Advanced Statistical Methods in Data Science by Ding-Geng Chen Jiahua Chen Xuewen Lu Grace Y. Yi & Hao Yu

Author:Ding-Geng Chen, Jiahua Chen, Xuewen Lu, Grace Y. Yi & Hao Yu
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

(6.6)

where is the vector of local normalized B-spline basis functions, , is the vector of B-spline coefficients. Having K B interior knots and B-splines of order ρ B , the degree of freedom of the B-spline would be df B  = K B +ρ B .

For identifiability purpose we assume ∥ β ∥  = 1 and perform the delete-one-component method by defining as a ( p − 1)-dimensional vector deleting the first component β 1. We also assume β 1 > 0 which could be implemented by considering . Then, we have where the true parameter . Therefore, β is infinitely differentiable in a neighborhood of the true parameter . Since we use B-spline to estimate , the baseline hazard function is which has to be positive and nondecreasing. The positivity is guaranteed by property of a logarithmic function and we just need to satisfy the condition of being nondecreasing by putting a constraint on the coefficients of the basis functions, that is .

Under suitable smoothness assumptions logit{1 − S 0(⋅ )} and ψ(⋅ ) can be well approximated by functions in and , respectively. Therefore, we have to find members of and along with values for α and β that maximize the semiparametric log-likelihood function.

Now in Eq. (6.4) we replace B-spline approximations for logit{1 − S 0(t)} and ψ(β ⊤ X) from (6.5) and (6.6), respectively. Considering logit{1 − S(t | Z)} = −logit{S(t | Z)}, (6.4) is equivalent to

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