Inference in the statistical models for counting processes information with incomplete data and studies of three types of partial likelihoods

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We consider methods for statistical inference in order to confirm or discovery a scientific knowledge under investigation from the data collected by following up times at which individuals sampled or selected from the population experience the event of interest. Then, if the observed data include uncertain concomitant information for individuals, it is founded that there are at least three types of statistical inference methods, based on outlooks in mathematical science. We studied whether the 3-types are truly valid methods in science, and how the 3-types are related to each other, so that we could obtain several useful findings on contributions in mathematical science and applications to substantial science fields.