Exponential Models, Maximum Likelihood Estimation, and the Haar Condition
Exponential Models, Maximum Likelihood Estimation, and the Haar Condition
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指数模型、最大似然估计和 Haar 条件
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发表时间：
1976
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通讯作者：
B. R. Crain
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作者：
B. R. Crain
设f（x）是集合x上关于非原子测度μ的概率密度函数。设f（x）= exp [<$m i = 1 τi <$i（x）− <$m（τ）]，其中<$m（τ）=<$m（τ1，τ2，.，τm）定义如下：exp [<$m（τ）] =<$exp [<$m i = 1，τi <$i，（x）] dμ（x），当右边是有限的，积分在x上。设x上的函数是满足Haar条件的函数的集合，且{x_i（X）} m_i = 0.利用凸性性质，我们得到了τ的极大似然估计的几乎必然存在性和不存在性的一些结果.
Abstract 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.