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 Incoming links: Fisher information per observation and per trial

Past exam of the options I course of the University of Oxford / 2012 / O / O1 / d / Solution 2026-10-31

For the same sample size

n

, the geometric distribution provides more Fisher information:

\frac{I _{n}^{geom} ( θ )}{I _{n}^{Bern} ( θ )} = \frac{1}{θ} > 1 (0 < θ < 1) .

The corresponding asymptotic variances of the maximum-likelihood estimators are

Var (θ_{geom}) \approx \frac{θ ^{2} ( 1 - θ )}{n}, Var (θ_{Bern}) \approx \frac{θ ( 1 - θ )}{n} .

Hence choose the geometric sample for more precise inference at equal numbers of observations. Its asymptotic variance is

θ

times as large, and its standard error is

θ

times as large. This illustrates Fisher information per observation and per trial. It is a per-observation comparison: one geometric observation records a waiting time containing, on average,

1/ θ

Bernoulli trials. A constraint on the total number of underlying trials would be a different sampling-budget comparison.

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