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Bernoulli score, information, and attainment of the scalar bound

Example

Assume AC. Let $X$ take values in ${0, 1}$ , with $P_{p} (X = 1) = p$ , $P_{p} (X = 0) = 1 - p$ , and $p \in (0, 1)$ . With counting measure as dominator, this is a one-parameter common-support model. The statistic $T = X$ is unbiased for $ψ (p) = p$ , and it attains the scalar Cramér–Rao bound.

Facts & Assumptions

Given: AC; $p \in (0, 1)$ ; the finite probability law on ${0, 1}$ with masses $p$ and $1 - p$ ; and the statistic $T (x) = x$ .

[A1]

The Axiom of Choice: AC supplies the arbitrary-index density representatives used by the general score definition.

[F1]

Statistics and estimators: an estimator is a measurable statistic with a specified target space.

[F2]

Expectation agrees with the published finite weighted sum: on a finite probability space, expectation is the sum of values weighted by their probabilities.

[F3]

Score and Fisher information: the score is the parameter derivative of the log density, and scalar information is its expected square.

[F4]

Mean-zero score and the information identity: fixed positive support, continuous first and second derivatives with common integrable envelopes on a neighborhood, and a finite score second moment imply mean-zero score and the information identity.

[F5]

Moments, variance, and covariance on a probability space: variance is the expected square of the centered variable, and covariance is the expected product of centered variables.

[F6]

Scalar Cramér–Rao lower bound: assume AC and an open scalar parameter interval. For a real-valued estimator $T$ with finite second moment at $p$ and $E_{q} T = ψ (q)$ on a neighborhood of $p$ , where $ψ$ is differentiable at $p$ , a mean-zero score with finite positive information and ${Cov}_{p} (T, S_{p}) = ψ^{'} (p)$ gives the scalar variance bound and its almost-sure equality condition.

[F8]

The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set: the derivative is the limit of the difference quotient $(f (x) - f (c)) / (x - c)$ at $c$ .

Verification

technique · direct

1.1F4given

Put $ε : = \frac{1}{2} \min {p, 1 - p} > 0$ , the open neighborhood $U : = (p - ε, p + ε)$ , and its compact closure $K : = [p - ε, p + ε] \subset (0, 1)$ . For $q \in K$ , the masses $p_{q} (1) = q$ and $p_{q} (0) = 1 - q$ are positive and twice continuously differentiable on the fixed support $S = {0, 1}$ , with common exceptional set $N = \emptyset$ . With counting measure $#$ , their first derivatives are $1$ and $- 1$ , and their second derivatives are $0$ . Thus the common envelopes $g_{1} (0) = g_{1} (1) = 1$ and $g_{2} \equiv 0$ dominate all first and second derivatives on $U$ ; they are integrable because $\int g_{1} d # = 2$ and $\int g_{2} d # = 0$ .

1.2A1F1F2F3F4givenalgebra

The map $T (x) = x$ is real-valued and measurable on the full power set of ${0, 1}$ . For every $q \in (0, 1)$ , the masses sum to $1$ , and [F2] gives $E_{q} T = q$ and $E_{q} T^{2} = q < \infty$ ; hence $T = X$ is unbiased for $ψ (q) = q$ throughout the parameter interval, including a neighborhood of $p$ . At $p$ , finite summation gives ${Var}_{p} (T) = p (1 - p)$ . The log density is $ℓ_{p} (x) = x \log p + (1 - x) \log (1 - p)$ . Differentiating gives $S_{p} (x) = x / p - (1 - x) / (1 - p) = (x - p) / (p (1 - p))$ and $ℓ_{p}^{''} (x) = - x / p^{2} - (1 - x) / (1 - p)^{2}$ . Since the two score values are finite, its second moment is finite. Finite summation yields $E_{p} S_{p} = 0$ and $I (p) = E_{p} S_{p}^{2} = 1 / [p (1 - p)] = - E_{p} ℓ_{p}^{''}$ , agreeing with [F4].

2.1F5F6F8step 1.1step 1.2algebra

Since $S_{p} = (X - p) / (p (1 - p))$ , [F5] gives ${Cov}_{p} (X, S_{p}) = {Var}_{p} (X) / [p (1 - p)] = 1$ . On the open parameter interval $(0, 1)$ , $ψ (p) = p$ has difference quotient $1$ at every $x \neq p$ , so [F8] gives $ψ^{'} (p) = 1$ ; the score-covariance identity required by [F6] holds.

3.1F6step 1.1step 1.2step 2.1algebra∎

The bound from [F6] is ${Var}_{p} (T) \geq 1 / I (p) = p (1 - p)$ , and step 1.2 gives equality. Moreover $T - ψ (p) = X - p = p (1 - p) S_{p} = (ψ^{'} (p) / I (p)) S_{p}$ , exactly its equality condition.

Depends on

Used by

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Sources

George Deligiannidis, SB2.1 Foundations of Statistical Inference, §3.1 and Bernoulli information calculation, PDF pp. 21–23 (standard reference, not scraped)
Anna Mikusheva, MIT 14.381 Lecture 6: Efficient Estimators, Rao-Cramer Bound, PDF pp. 4–6 (standard reference, not scraped)