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 Incoming links: Cramér-Rao lower bound

Differentiating the log-likelihood gives the unique stationary point

M_{0}^{*} = s_{1}, Δ M = \frac{1}{G} \sum_{g} s_{2 g}, M = \frac{1}{N _{SN}} \sum_{i} s_{3 i} .

Its Hessian is diagonal with entries

- 1/ V_{1}

- G / V_{2}

, and

- N_{SN} / σ_{SN}^{2}

, so it is the unique maximum. The estimators are unbiased and have variances

V_{1}, \frac{V _{2}}{G}, \frac{σ _{SN}^{2}}{N _{SN}},

which equal their Cramér-Rao lower bounds because the normal location statistics are efficient.

Since

θ = M_{0}^{*} + Δ M - M

θ = M_{0}^{*} + Δ M - M .

It is unbiased and normal with variance

σ_{θ}^{2} = V_{1} + \frac{V _{2}}{G} + \frac{σ _{SN}^{2}}{N _{SN}} .

Ignoring terms independent of

r_{0}

ℓ (r_{0}) = - 3 N lo g r_{0} - \frac{1}{r _{0}} \sum_{s = 1}^{N} r_{s} .

Its score vanishes at

r_{0} = \frac{1}{3 N} \sum_{s = 1}^{N} r_{s} .

I_{N} (r_{0}) = - E ℓ^{''} (r_{0}) = \frac{3 N}{r _{0}^{2}} .

Because a shape-three gamma variable has mean

3 r_{0}

3 r_{0}^{2}

E r_{0} = r_{0}, Var (r_{0}) = \frac{r _{0}^{2}}{3 N} .

The estimator is unbiased and attains the Cramér-Rao lower bound

I_{N} (r_{0})^{- 1}