Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-29T23:38:11.387Z Has data issue: false hasContentIssue false
  • English
  • Français

NONNEGATIVITY OF COVARIANCES BETWEEN FUNCTIONS OF ORDERED RANDOM VARIABLES

Published online by Cambridge University Press:  22 October 2007

Taizhong Hu ,
Junchao Yao  and
Qingshu Lu
Taizhong Hu
Affiliation:
Department of Statistics and FinanceUniversity of Science and Technology of ChinaHefei, Anhui 230026People's Republic of China E-mail: thu@ustc.edu.cn
Junchao Yao
Affiliation:
Department of Statistics and FinanceUniversity of Science and Technology of ChinaHefei, Anhui 230026People's Republic of China E-mail: thu@ustc.edu.cn
Qingshu Lu
Affiliation:
Department of Statistics and FinanceUniversity of Science and Technology of ChinaHefei, Anhui 230026People's Republic of China E-mail: thu@ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article we investigate conditions by a unified method under which the covariances of functions of two adjacent ordered random variables are nonnegative. The main structural results are applied to several kinds of ordered random variable, such as delayed record values, continuous and discrete ℓ∞⩽-spherical order statistics, epoch times of mixed Poisson processes, generalized order statistics, discrete weak record values, and epoch times of modified geometric processes. These applications extend the main results for ordinary order statistics in Qi [28] and for usual record values in Nagaraja and Nevzorov [25].

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 21 , Issue 4 , October 2007 , pp. 557 - 577
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ahsanullah, M. (1988). Introduction to record values. Needham Heights, MA: Ginn Press.Google Scholar
2.Ahsanullah, M. (1995). Record statistics. New York: Nova Science Publishers.Google Scholar
3.Arnold, B.C., Balakrishnan, N., & Nagaraja, H.N. (1992). A first course in order statistics. New York: Wiley.Google Scholar
4.Arnold, B.C., Balakrishnan, N., & Nagaraja, H.N. (1998). Records. New York: Wiley.Google Scholar
5.Bairamov, I. & Stepanov, A. (2006). A note on large deviations for weak records. Statistics and Probability Letters 76: 1449–1453.Google Scholar
6.Balakrishnan, N. & Aggarwala, R. (2000). Progressive censoring. Boston: Birkhauser.CrossRefGoogle Scholar
7.Belzunce, F., Mercader, J.A., & Ruiz, J.M. (2005). Stochastic comparisons of generalized order statistics. Probability in the Engineering and Informational Sciences 19: 99–120.Google Scholar
8.Belzunce, F., Ortega, E.-M., & Ruiz, J.M. (2006). Stochastic orderings of discrete-time processes and discrete record values. Probability in the Engineering and Informational Sciences 20: 447–464.CrossRefGoogle Scholar
9.Cramer, E. (2006). Dependence structure of generalized order statistics. Statistics 40: 409–413.CrossRefGoogle Scholar
10.Cramer, E. & Kamps, U. (2001). Sequential k-out-of-n systems. In Balakrishnan, N. & Rao, C.R. (eds.), Handbook of staistics: Advances in reliability, Vol. 20, Amsterdam: Elsevier, pp. 301–372.Google Scholar
11.Cramer, E. & Kamps, U. (2003). Marginal distributions of sequential and generalized order statistics. Metrika 58: 293–310.CrossRefGoogle Scholar
12.Danielak, K. & Dembińska, A. (2006). On characterizing discrete distributions via conditional expectations of weak record values. Metrika DOI 10.1007/s00184-006-0100-9.Google Scholar
13.Dembińska, A. & López-Blázquez, F. (2005). A characterization of geometric distribution through kth weak records. Communication in Statistics: Theory and Methods 34: 2345–2351.CrossRefGoogle Scholar
14.Dembińska, A. & Stepanov, A. (2006). Limit theorems for the ratio of weak records. Statistics and Probability Letters 76: 1454–1464.CrossRefGoogle Scholar
15.Feigin, P.D. (1979). On the characterization of point processes with the order statistic property. Journal of Applied Probability 16: 297–304.CrossRefGoogle Scholar
16.Grandell, J. (1997). Mixed Poisson processes. London: Chapman & Hall.Google Scholar
17.Hayakawa, Y. (2000). A new characterization property of mixed Poisson process via Berman's theorem. Journal of Applied Probability 37: 261–268.CrossRefGoogle Scholar
18.Huang, W.-J. & ShoungHuang, J.-M. (1994). On a study of some properties of point processes. Sankhyā A 56: 67–76.Google Scholar
19.Kamps, U. (1995). A concept of generalized order statistics. Stuttgard: Teubner.Google Scholar
20.Kamps, U. (1995). A concept of generalized order statistics. Journal of Statistical Planning and Inference 48: 1–23.Google Scholar
21.Khaledi, B.-E. & Kochar, S.C. (2005). Dependence orderings for generalized order statistics. Statistics and Probability Letters 73: 357–367.CrossRefGoogle Scholar
22.Li, L. (1994). A counterexample to a conjecture on order statistics. Statistics and Probability Letters 19: 129–130.CrossRefGoogle Scholar
23.Ma, C. (1992). Variance bound of function of order statistics. Statistics and Probability Letters 13: 25–27.Google Scholar
24.Ma, C. (1992). Moments of functions of order statistics. Statistics and Probability Letters 15: 57–62.Google Scholar
25.Nagaraja, H.N. & Nevzorov, V.B. (1996). Correlations between functions of records can be negative. Statistics and Probability Letters 29: 95–100.CrossRefGoogle Scholar
26.Pfeifer, D. & Heller, U. (1987). A martingale characterization of mixed Poisson processes. Journal of Applied Probability 24: 246–251.Google Scholar
27.Puri, P.S. (1982). On the characterization of point processes with the order statistics property without the moment condition. Journal of Applied Probability 19: 39–51.Google Scholar
28.Qi, Y. (1994). Some results on covariance of function of order statistics. Statistics and Probability Letters 19: 111–114.CrossRefGoogle Scholar
29.Rinott, Y. & Pollak, M. (1980). A strong ordering induced by a concept of positive dependence and monotonicity of asymptotic test sizes. Annals of Statistics 8: 190–198.CrossRefGoogle Scholar
30.Shaked, M. (1979). Some concepts of positive dependence for bivariate interchangeable distributions. Annals of the Institute of Statistical Mathematics 31: 67–84.Google Scholar
31.Shaked, M. & Shanthikumar, J.G. (1986). Multivariate imperfect repair. Operations Research 34: 437–448.CrossRefGoogle Scholar
32.Shaked, M., Spizzichino, F., & Suter, F. (2002). Nonhomogeneous birth processes and l ∞-sperical densities, with applications in reliability theory. Probability in the Engineering and Informational Sciences 16: 271–288.CrossRefGoogle Scholar
33.Shaked, M., Spizzichino, F., & Suter, F. (2004). Uniform order statistics property and l ∞-sperical densities. Probability in the Engineering and Informational Sciences 18: 275–297.Google Scholar
34.Spizzichino, F. (2001). Subjective probability models for lifetimes. New York: Chapman & Hall.CrossRefGoogle Scholar
35.Stepanov, A. (1992). [Limit theorems for weak records.] Theory Probability and Its Applications 38: 762–764.Google Scholar
36.Stepanov, A., Balakrishnan, N., & Hofmann, G. (2003). Exact distributions and Fisher information of weak record values. Statistics and Probability Letters 64: 69–81.CrossRefGoogle Scholar
37.Vervaat, W. (1973). Limit theorems for records from discrete distributions. Stochastic Processes and Their Applications 1: 317–334.Google Scholar
38.Wei, G. & Hu, T. (2006). Characterizations of aging classes in terms of spacings between record values. Technical report, Department of Statistics and Finance, University of Science and Technology of China, Hefei.Google Scholar
39.Wesolowski, J. & López-Blázquez, F. (2004). Linearity of regression for the past weak and ordinary records. Statistics 38: 457–464.CrossRefGoogle Scholar