Abstract
We study a well-known problem concerning a random variable uniformly distributed between two independent random variables. A new extension has been introduced for this problem and fairly large classes of randomly weighted average distributions are identified by their generalized Stieltjes transforms. In this article we employ the Schwartz distribution theory for finding the distributions of the random variable in question; we also study some properties of these distributions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The addition of this subsection and citing the reference (Johnson and Kotz 1990) was due to the suggestion of an anonymous referee of the journal.
Let us note that some of the presented examples of applications may not satisfy the conditions of the defined \((\mathsf{RWA})_{n}\), similar to some of the examples and applications presented by Soltani and Roozegar (2012). Studying the cases in which the \(R_i\)’s (\(i=1,\ldots ,n\)) are not necessarily weights (be positive and sum to one), or some other weaker conditions in the \((\mathsf{RWA})_{n}\), can be a new line of research for future.
See also “Solid-State Lighting Technology Fact Sheet”, Building Technologies Program, Energy Efficiency & Renewable Energy, U.S. Department of Energy, August 2013, available on the net at the web address http://apps1.eere.energy.gov/buildings/publications/pdfs/ssl/life-reliability_fact-sheet.
Thanks go to a referee whose suggestions improved the theorem and corrected some of its errors in the previous version of the paper.
Thanks go to a couple of referees whose suggestions improved the theorem and corrected some of its errors in the previous version of the paper.
Again more thanks go to an anonymous referee of the journal whose only suggestion was providing a more detailed proof of this theorem here (and also of Theorem 2), emphasizing that our argument is not the only available proof, and some other simpler demonstrations (especially of Lemma 4) may be found.
This last section was added due to the kind suggestion of the last anonymous referee of the journal.
References
Abourizk SM, Halpin DW, Wilson JR (1994) Fitting beta distributions based on sample data. J Constr Eng Manag 120:288–305
Aitchison J (1986) The statistical analysis of compositional data. Chapman and Hall, London
Belassi W, Tukel OI (1996) A new framework for determining critical success/failure factors in projects. Int J Proj Manag 14:141–151
Dababneh AB (2013) A virtual Predictive environment for monitoring reliability, life time, and maintainability of printed circuit boards. Master Thesis, Department of Industrial Engineering, University of Iowa. http://ir.uiowa.edu/etd/2470/
Devroye L (1981) Laws of the iterated logarithm for order statistics of uniform spacings. Ann Probab 9:860–867
Diniz MC, de Souza e Silva E, Gail HR (2002) Calculating the distribution of a linear combination of uniform order statistics. INFORMS J Comput 14:124–131
Fang K-T, Kotz S, Ng KW (1990) Symmetric multivariate and related distributions. Chapman and Hall, London
Han H, Yu J, Zhu H, Chen Y, Yang J, Zhu Y, Xue G, Li M (2014) SenSpeed: sensing driving conditions to estimate vehicle speed in urban environments. In: Proceedings IEEE INFOCOM 2014, pp 727–735. doi:10.1109/INFOCOM.2014.6847999. http://www.cs.sjtu.edu.cn/~jdyu/research/SenSpeed/papers/SenSpeed (to appear)
Homei H (2012) Randomly weighted averages with beta random proportions. Stat Probab Lett 82:1515–1520
Johnson NL, Kotz S (1990) Randomly weighted averages: some aspects and extensions. Am Stat 44:245–249
Kubo I, Kuo H-H, Namli S (2011) MRM-applicable measures for the power function of the second order. Commun Stoch Anal 5:647–670
Kvam PH, Vidakovic B (2007) Nonparametric statistics with applications to science and engineering. Wiley-Interscience, New York
Patil GP, Gore SD, Taillie C (2011) Composite sampling: a novel method to accomplish observational economy in environmental studies. Springer, New York
Pruitt WE (1966) Summability of independent random variables. J Math Mech 15:769–776
Soltani AR, Homei H (2009a) Weighted averages with random proportions that are jointly uniformly distributed over the unit simplex. Stat Probab Lett 79:1215–1218
Soltani AR, Homei H (2009b) A generalization for two-sided power distributions and adjusted method of moments. Statistics 43:611–620
Soltani AR, Roozegar R (2012) On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Stat Probab Lett 82:1012–1020
Van Assche W (1986) Products \(2\times 2\) stochastics matrices with random entries. J Appl Probab 23:1019–1024
Van Assche W (1987) A random variable uniformly distributed between two independent random variables. Sankhyā 49:207–211
Weisberg H (1971) The distribution of linear combinations of order statistics from the uniform distribution. Ann Math Stat 42:704–709
Acknowledgments
The author warmly thanks the valuable comments and suggestions of Professors M.H. Alamatsaz and M. Asadi (from Isfahan University).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Homei, H. A novel extension of randomly weighted averages. Stat Papers 56, 933–946 (2015). https://doi.org/10.1007/s00362-014-0615-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-014-0615-5