
NonAsymptotic Behavior of the Maximum Likelihood Estimate of a Discrete Distribution
In this paper, we study the maximum likelihood estimate of the probabili...
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On some properties of the bimodal normal distribution and its bivariate version
In this work, we derive some novel properties of the bimodal normal dist...
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Asymptotic accuracy of the saddlepoint approximation for maximum likelihood estimation
The saddlepoint approximation gives an approximation to the density of a...
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Some characterisation results on classical and free Poisson thinning
Poisson thinning is an elementary result in probability, which is of gre...
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A Spline Chaos Expansion
A spline chaos expansion, referred to as SCE, is introduced for uncertai...
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A Structural Approach to CoordinateFree Statistics
We consider the question of learning in general topological vector space...
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MultiNormex Distributions for the Sum of Random Vectors. Rates of Convergence
We build a sharp approximation of the whole distribution of the sum of i...
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A Note on Taylor's Expansion and Mean Value Theorem With Respect to a Random Variable
We introduce a stochastic version of Taylor's expansion and Mean Value Theorem, originally proved by Aliprantis and Border (1999), and extend them to a multivariate case. For a univariate case, the theorem asserts that "suppose a realvalued function f has a continuous derivative f' on a closed interval I and X is a random variable on a probability space (Ω, ℱ, P). Fix a ∈ I, there exists a random variable ξ such that ξ(ω) ∈ I for every ω∈Ω and f(X(ω)) = f(a) + f'(ξ(ω))(X(ω)  a)." The proof is not trivial. By applying these results in statistics, one may simplify some details in the proofs of the Delta method or the asymptotic properties for a maximum likelihood estimator. In particular, when mentioning "there exists θ ^ * between θ̂ (a maximum likelihood estimator) and θ_0 (the true value)", a stochastic version of Mean Value Theorem guarantees θ ^ * is a random variable (or a random vector).
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