Asymptotic distributions for weighted power sums of extreme values

2021/04/10 by Oluoch, Lillian Achola, Viharos, László
#60F05 #62G32 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2104.04863

Abstract

Let X1,n≤⋯≤ Xn,n be the order statistics of n independent random variables with a common distribution function F having right heavy tail with tail index γ. Given known constants di,n, 1≤ i≤ n, consider the weighted power sums ∑kni=1dn+1-i,nlogpXn+1-i,n, where p>0 and the kn are positive integers such that kn→∞ and kn/n→0 as n→∞. Under some constraints on the weights di,n, we prove asymptotic normality for the power sums over the whole heavy-tail model. We apply the obtained result to construct a new class of estimators for the parameter γ.

Related

• Uniform asymptotics for the tail probability of weighted sums with heavy tails
• Limit laws for the norms of extremal samples
• Asymptotics for Weighted Random Sums
• Almost Sure Convergence of Extreme Order Statistics
• Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks