Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T19:47:05.093Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Convolution of Heterogeneous Bernoulli Random Variables

Part of: Nonparametric inference Distribution theory - Probability Survival analysis and censored data Sampling theory, sample surveys

Published online by Cambridge University Press:  14 July 2016

Maochao Xu  and
N. Balakrishnan
Maochao Xu*
Affiliation:
Illinois State University
N. Balakrishnan*
Affiliation:
McMaster University
*
∗ Postal address: Department of Mathematics, Illinois State University, Normal, IL, USA. Email address: mxu2@ilstu.edu
∗∗ Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada.
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.

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

Keywords

Bernoulliheterogeneous variablelikelihood ratio ordermajorizationreversed hazard rate order

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 62N05: Reliability and life testing 62G30: Order statistics; empirical distribution functions 62D05: Sampling theory, sample surveys
Type
Short Communications
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 877 - 884
Copyright
Copyright © Applied Probability Trust 2011 

References

Boland, P. J. (2007). The probability distribution for the number of successes in independent trials. Commun. Statist. Theory Meth. 36, 1327–1331.Google Scholar
Boland, P. J. and Proschan, F. (1983). The reliability of k-out-of-n systems. Ann. Prob. 11, 760–764.Google Scholar
Boland, P. J. and Singh, H. (2006). Stochastic comparisons of Bernoulli sums and binomial random variables. In Advances in Distribution Theory, Order Statistics, and Inference, eds Balakrishnan, N., Castillo, E. and Sarabia, J. M., Birkhäuser, Boston, MA, pp. 3–11,Google Scholar
Boland, P. J., Singh, H. and Cukic, B. (2002). Stochastic orders in partition and random testing of software. J. Appl. Prob. 39, 555–565.Google Scholar
Boland, P. J., Singh, H. and Cukic, B. (2004). The stochastic precedence ordering with applications in sampling and testing. J. Appl. Prob. 41, 73–82.Google Scholar
Bon, J.-L. and Paltanea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Anal. 5, 185–192.Google Scholar
Dharmadhikari, S. W. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.Google Scholar
Gleser, L. J. (1975). On the distribution of the number of successes in independent trials. Ann. Prob. 3, 182–188.Google Scholar
Hoeffding, W. (1956). On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27, 713–721.Google Scholar
Hu, T. and Ruan, L. (2004). A note on multivariate stochastic comparisons of Bernoulli random variables. J. Statist. Planning Infer. 126, 281–288.Google Scholar
Karlin, S. and Novikoff, A. (1963). Generalized convex inequalities. Pacific J. Math. 13, 1251–1279.Google Scholar
Kochar, S. and Xu, M. (2011). The tail behavior of the convolutions of gamma random variables. J. Statist. Planning Infer. 141, 418–428.Google Scholar
Ma, C. (2000). Convex orders for linear combinations of random variables. J. Statist. Planning Infer. 84, 11–25.Google Scholar
Mao, T., Hu, T. and Zhao, P. (2010). Ordering convolutions of heterogeneous exponential and geometric distribution revisited. Prob. Eng. Inf. Sci. 24, 329–348.Google Scholar
Marshall, A. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Pledger, G. and Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Optimizing Methods in Statistics, Academic Press, New York, pp. 89–113.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders and Their Applications. Springer, New York.Google Scholar
Wang, Y. H. (1993). On the number of successes in independent trials. Statistica Sinica 3, 295–312.Google Scholar
Xu, M. and Balakrishnan, N. (2010). On the convolution of heterogeneous Bernoulli random variables with applications. Tech. Rep., Illinois State University.Google Scholar
Zhao, P. and Balakrishnan, N. (2010). Ordering properties of convolutions of heterogeneous Erlang and Pascal random variables. Statist. Prob. Lett. 80, 969–974.Google Scholar