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Skewness and elongation are two factors that directly determine the shape of a probability distribution. Thus, to obtain a flexible distribution it is always desirable that the parameters of the distribution directly determine the skewness and elongation. To meet this purpose, Tukey (1977) introduced a family of distributions called g-and-h family (gh family) based on a transformation of the standard normal variable where g and h determine the skewness and the elongation, respectively. The gh family of distributions was extensively studied by Hoaglin (1985) and Martinez and Iglewicz (1984). For its flexibility in shape He and Raghunathan (2006) have used this distribution for multiple imputations. Because of the complex nature of this family of distributions, it is not possible to have an explicit mathematical form of the density function and the estimates of the parameters g and h fully depend on extensive numerical computations.In this study, we have developed algorithms to numerically compute the density functions. We present algorithms to obtain the estimates of g and h using method of moments, quantile method and maximum likelihood method. We analyze the performance of each method and compare them using simulation technique. Finally, we study some special cases of gh family and their properties. / Department of Mathematical Sciences

There is no abstract available for this thesis.

The family of univariate skew-normal probability distributions, an extension of symmetric normal distribution to a general case of asymmetry, was originally proposed by Azzalani [1]. Since its introduction, very limited research has been conducted in this area. An extension of the univariate skew-normal distribution to the multivariate case was considered by Azzalani and Dalla Valle [4]. Its application in statistics was recently considered by Azzalani and Capitanio [3]. As a general result, Azzalani (1985) [See [1]] showed that, any symmetric distribution can be viewed as a member of a more general class of skewed distributions.In this study we establish some properties of general family of skewed distributions. Examples of general family of asymmetric distributions is presented in a way to show their differences from the corresponding symmetric distributions. The skew-logistic distribution and its properties are considered in great details. / Department of Mathematical Sciences

The set of non-crossing partitions was first studied by Kreweras in 1972 and was known to play an important role in combinatorics, geometric group theory, and free probability. In particular, it has a natural embedding into the symmetric group, and there is an extensive literature on the asymptotic cycle structures of random permutations. This motivates our study on analogous results regarding the asymptotic block structure of random non-crossing partitions. We first investigate an analogous result of the asymptotic distribution for the total number of cycles of random permutations due to Goncharov in 1940's: Goncharov showed that the total number of cycles in a random permutation is asymptotically normally distributed with mean log(n) and variance log(n). As a analog of this result, we show that the total number of blocks in a random non-crossing partition is asymptotically normally distributed with mean n/2 and variance n/8. We also investigate the outer blocks, which arise naturally from non-crossing partitions and has many connections in combinatorics and free probability. It is a surprising result that among many blocks of non-crossing partitions, the expected number of outer blocks is asymptotically 3. We further computed the asymptotic distribution for the total number of blocks, which is a shifted negative binomial distribution.

We derive analytic expressions for the biases, to O(n^(-1)) of the maximum likelihood estimators of the parameters of the generalized Rayleigh distribution family. Using these expressions to bias-correct the estimators is found to be extremely effective in terms of bias reduction, and generally results in a small reduction in relative mean squared error. In general, the analytic bias-corrected estimators are also found to be superior to the alternative of bias-correction via the bootstrap. / Graduate

Lake size is a strong control on lake function and on how lakes interact with the environment. For example, lake size is related to carbon burial rates in lake sediments. Lake size distribution (the number of small, medium, and large lakes per unit area) can be used to extrapolate lake function to landscapes at local, regional and global scales. This research examined the utility of using radar satellite imagery (ALOS PALSAR) and existing spatial data (CanVec) for the construction of a new Canadian lake database, which was then used to estimate carbon accumulation in Canadian boreal lake sediments. The capability of ALOS PALSAR images for classifying lakes from eight pilot regions across Canada was assessed by direct comparison to existing CanVec data. The PALSAR lake classification differed between -1.8% to 18.0% for overall lake area and -56.0% to 196.0% for overall lake count compared to CanVec. The wide range in difference can be explained by limitations in resolution, classification method, and how a lake was defined. While the temporal resolution of PALSAR was superior, it did not provide better spatial resolution and accuracy than existing datasets. PALSAR’s utility therefore is in short term change determination. Consequently, CanVec was used to construct the final database describing lake distribution in Canada, resulting in over 13.2 million features with a total area of almost 1.2 million km2. Lake database results suggest that the scaling rules used in previous studies to estimate the number of very small lakes regionally and globally have limits. The use of real lake data allowed for a better understanding of regional differences in lake distribution across Canada that was not possible with scaling rule approaches. Estimates of carbon accumulation in boreal Canada lake sediments based on the new CanVec lake distribution and literature-based accumulation rates ranged from 1.65 and 2.34 Mt C yr-1, or roughly equal to the carbon emissions of 300,000-450,000 cars per year. Similarly, it would require only 36 years for Canada’s total annual emissions to account for all the carbon accumulation in Canadian boreal lakes over the Holocene (last 10,000 years). Thematic products derived from the lake database suggest that number of lakes is more important than the distribution of small, medium and large lakes when estimating carbon accumulation in the lake sediments of boreal Canada. / Graduate

Classification using mixture distributions to model each class has not received too much attention in the literature. The most important attempts use normal distributions as com- ponents in these mixtures. Recently developed methods have allowed the use of these kinds of models as a flexible approach for density estimation. Most of the methods de- veloped so far use plug-in estimates for the parameters and assume that the number of components in the mixture is known. We obtain a predictive classifier for the classes by using Markov Chain Monte Carlo techniques which allow us to obtain a sampling chain for the parameters. This fully Bayesian approach to classification has the advantage that the number of components for each class is taken as another variable parameter and integrated out of the classification. To achieve this we use a birth-and-death/Gibbs sampler algorithm developed by Stephens (1997). We use five different datasets, two simulated ones to test the methods on a single class and three real datasets to test the methods for classification. We look at different models to de- fine which gives better flexibility in the modelling and an overall better classification. We look at different types of priors for the means and dispersion matrices of the components. Joint conjugate priors and an independent conjugate priors for the means and dispersion matrices for the components are used. We use a model with a common dispersion matrix for all the components and another one with a reparametrisation of these dispersion ma- trices into size, shape and orientation (Banfield and Raftery (1993)). We allow the sizes to differ while keeping a common shape and orientation for the dispersion matrices of the components in a class. We found that this type of modelling with independent conjugate priors for the means and dispersions while allowing the sizes of the dispersions to vary gave the best results for classification purposes as it allowed great flexibility and separation between the compo- nents of the classes.

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No description available.