Bartlett's Identity
Theorem
- (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: $$ E \left[ {{ \partial^{2} \log f ( X ; \theta ) } \over { \partial \theta^{2} }} \right] + \text{Var} \left( {{ \partial \log f ( X ; \theta ) } \over { \partial \theta }} \right) = 0 $$
Derivation
Direct deduction using regular conditions.
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