Non-Steepness and Maximum Likelihood Estimation Properties of the Truncated Multivariate Normal Distributions
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We consider the truncated multivariate normal distributions for which every component is one-sided truncated. We show that this family of distributions is an exponential family. We identify 𝒟, the corresponding natural parameter space, and deduce that the family of distributions is not regular. We prove that the gradient of the cumulant-generating function of the family of distributions remains bounded near certain boundary points in 𝒟, and therefore the family also is not steep. We also consider maximum likelihood estimation for μ, the location vector parameter, and Σ, the positive definite (symmetric) matrix dispersion parameter, of a truncated non-singular multivariate normal distribution. We prove that each solution to the score equations for (μ,Σ) satisfies the method-of-moments equations, and we obtain a necessary condition for the existence of solutions to the score equations.
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