Some Limiting Results for Randomly Weighted Average

Document Type : Scientific Research

Authors

Abstract

Randomly weighted average (RWA) is a suitable alternative to the sample mean in estimating unknown parameter in population, especially when the random weights are unequal. Establishing good limiting results for a sequence of random variables is one of the most important features in theoretical, applied probability and statistical inference. In this paper for randomly weighted average, some limiting results specially weak and strong law of large numbers is obtained. Central limit theorem for randomly weighted average in some special cases and generally is investigated.

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Arizmendi, O. and Pérez-Abreu, V. (2010). On the non-classical infinite divisibility of power semicircle distributions. Communications on Stochastic Analysis, 4(2), 161-178.‏
Dempster, A.P. and Kleyle, R.M. (1968). Distributions determined by cutting a simplex with hyperplanes. The Annals of Mathematical Statistics, 1473-1478.‏
Devroye, L. (1981). Laws of the iterated logarithm for order statistics of uniform spacings. The Annals of Probability, 860-867.‏
Feller, W. (2008). An introduction to probability theory and its applications (Vol. 2). John Wiley & Sons.
Ferguson, T.S. (1996). A course in large sample theory (Vol. 49). London: Chapman & Hall.‏
Gao, S., Zhang, J. and Zhou, T. (2003). Law of large numbers for sample mean of random weighting estimate. Information Sciences, 155(1), 151-156.‏
Johnson, N.L. and Kotz, S. (1990). Randomly weighted averages: Some aspects and extensions. The American Statistician, 44(3), 245-249.‏
Koul, H.L. (2002). Weighted empirical processes in dynamic nonlinear models (Vol. 166). Springer Science & Business Media.‏
Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic processes and their applications, 95(1), 83-107.‏
Nadaraya, E.A. (1964). On estimating regression. Theory of Probability & Its Applications, 9(1), 141-142.‏
Nyrhinen, H. (2001). Finite and infinite time ruin probabilities in a stochastic economic environment. Stochastic Processes and their Applications, 92(2), 265-285.‏
Pingyan, C. and Shixin, G. (2007). Limiting behavior of weighted sums of iid random variables. Statistics & Probability Letters, 77(16), 1589-1599.‏
Pruitt, W.E. (1966). Summability of independent random variables (Summ-ability of independent random variables, discussing convergence properties of sequence). Journal of Mathematical and Mechanics, 15, 769-776.‏
Rohatgi, V.K. (1971). Convergence of weighted sums of independent random variables. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 69, No. 02, pp. 305-307). Cambridge University Press.‏
Roozegar, R. and Soltani, A.R. (2014). Classes of power semicircle laws that are randomly weighted average distributions. Journal of Statistical Computation and Simulation, 84(12), 2636-2643.‏
Rosalsky, A. and Sreehari, M. (1998). On the limiting behavior of randomly weighted partial sums. Statistics & probability letters, 40(4), 403-410.‏
Soltani, A.R. and Roozegar, R. (2012). On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach. Statistics & Probability Letters, 82(5), 1012-1020.‏
Watson, G.S. (1956). On the joint distribution of the circular serial correlation coefficients. Biometrika, 161-168.‏
Xiru, C. and Mingzhong, J. (1994). A randomly weighted estimate of the population mean. Acta Mathematicae Applicatae Sinica, 10(3), 274-287.‏
Yang, Y., Leipus, R. and Šiaulys, J. (2012). Tail probability of randomly weighted sums of subexponential random variables under a dependence structure. Statistics & Probability Letters, 82(9), 1727-1736.‏