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Hammou El Barmi, Ganesh Malla, and Hari Mukerjee Estimation of a star-shaped distribution function Journal Of Nonparametric Statistics Vol. 29 , Iss. 1, pp 22-39
A life distribution function (DF) F is said to be star-shaped if F(x)/x is nondecreasing on its support. This generalises the model of a convex DF, even allowing for jumps. The nonparametric maximum likelihood estimation is known to be inconsistent. We provide a uniformly strongly consistent least-squares estimator. We also derive the convergence in distribution of the estimator at a point where F(x)/x is increasing using the arg max theorem.
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