The statistics literature has mostly focused on the case when the data available is in the form of a random sample. In many cases, the observations are not identically distributed. Such samples are called heterogeneous samples. The study of heterogeneous samples is of great interest in many areas, such as statistics, econometrics, reliability engineering, operation research and risk analysis. Stochastic orders between probability distributions is a widely studied concept. There are several kinds of stochastic orders that are used to compare different aspects of probability distributions like location, variability, skewness, dependence, etc. In this dissertation, most of the work is devoted to investigating the properties of statistics based on heterogeneous samples with the aid of stochastic orders. We will see the effect of the change in the stochastic properties of various functions of observations as their parameters change. The theory of majorization will be used for this purpose. First, order statistics from heterogeneous samples will be investigated. Order statistics appear everywhere in statistics and related areas. The k-out-of-n systems are building blocks of a coherent system. The lifetime of such a system is the same as that of the (n-k+1)th order statistic in a sample size of n. Stochastic comparisons between order statistics have been studied extensively in the literature in case the parent observations are independent and identically distributed. However, in practice this assumption is often violated as different components in a system may not have the same distribution. Comparatively less work has been done in the case of heterogeneous random variables, mainly because of the reason that their distribution theory is very complicated. Some open problems in the literature have been solved in the dissertation. Some new problems associated with order statistics have been investigated in the thesis. Next, stochastic properties of spacings based on heterogeneous observations are studied. Spacings are of great interest in many areas of statistics, in particular, in the characterizations of distributions, goodness-of-fit tests, life testing and reliability models. In particular, the stochastic properties of the sample range are investigated in detail. Applications in reliability theory are highlighted. The relative dependence between extreme order statistics will be investigated in Chapter 4. In particular, the open problem discussed in Dolati, et al. (2008) is solved in this Chapter. In the last Chapter, convolutions of random variables from heterogeneous samples will be investigated. Convolutions have been widely used in many areas to model many practical situations. For example, in reliability theory, it arises as the lifetime of a redundant standby system; in queuing theory, it is used to model the total service time by an agent in a system; in insurance, it is used to model total claims on a number of policies in the individual risk model. I will compare the dispersion and skewness properties of convolutions of different heterogeneous samples. The tail behavior of convolutions are investigated as well. The work in this dissertation has significant applications in many diverse areas of applied probability and statistics. For example, statistics based on order statistics and spacings from heterogeneous samples arise in studying the robust properties of statistical procedures; the work on order statistics will also provide a better estimation of lifetime of a coherent system in reliability engineering; convolution results will be of great interest in insurance and actuarial science for evaluating risks.

published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2009. / Includes bibliographical references (leaves 82-87). Also available in print.

We propose a novel analytic approach for the comparative statics analysis of operations management problems on the capacity investment decision and the influenza (flu) vaccine composition decision. Our approach involves exploiting the properties of the underlying mathematical models, and linking those properties to the concept of stochastic orders relationship. The use of stochastic orders allows us to establish our main results without restriction to a specific distribution. A major strength of our approach is that it is "scalable," i.e., it applies to capacity investment decision problem with any number of non-independent (i.e., demand or resource sharing) products and resources, and to the influenza vaccine composition problem with any number of candidate strains, without a corresponding increase in computational effort. This is unlike the current approaches commonly used in the operations management literature, which typically involve a parametric analysis followed by the use of the implicit function theorem. Providing a rigorous framework for comparative statics analysis, which can be applied to other problems that are not amenable to traditional parametric analysis, is our main contribution. We demonstrate this approach on two problems: (1) Capacity investment decision, and (2) influenza vaccine composition decision. A comparative statics analysis is integral to the study of these problems, as it allows answers to important questions such as, "does the firm acquire more or less of the different resources available as demand uncertainty increases? does the firm benefit from an increase in demand uncertainty? how does the vaccine composition change as the yield uncertainty increases?" Using our proposed approach, we establish comparative statics results on how the newsvendor's expected profit and optimal capacity decision change with demand risk and demand dependence in multi-product multi-resource newsvendor networks; and how the societal vaccination benefit, the manufacturer's profit, and the vaccine output change with the risk of random yield of strains. / Ph. D.

Este trabalho reporta sobre a avaliação da confiabilidade de networks, uma representação analítica para diversos sistemas de engenharia e de comunicação, cujas falhas de seus componentes (links) ocorrem segundo um Processo de Poisson Não Homogêneo. Concluiremos que, na comparação de dois networks com a mesma quantidade de links, as desigualdades estocásticas de seus D-espectros serão preservadas em seus tempos de vidas e a preservação das classes de distribuições do D-espectro para o tempo de vida de um network ocorrerá com restrições na função de risco do Processo de Poisson Não Homogêneo. / This work reports networks reliability evaluation, an analytic representation to several engineering and comunication systems, whose components (links) failures occur according to a Nonhomogeneous Poisson Process. We will conclude that, on comparison of two networks with same number of links, stochastic orders of their D-spectra will be preserved to their lifetimes and distribution classes preservation of network D-spectrum to its lifetime will occur with restrictions in hazard function of Nonhomogeneous Poisson Process.

In this thesis we consider two comparative statics questions of changes in risk. The first question concerns situations where an individual faces some risk and has no control over the uncertain environment. In these situations we ask what kind of changes in risk will cause the individual's expected utility to increase. The second comparative statics question concerns situations where an individual faces some risk and has some control over the uncertain environment. In particular, we consider situations where the individual maximizes her expected utility with respect to some control parameter. Here we ask what kind of changes in risk will cause the individual's optimal value of the control parameter to increase. The existing literature has answered these questions for a class of individuals (for example, the class of risk averse individuals). This thesis differs from existing literature as it focuses on a given individual, and thus reveals some of the person-specific factors that affect individual?s responses to changes in risk. The aim of the thesis is to show how an order on distributions, termed single crossing likelihood ratio (SCLR) order, can intuitively answer both questions for a given individual. The main contributions of the thesis are as follows. First, the thesis presents the SCLR order and its main properties. Second, the thesis shows that the SCLR order can answer the above comparative statics questions in an intuitive way. In particular, the thesis shows that the answer to the above questions, with the use of the SCLR order, depends on a risk reference point which can be interpreted as a "certainty equivalent" point. Thus it is demonstrated that individual's responses to changes in risk are affected by her "certainty equivalent" point. Lastly, the results of the thesis can be used to provide an intuitive explanation of related existing results that were obtained for a class of individuals.

The increasing complexity of heterogeneous cellular networks (HetNets) due to the irregular deployment of small cells demands significant rethinking in the way cellular networks are perceived, modeled and analyzed. In addition to threatening the relevance of classical models, this new network paradigm also raises questions regarding the feasibility of state-of-the-art simulation-based approach for system design. This dissertation proposes a fundamentally new approach based on random spatial models that is not only tractable but also captures current deployment trends fairly accurately. First, this dissertation presents a general baseline model for HetNets consisting of K different types of base stations (BSs) that may differ in terms of transmit power, deployment density and target rate. Modeling the locations of each class of BSs as an independent Poisson Point Process (PPP) allows the derivation of surprisingly simple expressions for coverage probability and average rate. One interpretation of these results is that adding more BSs or tiers does not necessarily change the coverage probability, which indicates that fears of "interference overload" in HetNets are probably overblown. Second, a flexible notion of BS load is incorporated by introducing a new idea of conditionally thinning the interference field. For this generalized model, the coverage probability is shown to increase when lightly loaded small cells are added to the existing macrocellular networks. This is due to the fact that owing to the smaller loads, small cells typically transmit less often than macrocells, thus contributing less to the interference power. The same idea of conditional thinning is also shown to be useful in modeling the non-uniform user distributions, especially when the users lie closer to the BSs. Third, the baseline model is extended to study multi-antenna HetNets, where BSs across tiers may additionally differ in terms of the number of transmit antennas, number of users served and the multi-antenna transmission strategy. Using novel tools from stochastic orders, a tractable framework is developed to compare the performance of various multi-antenna transmission strategies for a fairly general spatial model, where the BSs may follow any general stationary distribution. The analysis shows that for a given total number of transmit antennas in the network, it is preferable to spread them across many single-antenna BSs vs. fewer multi-antenna BSs. Fourth, accounting for the load on the serving BS, downlink rate distribution is derived for a generalized cell selection model, where shadowing, following any general distribution, impacts cell selection while fading does not. This generalizes the baseline model and all its extensions, which either ignore the impact of channel randomness on cell selection or lumps all the sources of randomness into a single random variable. As an application of these results, it is shown that in certain regimes, shadowing naturally balances load across various tiers and hence reduces the need for artificial cell selection bias. Fifth and last, a slightly futuristic scenario of self-powered HetNets is considered, where each BS is powered solely by a self-contained energy harvesting module that may differ across tiers in terms of the energy harvesting rate and energy storage capacity. Since a BS may not always have sufficient energy, it may not always be available to serve users. This leads to a notion of availability region, which characterizes the fraction of time each type of BS can be made available under variety of strategies. One interpretation of this result is that the self-powered BSs do not suffer performance degradation due to the unreliability associated with energy harvesting if the availability vector corresponding to the optimal system performance lies in the availability region. / text

In this thesis we consider two comparative statics questions of changes in risk. The first question concerns situations where an individual faces some risk and has no control over the uncertain environment. In these situations we ask what kind of changes in risk will cause the individual's expected utility to increase. The second comparative statics question concerns situations where an individual faces some risk and has some control over the uncertain environment. In particular, we consider situations where the individual maximizes her expected utility with respect to some control parameter. Here we ask what kind of changes in risk will cause the individual's optimal value of the control parameter to increase. The existing literature has answered these questions for a class of individuals (for example, the class of risk averse individuals). This thesis differs from existing literature as it focuses on a given individual, and thus reveals some of the person-specific factors that affect individual?s responses to changes in risk. The aim of the thesis is to show how an order on distributions, termed single crossing likelihood ratio (SCLR) order, can intuitively answer both questions for a given individual. The main contributions of the thesis are as follows. First, the thesis presents the SCLR order and its main properties. Second, the thesis shows that the SCLR order can answer the above comparative statics questions in an intuitive way. In particular, the thesis shows that the answer to the above questions, with the use of the SCLR order, depends on a risk reference point which can be interpreted as a "certainty equivalent" point. Thus it is demonstrated that individual's responses to changes in risk are affected by her "certainty equivalent" point. Lastly, the results of the thesis can be used to provide an intuitive explanation of related existing results that were obtained for a class of individuals.