The NPMLE for doubly censored current status data (Q2771557)

Let \(I\leq J\) be the (random) times of two events which are not observed. The observed times \(A\) and \(B\) are such that \(A\leq I\leq B\). It is supposed that the conditional distribution of \(I\) for fixed \(A\) and \(B\) is uniform on \([A,B]\). The aim is to estimate the CDF \(G\) of \(T=J-I\) by i.i.d. observations \(Y_j=(A_j,B_j,\Delta_j)\), where \(\Delta_j={\mathbf 1}\{J_j\leq B_j\}\). Note, that for \(C=B-A\): NEWLINE\[NEWLINE\Pr(\Delta=1 \;|\;A,B)=F_G(C)=C^{-1}\int_0^C G(t) dt. NEWLINE\]NEWLINE The authors derive estimators \(G_n\) and \(F_{G_n}\) for \(G\) and \(F_G\) using nonparametric maximum likelihood techniques and iterative weighted pool-adjacent violator algorithms. \(L_2\) and uniform consistency results are obtained. It is shown that \(G_n\) converges at rate \(n^{-1/5}\), whence \(F_{G_n}\) at rate \(n^{-2/5}\).