A goodness-of-fit test for parametric models based on dependently truncated data

Suppose that one can observe bivariate random variables (L,X)(L,X) only when L≤XL≤X holds. Such data are called left-truncated data and found in many fields, such as experimental education and epidemiology. Recently, a method of fitting a parametric model on (L,X)(L,X) has been considered, which can easily incorporate the dependent structure between the two variables. A primary concern for the parametric analysis is the goodness-of-fit for the imposed parametric forms. Due to the complexity of dependent truncation models, the traditional goodness-of-fit procedures, such as Kolmogorov–Smirnov type tests based on the Bootstrap approximation to null distribution, may not be computationally feasible. In this paper, we develop a computationally attractive and reliable algorithm for the goodness-of-fit test based on the asymptotic linear expression. By applying the multiplier central limit theorem to the asymptotic linear expression, we obtain an asymptotically valid goodness-of-fit test. Monte Carlo simulations show that the proposed test has correct type I error rates and desirable empirical power. It is also shown that the method significantly reduces the computational time compared with the commonly used parametric Bootstrap method. Analysis on law school data is provided for illustration. R codes for implementing the proposed procedure are available in the supplementary material.