Non-parametric estimation for the location of a change-point in an otherwise smooth hazard function under random censoring (Q2742763)

The following problem is considered. Let \(T_1, \ldots, T_n\) be i.i.d. lifetimes with cumulative distribution function (c.d.f.) \(F\) and let \(C_1, \ldots, C_n\) be i.i.d. censoring times with c.d.f. \(G\), \(T_i\) and \(C_i\) are independent. The observed data is a realization of \(\{ (X_i,\;\delta_i)\), \(i=1,\ldots,n\}\) with \(X_i = \min(T_i, C_i)\), \(\delta_i =I\{ X_i =T_i\}.\) Let \(\Lambda\) be cumulative hazard function \(\Lambda (x) = -\log(1-F(x))\) and let \(\lambda\) be the associated hazard rate function \(\lambda(x) =(d/dx)\Lambda(x).\) A change-point in a hazard function refers here to a localized jump or a sharp cusp in an otherwise smooth hazard function. The authors consider a nonparametric wavelet based estimator for the location of a change point in an otherwise smooth hazard function. Let \(\hat\Lambda_n\) denote the Nelson estimator of the cumulative hazard function \(\Lambda\), given by NEWLINE\[NEWLINE\hat\Lambda_n(x)=\sum_{X_{(i)}\leq x}\delta_{(i)}(n-i+1)^{-1}.NEWLINE\]NEWLINE Let \(\phi_{jk}(x) =2^{j/2} \phi(2^j x -k)\), \(\psi_{jk}(x) =2^{j/2} \psi(2^j x -k)\) be a wavelet basis. Let NEWLINE\[NEWLINE\Delta_{jk}(\lambda)=\int \Phi_{jk} d\Lambda_n(x),\quad \Phi_{jk}(x)=\phi_{j,2k+1} -\phi_{j,2k+3}(x).NEWLINE\]NEWLINE The authors propose the estimator NEWLINE\[NEWLINE\hat\theta_n = 2^{-j_1-1}( \tau(\hat k_n)+\tau(\hat k_n +2) +2),NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\hat k_n=\text{arg} \max_{k \in \{ 0,\ldots,(2^{j_1} -6)/2\}} |\Delta_{j_1 k} |.NEWLINE\]NEWLINE They study the properties of this estimator under some assumptions. The performance of the estimator is checked via simulations and two real examples.