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Abstract:
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The weighed version g(x ) of the original p.d.f. f(x) of a r.v. X is defined as g(x)=w(x) f(x)/E[w(X)] where w(x) > 0 and E[w(X)] < infinity. The weighted distributions are considered for modelling the size-biased data where the probability of inclusion of the observation in the sample depends on its magnitude. In most of the data analyses studies, that are reported in the related publications, the weight function w(x) = x^k where k is a suitable constant that is thought as useful for accommodating the size-biasedness. In this presentation, k is considered as a parameter instead of a known constant. In particular, a number of different weighted versions of the lognormal distribution with usual parameters (mu, sigma) are defined using the weight function w(x,k) where k is a function of mu or sigma or both. Further, to demonstrate their usefulness these weighed lognormal distributions are fitted to the data arising in (1) oil -field exploration, and(2)the survival data analysis. The corresponding estimates of the parameters are obtained and compared. The effect of weight function on KL divergence is investigated for related applications.
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