paper, we show that log concavity of the reversed hazard rate implies that of the density function. Using this fundamental result, we establish some convexity properties of such random variables. To do this, we first provide a counterexample showing that a claim of Basak & Basak [7] about the lower record values is not valid. Then, we provide conditions under which unimodality properties of the distribution of lower k-record values would hold. Finally, some extensions to dual generalized order statistics in both univariate and multivariate cases are discussed.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 3 , July 2014 , pp. 389 - 399
Copyright
Copyright © Cambridge University Press 2014 

References

1.Alam, K. (1972). Unimodality of the distribution of an order statistic. The Annals of Mathematical Statistics 43: 2041–2044.Google Scholar
2.Aliev, F.A. (2003). A comment on ‘Unimodality of the distribution of record statistics’. Statistics and Probability Letters 64: 39–40.CrossRefGoogle Scholar
3.Alimohammadi, M. & Alamatsaz, M.H. (2011). Some new results on unimodality of generalized order statistics and their spacing. Statistics and Probability Letters 81: 1677–1682.CrossRefGoogle Scholar
4.An, M.Y. (1998). Logconcavity versus logconvexity: a complete characterization. Journal of Economic Theory 80: 350–369.Google Scholar
5.Arnold, B.C., Balkrishnan, N. & Nagaraja, H.N. (1998). Records. New York: Wiley.Google Scholar
6.Barlow, R.E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston.Google Scholar
7.Basak, P. & Basak, I. (2002). Unimodality of the distribution of record statistics. Statistics and Probability Letters 56: 395–398.Google Scholar
8.Block, H., Savits, T.H. & Singh, H. (1998). The reversed hazard rate function. Probability in the Engineering and Informational Sciences 12: 69–90.Google Scholar
9.Burkschat, M., Cramer, E. & Kamps, U. (2003). Dual generalized order statistics. Metron 61: 13–26.Google Scholar
10.Chandra, N.K. & Roy, D. (2001). Some results on reverse hazard rate. Probability in the Engineering and Informational Sciences 15: 95–102.Google Scholar
11.Chechile, R.A. (2011). Properties of reverse hazard functions. Journal of Mathematical Psychology 55: 203–222.Google Scholar
12.Chen, H., Xie, H. & Hu, T. (2009). Log-concavity of generalized order statistics. Statistics and Probability Letters 79: 396–399.Google Scholar
13.Cramer, E. (2004). Logconcavity and unimodality of progressively censored order statistics. Statistics and Probability Letters 68: 83–90.Google Scholar
14.Cramer, E. & Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58: 293–310.Google Scholar
15.Cramer, E., Kamps, U. & Rychlik, T. (2004). Unimodality of uniform generalized order statistics, with applications to mean bounds. Annals of the Institute of Statistical Mathematics 56: 183–192.Google Scholar
16.Cuculescu, I. & Theodorescu, R. (1998). Multiplicative strong unimodality. Australian and New Zealand Journal of Statistics 40: 205–214.Google Scholar
17.Dharmadhikari, S. & Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Boston: Academic Press.Google Scholar
18.Finkelstein, M.S. (2002). On the reversed hazard rate. Reliability Engineering and System Safety 78: 71–75.Google Scholar
19.Huang, J.S. & Ghosh, M. (1982). A note on strong unimodality of order statistics. Journal of the American Statistical Association 77: 929–930.CrossRefGoogle Scholar
20.Ibragimov, I.A. (1956). On the composition of unimodal distributions. Theory of Probability and Its Applications 1: 255–260.CrossRefGoogle Scholar
21.Kamps, U. (1995a). A Concept of Generalized Order Statistics. Stuttgart: B.G. Teubner.Google Scholar
22.Kamps, U. (1995b). A concept of generalized order statistics. Journal of Statistical Planning and Inference 48: 1–23.Google Scholar
23.Kamps, U. & Cramer, E. (2001). On distributions of generalized order statistics. Statistics 35: 269–280.Google Scholar
24.Karlin, S. (1968). Total Positivity. Stanford, CA: Stanford University Press.Google Scholar
25.Marshall, A.W. & Olkin, I. (2007). Life Distributions. New York: Springer.Google Scholar
26.Nevzorov, V. (2001). Records: Mathematical Theory. Providence, RI: American Mathematical Society.Google Scholar
27.Pellerey, F., Shaked, M. & Zinn, J. (2000). Nonhomogeneous Poisson processes and logconcavity. Probability in the Engineering and Informational Sciences 14: 353–373.CrossRefGoogle Scholar
28.Prékopa, A. (1973). On logarithmic concave measures and functions. Acta Scientiarum Mathematicarum 34: 335–343.Google Scholar
29.Raqab, M.Z. & Amin, W.A. (1997). A note on reliability properties of k-record statistics. Metrika 46: 245–251.Google Scholar
30.Shaked, M. & Shanthikumar, J.G. (2007). Stochastic Orders. New York: Springer.CrossRefGoogle Scholar
31.Tavangar, M. & Asadi, M. (2012). Some unified characterization results on the generalized Pareto distributions based on generalized order statistics. Metrika 75: 997–1007.Google Scholar