Academic literature on the topic 'Distribution (probability theory)'

In this thesis, we have studied L-moments and trimmed L-moments (TL-moments) which are both linear functions of order statistics. We have derived expressions for exact variances and covariances of sample L-moments and of sample TL-moments for any sample size n in terms of first and second-order moments of order statistics from small conceptual sample sizes, which do not depend on the actual sample size n. Moreover, we have established a theorem which characterises the normal distribution in terms of these second-order moments and the characterisation suggests a new test of normality. We have also derived a method of estimation based on TL-moments which gives zero weight to extreme observations. TL-moments have certain advantages over L-moments and method of moments. They exist whether or not the mean exists (for example the Cauchy distribution) and they are more robust to the presence of outliers. Also, we have investigated four methods for estimating the parameters of a symmetric lambda distribution: maximum likelihood method in the case of one parameter and L-moments, LQ-moments and TL-moments in the case of three parameters. The L-moments and TL-moments estimators are in closed form and simple to use, while numerical methods are required for the other two methods, maximum likelihood and LQ-moments. Because of the flexibility and the simplicity of the lambda distribution, it is useful in fitting data when, as is often the case, the underlying distribution is unknown. Also, we have studied the symmetric plotting position for quantile plot assuming a symmetric lambda distribution and conclude that the choice of the plotting position parameter depends upon the shape of the distribution. Finally, we propose exponentially weighted moving average (EWMA) control charts to monitor the process mean and dispersion using the sample L-mean and sample L-scale and charts based on trimmed versions of the same statistics. The proposed control charts limits are less influenced by extreme observations than classical EWMA control charts, and lead to tighter limits in the presence of out-of-control observations.

Nontrivial domains can be difficult to understand and the task of encoding a model of such a domain can be difficult for a human expert, which is one of the fundamental problems of knowledge acquisition. Model learning provides a way to address this problem by allowing a predictive model of the domain's dynamics to be learnt algorithmically, without human supervision. Such models can provide insight about the domain to a human or aid in automated planning or reinforcement learning. This dissertation addresses the problem of how to learn a model of a continuous, dynamic domain, from sensory observations, through the discretisation of its continuous state space. The learning process is unsupervised in that there are no predefined goals, and it assumes no prior knowledge of the environment. Its outcome is a model consisting of a set of predictive cause-and-effect rules which describe changes in related variables over brief periods of time. We present a novel method for learning such a model, which is centred around the idea of discretising the state space by identifying clusters of uniform density in the probability density function of variables, which correspond to meaningful features of the state space. We show that using this method it is possible to learn models exhibiting predictive power. Secondly, we show that applying this discretisation process to two-dimensional vector variables in addition to scalar variables yields a better model than only applying it to scalar variables and we describe novel algorithms and data structures for discretising one- and two-dimensional spaces from observations. Finally, we demonstrate that this method can be useful for planning or decision making in some domains where the state space exhibits stable regions of high probability and transitional regions of lesser probability. We provide evidence for these claims by evaluating the model learning algorithm in two dynamic, continuous domains involving simulated physics: the OpenArena computer game and a two-dimensional simulation of a bouncing ball falling onto uneven terrain.

The inverse Gaussian distribution is investigated as a basis for statistical analyses of skewed and possibly censored response times. This distribution arises from a random walk process, is a member of the exponential family, and admits the sample arithmetic and harmonic means as complete sufficient statistics. In addition, the inverse Gaussian provides a reasonable alternative to the more commonly used lognormal statistical model due to the attractive properties of its parameter estimates.<br>Three modifications were made to the basic distribution definition: adding a shift parameter to account for minimum latencies, allowing for Type I censoring, and convoluting two inverse Gaussian random variables in order to model components of response times. Corresponding parameter estimation and large sample test procedures were also developed.<br>Results from analysing two extensive sets of simple and two-choice reaction times suggest that shifting the origin and accounting for Type I censoring can substantially improve the reliability of inverse Gaussian parameter estimates. The results also indicate that the convolution model provides a convenient medium for probing underlying psychological processes.

Counting data without zero category often occur in various _elds. Examples include days of hospital stay for patients, numbers of publication for tenure-tracked faculty in a university, numbers of tra_c violation for drivers during a certain period and so on. A class of zero-truncated discrete models such as zero-truncated Poisson, zero-truncated binomial and zero-truncated negative-binomial distributions are proposed in literature to model such count data. In this thesis, firstly, literature review is presented in Chapter 1 on a class of commonly used univariate zero-truncated discrete distributions. In Chapter 2, a unified method is proposed to derive the distribution of the sum of i.i.d. zero-truncated distribution random variables, which has important applications in the construction of the shortest Clopper-Person confidence intervals of parameters of interest and in the calculation of the exact p-value of a two-sided test for small sample sizes in one sample problem. These problems are discussed in Section 2.4. Then a novel expectation-maximization (EM) algorithm is developed for calculating the maximum likelihood estimates (MLEs) of parameters in general zero-truncated discrete distributions. An important feature of the proposed EM algorithm is that the latent variables and the observed variables are independent, which is unusual in general EM-type algorithms. In addition, a unified minorization-maximization (MM) algorithm for obtaining the MLEs of parameters in a class of zero-truncated discrete distributions is provided. The first objective of Chapter 3 is to propose the multivariate zero-truncated Charlier series (ZTCS) distribution by developing its important distributional properties, and providing efficient MLE methods via a novel data augmentation in the framework of the EM algorithm. Since the joint marginal distribution of any r-dimensional sub-vector of the multivariate ZTCS random vector of dimension m is an r-dimensional zero-deated Charlier series (ZDCS) distribution (1 6 r < m), it is the second objective of Chapter 3 to propose a new family of multivariate zero-adjusted Charlier series (ZACS) distributions (including the multivariate ZDCS distribution as a special member) with a more flexible correlation structure by accounting for both inflation and deflation at zero. The corresponding distributional properties are explored and the associated MLE method via EM algorithm is provided for analyzing correlated count data.<br>published_or_final_version<br>Statistics and Actuarial Science<br>Master<br>Master of Philosophy

Thesis (Ph. D.)--University of California, San Diego, 2009.<br>Title from first page of PDF file (viewed July 23, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 104-109).

The Frechet distribution was applied to a set of earthquake data in order to test its validity as a practical alternative distribution for extreme value data. It was concluded that the Frechet distribution was the best model representing that data set. Also, a Poisson model of occurrence could not be rejected for that data set. The combination of these two models resulted in a closed form unconditional extreme value distribution which was developed analytically. The appropriate statistical tests and sensitivity analyses were performed on the obtained model.<br>Applied Science, Faculty of<br>Mechanical Engineering, Department of<br>Graduate