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Freund [1961] introduced a bivariate extension of the exponential distribution that provides a model in which the exponential residual lifetime of one component depends on the working status of another component. We define and study an extension of the Freund distribution in this dissertation. In the first chapter we define some basic concepts that are needed for later developments. We give the definition of the multivariate conditional hazard rate functions of a nonnegative absolutely continuous random vector and study a characterization of these functions in Section 1.1. Then we study some notions of aging: an increasing failure rate (IFR) distribution, a decreasing failure rate (DFR) distribution, an increasing failure rate average (IFRA) distribution, and a decreasing failure rate average (DFRA) distribution in Section 1.2. In Section 1.3 we study two concepts of multivariate dependence: association and positive quadrant dependence. In Chapter 2 we construct a shock model and the new bivariate distribution is the joint distribution of the resulting lifetimes. We explicitly compute the density function, survival function, moment generating function, marginal density functions and marginal survival functions. Also in this chapter, we study the correlation coefficient and other senses of positive dependence of the two random variables of the new bivariate distribution. Then we extend the new distribution to multivariate case. In Chapter 3 we study some aging properties. We obtain two results about the new distribution in n dimensions. The first result says that the marginal distributions of the new multivariate distribution have decreasing failure rate if the conditional hazard rates are decreasing and bounded above by 1. The second one concerns an (n-1 )-out-of-n system such that the joint distribution of the lifetimes of each component is the new distribution in n dimensions. It gives conditions on the parameters under which the system has an IFRA distribution. In Chapter 4 we develop some estimation procedure for the parameters a and b of the new bivariate distribution. We apply the method of moments and the maximum likelihood principle to estimate the parameters. We prove that the method of moments estimator is a consistent asymptotically normal estimator. Then we use Mathematica to run simulation and compare the method of moments estimator with the maximum likelihood estimator. We also compute the 95% confidence interval for a and b from the method of moments estimator. In the last chapter we study a stochastic ordering problem. We have two nonnegative n dimensional random vectors X and Y. We assume that X and Y have the same conditional hazard rates up to a certain level. We give a condition under which the two vectors X and Y are stochastically ordered. |
