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Abstract
Here three different problems are addressed in Statistics and its applications. The first chapter addresses a statistical estimation problem in controlled branching processes, then a Bayesian test is developed for testing zero-inflation in count and it ends with an application of Bayesian techniques in Bio-medical imaging. Controlled branching processes (CBP) with a random control function provide a useful way to model generation sizes in population dynamics studies, where control on the growth of the population size is necessary at each generation. Motivated by the work of Wei andWinnicki (1990), we develop a weighted conditional least squares estimator of the offspring mean of the CBP and derive the asymptotic limit distribution of the estimator when the process is subcritical, critical and supercritical, respectively. The results obtained here extend those of Wei and Winnicki (1990) for branching processes with immigration and provide aunified limit theory of estimation. A zero-inflated power series distribution is a mixture of a power series distribution and a degenerate distribution at zero, with a mixing probability p for the degenerate distribution. This distribution is useful for modeling count data that may have extra zeros. One question is whether the mixture model can be reduced to the power series portion, corresponding to p = 0, or whether there are so many zeros in the data that zero inflation relative to the pure power series distribution must be included in the model i.e., p 0. The problem is difficult partially because p = 0 is a boundary point. Here, we present a Bayesian test for this problem. We compare our Bayesian solution to two standard frequentist testing procedures. The next topic addresses two clinically challenging problems in the domain of virtual mandibular surgery, namely, (a) detection of minor/hairline fractures and (b) generation of target pattern (reconstructed mandible) with accompanied prognosis of fracture healing. Identification of hairline fractures in input X-ray or Computer Tomography (CT) images isvery difficult, especially, in the presence of noise.We propose aMarkov Random Field (MRF)-Maximum A Posteriori (MAP) probability based two-phase approach using the principles of Bayesian image restoration to solve both the aforementioned problems.