An insurer records claim sizes above a policy excess of (in US dollars)

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Many Parts, One Behaviour, Ch Heavy tails: power laws and how to measure them

Problem

An insurer records

n=1000

claim sizes above a policy excess of

x_{\min} = 500

(in US dollars), and finds

\sum_{i=1}^{n}\ln(x_i/x_{\min}) = 800

. Assume the claims above the excess follow a power-law density

p(x)\propto x^{-\alpha}

. ((a)) Compute the maximum-likelihood estimate

\hat\alpha

and its standard error. ((b)) State, with the condition you use, whether the mean and the variance of the claim-size distribution are finite at this exponent. ((c)) Using

W = P^{(\alpha-2)/(\alpha-1)}

for the fraction

W

of total claim value held by the largest fraction

P

of claims, find the fraction of total claim value held by the largest

1\%

of claims.

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