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Abstract The uses of the binomial and multinomial distributions in statistical modelling and analyzing discrete data are very well understood, with a huge variety of applications and appropriate software, but there are plenty of real-life examples where these simple models are inadequate. Therefore, it seems wise to consider flexible alternative models to take into account the overdispersion or underdispersion (Hinde & Demetrio (1998)). Thus, the binomial and Poisson distributions have been generalized in several ways to handle the problem of dispersion inherent in the analysis of discrete data that may arise with the presence of aggregation of the individuals. The binomial distribution has been generalized in various ways. Rudolfer (1990), Madsen (1993) and Luceno & Ceballos (1995) have summarized most of these generalizations. Among these extensions, there are the generalized binomial distribution introduced by Edwards (1960) and the multiplicative and the additive generalized binomial distributions which were derived by Altham (1978). As finite Markovian models are extensively used in varies application fields, the generalized Markov-binomial model (Markov- Bernoulli model MBM, also called Markov modulated Bernoulli process ( Ozekici(1997))) introduced by Edwards (1960) have been studied by many researchers from the various aspects of probability, statistics and their applications, in particular the classical problems related to the usual Bernoulli model (Anis & Gharib (1982), Arvidsson & Francke (2007), Cmey et al (2008) , Cekanavicius & Vellaisamy (2010) , Gharib & Yehia(1987), Inal(1987), Maillart et al.(2008), Minkova & Omey (2011), Ozekici (1997), Ozekici et al (2003),Pacheco et al. (2009), Yehia & Gharib(1993) and others.). Further, due to the fact that the MBM operates in a random environment depicted by a Markov chain so that the probability of success at each trial depends on the state of the environment, this model represents an interesting application of stochastic processes, and thus used by numerous authors in stochastic modelling (see for example, Switzer (1967, 1971), Pedler (1980), Xekalaki & Panaretos (2004), Arvidsson, & Francke (2007), Pires & Diniz (2012)). The present thesis is devoted to study the probability distributions related to the Markov- Bernoulli sequence of random variables (the MBM) from some aspects such as distributional properties, characterizations, limit theorems, generalizations, and throw the light on some applications. The thesis consists of five chapters and an introduction. The introduction is devoted to show the actuality of the subject of study and to give a historical survey about it. Chapter one is devoted to give the basic definitions, properties and preliminary results concerning the Markov- Bernoulli sequence of random variables (MBM). Chapter one, also, throw light on some generalizations of MBM and some examples of its applications. Chapter two is concerned with the Markov binomial and Markov negative binomial distributions, exploring their properties, characteristic functions and relations to other distributions. This chapter contains, also, a numerical study to specify the descriptive characteristics of the Markov binomial distribution, besides a detailed investigation for the generalized Markov-binomial distribution introduced by Xekalaki & Panaretos (2004). Chapter three is devoted to investigate the properties of the Markov-Bernoulli geometric (MBG) distribution through characterizing it. It is worth mentioning that the results of both sections 3.2 and 3.3 are totally new and is published respectively, in Journal of Mathematics and Statistics (Vol. 10, No. 2, 186-191, 2014), and in International Journal of Statistics and Probability (Vol. 3, No. 3, 138-146, 2014). Chapter four is devoted to investigating the limiting behavior of the sum of n- MarkovBernoulli random variables. In section 4.1 a new prove is given for the central limit theorem using generating functions technique. In section 4.2 we discuss in details the results of Gharib et al. (1987) concerning uniform estimates of the rate of convergence in the central limit theorem. Section 4.3 is devoted to discussing the results of Gharib et al. (1991) concerning limit theorem in the space 𝐿𝜋, (1 ≤ 𝜋 ≤ ∞). In chapter five, a new method is introduced for adding two parameters to an existing distribution. This new technique extends the methods of Edwards (1960) and Marshall and Olkin (1997) for adding a parameter to a family of distributions. The method is of direct relevance to the Markov-Bernoulli geometric distribution and is applied in particular, to a one parameter Burr XII distribution to yield a three parameter extended Burr XII distribution which may serve as a competitor to such commonly used three parameters families of distributions. The results of this chapter are totally new and are submitted for publication in an international specialized journal. |