A survey records meteorite impact masses above a recording threshold

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

A survey records n=450n=450 meteorite impact masses above a recording threshold xmin⁡=10 kgx_{\min}=10\,\mathrm{kg}, assumed to follow a power-law density p(x)∝x−αp(x)\propto x^{-\alpha} for x≥xmin⁡x\ge x_{\min}. The sum of the log-ratios of the recorded masses is ∑i=1nln⁡(xi/xmin⁡)=300\sum_{i=1}^{n}\ln(x_i/x_{\min}) = 300. ((a)) Compute the maximum-likelihood estimate α^\hat\alpha and its standard error. ((b)) State, with the condition you use, whether the mean and the variance of the mass distribution are finite at this exponent. ((c)) A future survey expects to record n=1×105n=1\times10^{5} meteorite masses above the same threshold. Estimate the largest mass expected in that survey, using the scaling of the sample maximum with sample size.

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