Abstract
Let X = (X 1, …, X n) be a random vector of observations which may not be independent or identically distributed, and let
be a linear function of X 1, …, X n, where θ i, i = 1, …, n, are constants. In this chapter we stochastically compare statistics of the above type as the coefficients θ i’s vary. The theory of majorization is used to find conditions on the vector of θ’s under which the weighted sums of X i’s are ordered according to various stochastic orders like the likelihood ratio, the hazard rate, the usual stochastic, the peakedness, the dispersive, and the right spread orders. The case of weighted sums of gamma distributions is studied in detail. The convolutions of Bernoulli and geometric random variables are stochastically compared as their parameters vary. The topic of weighted sums of dependent random variables is also considered.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Amiri, L., Khaledi, B. H. and Samaniego, F. J. (2011). On skewness and dispersion among convolutions of independent gamma random variables. Probability in the Engineering and Informational Sciences 25, 55–69.
Birnbaum, Z.W. (1948). On random variables with comparable peakedness. Annals of mathematical statistics 19, 76–81.
Bon, J. L. and Pǎltǎnea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5, 185–192.
Bock, M.E., Diaconis, P. , F. W. Huffer, F. W. and Perlman, M.D. (1987). Inequalities for linear combinations of gamma random variables. The Canadian Journal of Statistics 15, 387–395.
Boland, P. J., El-Neweihi, E. and Proschan, F. (1994). Schur properties of convolutions of exponential and geometric random variables. Journal of Multivariate Analysis 48, 157–167.
Cai, J and Wei, W (2014). Some new notions of dependence with applications in optimal allocation problems. Insurance: Mathematics and Economics 55, 200–209.
Diaconis,P. and Perlman, M.D. (1987). Bounds for tail probabilities of linear combinations of independent gamma random variables. The Symposium on dependence in Statistics and Probability, Hidden Valley, Pennsylvania, 1987.
Dharmadhikari, S. W. and Joag-Dev, K. (1988). Unimodality, Convexity and Applications, Academic Press, San Diego, C
Hitczenko, P. (1998). A note on a distribution of weighted sums of i.i.d. Rayleigh random variables. Sankhyā Series A 60, 171–175.
Hollander, M., Proschan, F. and Sethuraman, J. (1977). Functions decreasing in transposition and their applications in ranking problems. The Annals of Statistics 5, 722–733.
Kaas, R., Goovaerts, M. , Dhaene, J. and Denuit, M. (2001). Modern Actuarial Risk Theory. Kluwer Academic
Khaledi, B.E. and Kochar, S. (2002). Dispersive ordering among linear combinations of uniform random variables. Journal of Statistical Planning and Inference 100, 13–21.
Khaledi, B.E. and Kochar, S. (2004). Ordering convolutions of gamma random variables. Sankhyā 66, 466–473.
Khaledi, B.E and Amiri, K. (2011). On the mean residual life order of convolutions of independent uniform random variables. Journal of Statistical Planning and Inferences 141, 3716–3724.
Kochar, S. C. and Ma, C. (1999). Dispersive ordering of convolutions of exponential random variables. Statistics and Probability Letters 43, 321–324.
Kochar, S. and Xu, M. (2010). On the right spread order of convolutions of heterogeneous exponential random variables. Journal of Multivariate Analysis 101, 165–176.
Kochar, S. and Xu, M. (2012). Some unified results on comparing linear combinations of independent gamma random variables. Probability in Engineering and Information Sciences 26, 393–404.
Korwar, R.M. (2002). On stochastic orders for sums of independent random variables. Journal of Multivariate Analysis 80, 344–357.
Ma, C. (1998). On peakedness of distributions of convex combinations. Journal of Statistical Planning and Inference 70, 51–56
Manesh, S. F. and Khaledi, B. (2008). On the likelihood ratio order for convolutions of independent generalized Rayleigh random variables, Statistics and Probability Letters 78, 3139–3144.
Mao, T., Pan, X. and Hu, T. (2013). On orderings between weighted sums of random variables. Probability in the Engineering and Informational Sciences 27, 85–97.
Marshall, A. W., Olkin, I. and Arnold, B. (2011). Inequalities : Theory of Majorization and Its Applications, 2nd edition, Springer, New York.
Mi, J., Shi, W., and Zhou, Y. (2008). Some properties of convolutions of Pascal and Erlang random variables. Statistics and Probability Letters 78 2378–2387.
Pan, X., Yuan, M., and Kochar, S.C. (2015). Stochastic comparisons of weighted sums of arrangement increasing random variables. Statistics and Probability Letters 102, 42–50.
Proschan, F. (1965). Peakedness of distributions of convex combinations. Annals of Mathematical Statistics 36, 1703–1706.
Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
Shaked, M. and Shantikumar, J.G. (1998). Two variability orders. Probability in the Engineering and Informational Sciences 12, 1–23.
Xu, M. and Balakrishnan, N. (2011). On the convolution of hetergeneous Bernoulli random variables. Journal of Applied Probability 48, 877–884.
Xu, M. and Hu, T. (2011). Some inequalities of linear combinations of independent random variables. Journal of Applied Probability 48, 1179–1188.
You, Y. and Li, X. (2014). Optimal capital allocations to interdependent actuarial risks. Insurance: Mathematics and Economics 57, 104–113.
Yu, Y. (2009). Stochastic ordering of exponential family distributions and their mixtures. Journal of Applied Probability 46, 244–254.
Yu, Y. (2011). Some stochastic inequalities for weighted sums. Bernoulli 17, 1044–1053.
Yu, Y. (2017). On the unique crossing conjecture of Diaconis and Perlman on convolutions of gamma random variables. Annals of Applied Probability 27, 3893–3910.
Zhao, P. (2011). Some new results on convolutions of heterogeneous gamma random variables. Journal of Multivariate Analysis 102, 958–976.
Zhao, P. and Balakrishnan, N. (2009). Mean residual life order of convolutions of heterogeneous exponential random variables. Journal of Multivariate Analysis 100, 1792–1801.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kochar, S.C. (2022). Stochastic Comparisons of Weighted Sums of Random Variables. In: Stochastic Comparisons with Applications. Springer, Cham. https://doi.org/10.1007/978-3-031-12104-3_9
Download citation
DOI: https://doi.org/10.1007/978-3-031-12104-3_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-12103-6
Online ISBN: 978-3-031-12104-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)