Zero Inflated Exponential Distribution and It\u27s Variants

There have been a lot of studies with respect to the popular Zero Inflated versions of some discrete distributions, like Zero Inflated Poisson and Zero Inflated Negative Binomial distributions. They arise naturally in the literature when one aims to model count data sets having more than usual number of zeros by well known probability distributions. But it can be argued and established that Zero Inflated versions of continuous distributions also make sense. In this thesis, we first provide some motivations for studying Zero Inflated versions of exponential distribution and its variants and then study some parametric and Bayesian aspects related to the Zero Inflated Exponential Distribution

Fractional boundary value problems and Lyapunov-type inequalities with fractional integral boundary conditions

We discuss boundary value problems for Riemann-Liouville fractional differential equations with certain fractional integral boundary conditions. Such boundary conditions are different from the widely considered pointwise conditions in the sense that they allow solutions to have singularities, and different from other conditions given by integrals with a singular kernel since they arise from well-defined initial value problems. We derive Lyapunov-type inequalities for linear fractional differential equations and apply them to establish nonexistence, uniqueness, and existence-uniqueness of solutions for certain linear fractional boundary value problems. Parallel results are also obtained for sequential fractional differential equations. An example is given to show how computer programs and numerical algorithms can be used to verify the conditions and to apply the results

Lyapunov-type inequalities and applications to boundary value problems

Advisors: Qingkai Kong.Committee members: Sien Deng; Bernard Harris; Jeffrey Thunder.Includes bibliographical references.In this dissertation, we derive Lyapunov-type inequalities for integer and fractional order differential equations and use them to study the nonexistence, uniqueness, and existence-uniqueness criteria for several classes of boundary value problems.First, we consider third-order half-linear differential equations of the form ([phi][sub [alpha]2] (([phi][sub [alpha]1](x'))'))' + q(t) [phi][sub [alpha]1[alpha]2](x) = 0, where [phi][sub p](x) = |x|^[p-1]x, and [alpha]1, [alpha]2 > 0. We obtain Lyapunov-type inequalities which utilize integrals of both q+(t) and q-(t) rather than those of |q(t)| as in most papers in the literature. Furthermore, by combining these inequalities with the ``uniqueness implies existence'' theorems by many authors, we establish the uniqueness and hence existence-uniqueness for several classes of boundary value problems for third-order linear equations. This is the first time for Lyapunov-type inequalities to be used to deal with the existence-uniqueness of boundary value problems. These inequalities are further extended to higher order half-linear differential equations. Our results cover and improve many results in the literature when the equations become linear. For the third-order linear differential equation x''' + q(t)x = 0, using the Green's function method in a subtle way, we obtain the sharpest Lyapunov-type inequalities in the literature. We further extend these inequalities to more general third-order and higher order linear differential equations. We also discuss their applications to the existence-uniqueness of boundary value problems. Then we investigate boundary value problems for Riemann-Liouville fractional differential equations with certain fractional integral boundary conditions. Such boundary conditions are different from the widely considered pointwise conditions in the sense that they allow solutions to have singularities. We derive Lyapunov-type inequalities for linear fractional differential equations with order [alpha] [epsilon] (1,2] and [alpha] [epsilon] (2,3], respectively. Our results are good in the sense that they are consistent with the existing ones for the second-order and third-order problems when [alpha]=2,3. Finally, we establish some Lyapunov-type inequalities for Riemann-Liouville fractional differential equations with order [alpha] [epsilon] (2,3] and certain pointwise or mixed boundary conditions. Results are first given for univariate case, and then extended to multivariate case. All the results are new and one of them extends and improves substantially the one in the literature for third-order multivariate boundary value problems.Ph.D. (Doctor of Philosophy

Lyapunov-type inequalities for α\alpha-th order fractional differential equations with 2<α≤3\alpha\le3 and fractional boundary conditions

We study linear fractional boundary value problems consisting of an $\alpha$-th order Riemann-Liouville fractional differential equation with 2<$\alpha\leq 3$ and certain fractional boundary conditions. We derive several Lyapunov-type inequalities and apply them to establish nonexistence, uniqueness, and existence-uniqueness of solutions for related homogeneous and nonhomogeneous linear fractional boundary value problems. As a special case, our work extends some existing results for third-order linear boundary value problems