Exponential Models, Maximum Likelihood Estimation, and the Haar Condition
Bradford R. Crain · Journal of the American Statistical Association · 1976
Let f(x) be a probability density function with respect to the non-atomic measure μ, over the set χ. Suppose f(x) = exp [Σ m i = 1 τiϕi(x) − ψ m (τ)] for × ∈ χ, where ψ m (τ) = ψ m (τ1, τ2, …, τm ) is well-defined by exp [ψ m (τ)] = ∫ exp [Σ m i = 1, τiϕi,(x)] dμ(x) when the right side is finite, and the integration is over χ. Let ϕ0(x) ≡ 1 on χ and assume {ϕi(X)} m i = 0 is a collection of functions which satisfy the Haar Condition. Using convexity properties, we obtain some results on the almost sure existence or nonexistence of the MLE for τ.