Approximating the Density of a Sum of Random Variables Sample Clauses

Approximating the Density of a Sum of Random Variables. Σ A natural inclination when approximating the sum of random variables may be to apply the Central Limit Theorem (CLT), which states that, if Y1, . . . , Yn are independent and identically-distributed with mean µ and finite variance σ2, then √nσ n i=1 Xx − nµ! → N (0, 1) as n → ∞ (Bain and Xxxxxxxxxx, 1992). Xxxxxxx (2010, 2011) has dedicated several pa- pers to producing estimates of the mean of pooled, lognormally distributed data based on an extension of the CLT. He explores a moment matching technique with bias-correction methods based on characteristics of the lognormal distribution, as well as an application of the CLT for larger pool sizes (Caudill, Xxxxxx, and Xxxxxxxxx, 2007; Xxxxxxx, 2010, 2011). However, since the CLT requires large pool sizes in order to accurately approximate the distribution of a sum (or mean) of right-skewed random variables, we seek other methods that can also accommodate moderate to small pool sizes. The field of engineering has produced an abundance of literature concerning the approx- imation of the density of a sum of lognormal random variables. In electrical engineering, the sum of lognormal variables is often used to characterize applications in wireless communi- cations, such as co-channel interference and large-scale signal fading (Xxxxxxxx, Xxx-Dayya, and XxXxxx, 1995; Xxxxxxxx and Xxx, 2004; Xxxxxx Xxxxx, Xxxxxx, and Xxxxxxxx, 2006; Xx, 2007; Xx et al., 2011; Xxx et al., 2007; Xxxxxxxxxxx and Xxxxxxxxxxxxx, 2009; Tellambura and Xxxxxxxxx, 2010; Xxxxx and Xxxx, 2006). Initially, we considered these methods as a potential solution to the problem of estimating regression coefficients for a pooled, lognor- mal outcome. In particular, we focused on the modified-power-lognormal (MPLN) function proposed by Xxxxxxxxxxx and Xxxxxxxxxxxxx (2009). Let f (yi; µi, σi) denote the lognormal density, such that 1 − 1 ln yi−µi 2 √ f (yi; µi, σi) = e 0 xx , 0xxxxx
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