Maximum Likelihood Estimation of a Likelihood Ratio Ordered Family of Distributions

07/22/2020

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by   Alexandre Mösching, et al.

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We consider bivariate observations (X_1,Y_1), …, (X_n,Y_n)⊂𝔛×ℝ with a real set 𝔛 such that, conditional on the X_i, the Y_i are independent random variables with distribution P_X_i, where (P_x)_x∈𝔛 is unknown. Using an empirical likelihood approach, we devise an algorithm to estimate the unknown family of distributions (P_x)_x∈𝔛 under the sole assumption that this family is increasing with respect to likelihood ratio order. We review the latter concept and realize that no further assumption such as all distributions P_x having densities or having a common countable support is needed. The benefit of the stronger regularization imposed by likelihood ratio ordering over the usual stochastic ordering is evaluated in terms of estimation and predictive performances on simulated as well as real data.