SEMIPARAMETRIC SURVIVAL ANALYSIS VIA BERNSTEIN POLYNOMIALS
ResumoAccording to Lorentz (2012), Bernstein's polynomials (BP) were introduced by Bernstein around 1912 as an alternative to extreme value theorem proof. The Weierstrass’ extreme value theorem, in turn, has been an important tool and has been used for many applications in calculus and analysis. The theorem is widely used to state that any continuous function over an interval [a, b] in R is limited and, in addition to, there is a maximum value and a minimum value in that interval. First, Bernstein showed there exists two reals k and K such that k ≤ Bm(x) ≤ K . So, the mathematician proved that if f(x) is uniformly continuous on [0, 1] then lim (x) (x) . m →∞ Bm = f Similarly, kernel functions, approximation splines and Bernstein's Polynomials can used to approximate functions. In 2012, Osman and Gosh proposed baseline risk non-parametric modelling for survival proportional hazards regression model. The authors approximate baseline risk function using BP and provide, among other results, proofs on asymptotics. The likelihood log-concavity property shown in this article leads to less costly computational procedures to find bayesian estimators and guarantees the uniqueness of maximum likelihood estimator. Bayesian inference isn’t straightforward as numerical optimization methods already implemented in R that ease frequentist approach. However, this was elegantly done based on gibbs sampling and Adaptive Rejection Metropolis Sampling (ARMS) algorithm.
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