📂Mathematical Statistics

Bartlett's Identity

Theorem

Regular Conditions:

• (R0): The probability density function $f$ is injective with respect to $\theta$. Mathematically, it satisfies the following. $$\theta \ne \theta ' \implies f \left( x_{k} ; \theta \right) \ne f \left( x_{k} ; \theta ' \right)$$
• (R1): The probability density function $f$ has the same support for all $\theta$.
• (R2): The true value $\theta_{0}$ is an interior point of $\Omega$.
• (R3): The probability density function $f$ is twice differentiable with respect to $\theta$.
• (R4): The integral $\int f (x; \theta) dx$ is twice differentiable with respect to $\theta$, across the integration sign.

Let us assume Regular Conditions (R0)~(R4) are satisfied.

• [1] First identity: $$E \left[ {{ \partial \log f ( X ; \theta ) } \over { \partial \theta }} \right] = 0$$
• [2] Second identity: $$E \left[ {{ \partial^{2} \log f ( X ; \theta ) } \over { \partial \theta^{2} }} \right] + \operatorname{Var} \left( {{ \partial \log f ( X ; \theta ) } \over { \partial \theta }} \right) = 0$$

Derivation

Direct deduction using regular conditions.

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