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A goodness-of-fit test for parametric models based on dependently truncated data. (English) Zbl 1252.62052
Summary: Suppose that one can observe bivariate random variables \((L,X)\) only when \(L\leq X\) 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 \((L,X)\) 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 the null distribution, may not be computationally feasible. 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.

62G10 Nonparametric hypothesis testing
62F40 Bootstrap, jackknife and other resampling methods
62H12 Estimation in multivariate analysis
60F05 Central limit and other weak theorems
62F03 Parametric hypothesis testing
62N01 Censored data models
65C05 Monte Carlo methods
65C60 Computational problems in statistics (MSC2010)
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