| With the development of science and technology,the purpose of research and investigation tends to be diversified.In tracking and observation research in many fields such as economics,demography,and biomedicine,the cohort sampling design is a commonly-used approach.However,owing to the limitation of human and material resources,the cohort-sampling design has been gradually replaced by a variety of more targeted and more efficient sampling methods in actual research.Length-biased sampling design,case-cohort sampling design,and nested case-control sampling design are the three important methods.These three methods are extremely economical and have been widely used.However,these three methods are all biased sampling,and all obtained biased data.At the same time,in most cases,it is difficult to avoid censoring in the actual tracking observation data,thus the data structure obtained is often extremely complex.Therefore,if you directly use the existing simple statistical methods to analyze this type of data,the result will be biased and untrustworthy.The monotonic transformation model is a semi-parametric model that requires fewer assumptions and contains many models.It doesn’t need to make assumptions about the functional relationship between covariates and response variables,nor to specify the distribution of error terms,and thus it has been widely used.Based on this,this article utilizes it to analyze the complex data obtained by biased sampling.This article mainly studies two issues.The first problem studies the statistical inference and its application of the monotonic transformation model under case-cohort sampling and nested case-control sampling with right-censored data;the second problem studies the statistical inference and its application of the monotonic transformation model under the case-cohort studies and nested case-control studies with length-biased sampling and right-censored data.For the first problem,this paper uses the inverse probability weighted idea to construct a new extreme objective function based on the maximum rank correlation estimation.Moreover,due to the obtained objective function containing a nondifferentiable indicator function,a smoothed objective function is further suggested for the convenience of calculation and proof.For the second question,this paper proposes an extreme objective function based on the monotonic rank estimation approach and the inverse probability weighted technique.Similarly,a simple and easy-to-used smoothing objective function is further constructed.In addition,the asymptotic properties of the resulting estimators are all derived.Meanwhile,numerical simulation and real data analysis are also conducted to verify the performance of the proposed method under limited samples and prove its feasibility and practicability.The research results indicate that the proposed two estimation methods for the monotonic transformation models under the biased sampling data in this paper are reasonable,effective,and easy to calculate. |
