Contributions to Semiparametric Inference to Biased-Sampled and Financial Data
This thesis develops statistical models and methods for the analysis of life-time and financial data under the umbrella of semiparametric framework. The first part studies the use of empirical likelihood on Levy processes that are used to model the dynamics exhibited in the financial data. The second part is a study of inferential procedure for survival data collected under various biased sampling schemes in transformation and the accelerated failure time models. During the last decade Levy processes with jumps have received increasing popularity for modelling market behaviour for both derivative pricing and risk management purposes. Chan et al. (2009) introduced the use of empirical likelihood methods to estimate the parameters of various diffusion processes via their characteristic functions which are readily available in most cases. Return series from the market are used for estimation. In addition to the return series, there are many derivatives actively traded in the market whose prices also contain information about parameters of the underlying process. This observation motivates us to combine the return series and the associated derivative prices observed at the market so as to provide a more reflective estimation with respect to the market movement and achieve a gain in efficiency. The usual asymptotic properties, including consistency and asymptotic normality, are established under suitable regularity conditions. We performed simulation and case studies to demonstrate the feasibility and effectiveness of the proposed method. The second part of this thesis investigates a unified estimation method for semiparametric linear transformation models and accelerated failure time model under general biased sampling schemes. The methodology proposed is first investigated in Paik (2009) in which the length-biased case is considered for transformation models. The new estimator is obtained from a set of counting process-based unbiased estimating equations, developed through introducing a general weighting scheme that offsets the sampling bias. The usual asymptotic properties, including consistency and asymptotic normality, are established under suitable regularity conditions. A closed-form formula is derived for the limiting variance and the plug-in estimator is shown to be consistent. We demonstrate the unified approach through the special cases of left truncation, length-bias, the case-cohort design and variants thereof. Simulation studies and applications to real data sets are also presented.
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