A trigger signal reaches a recorder after a fixed but unknown latency plus a random…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Error, Chance and the Machine, Ch Estimators and the information in data

Problem

A trigger signal reaches a recorder after a fixed but unknown latency

\theta

plus a random delay that is exponentially distributed with known mean

\tau = 1.0\,\mathrm{ms}

. One arrival time

x

therefore has the density

f(x;\theta) = \tfrac1\tau e^{-(x-\theta)/\tau}

for

x\ge\theta

, and

f = 0

for

x<\theta

. The latency is estimated from

N = 25

independent arrival times. ((a)) A colleague computes the information as the mean square of the score

\partial\ln f/\partial\theta

and quotes the resulting bound

1/(NI)

on the variance of an unbiased estimate of

\theta

. Find that value, and show by computing the mean of the score that the argument behind the bound does not apply to this density. ((b)) Find the maximum-likelihood estimate

\hat\theta

and explain why it is not found from a likelihood equation. ((c)) Find the bias and the variance of

\hat\theta

. Compare the variance with the colleague's bound, and say whether the Cramér-Rao bound has been violated.

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