We first consider confidence intervals for a normal percentile, an exponential percentile and a uniform percentile. Then we develop confidence intervals for a difference of percentiles from two independent normal populations, two independent exponential populations and two independent uniform populations. In our study, we mainly focus on the maximum likelihood to develop our confidence intervals. The efficiency of this method is examined via coverage rates obtained in a simulation study done with the statistical software R.

YSCAT is a unique ultrawideband microwave scatterometer developed to investigate the sea surface under a variety of environmental and radar parameters. The YSCAT94 experiment consisted of a six month deployment on the WAVES research tower operated by the Canada Center for inland Waters (CCIW). Over 3500 hours of data were collected at 2Γ 3.05Γ 5.3Γ 10.02Γ and 14 GHz and at a variety of wind speeds, relative azimuth angles, and incidence angle. A low wind speed "rolloff" of the normalized radar cross section (σ°) in YSCAT94 data is found and quantified. The rolloff wind speedΓ γΓ is estimated through regression estimation analysis using an Epanechnikov kernel. For YSCAT94 data, the rolloff is most noticeable at mid-range incidence angles with γ values ranging from 3 to 6 m/s. In order to characterized YSCAT94 backscatter distributions, a second order polynomial in log space is developed as a model for the probability of the radar cross sectionΓρ(σ°). Following Gotwols and ThompsonΓρ(σ°) is found to adhere to a log-normal distribution for horizontal polarization and a generalized log-normal distribution for vertical polarization. If ρ(α|σ°) is assumed to be Rayleigh distributed, the instantaneous amplitude distribution ρ(α) is found to be the integral of a Rayleigh/generalized log-normal distribution. A robust algorithm is developed to fit this probability density function to YSCAT94 backscatter distributions. The mean and variance of the generalized log-normal distribution are derived to facilitate this algorithm. Over 2700 distinct data cases sorted according to five different frequencies, horizontal and vertical polarizations, upwind and downwind, eight different incidence angles Γ1-10 m/s wind speeds, and 0.1-0.38 mean wave slope are considered. Definite trends are recognizable in the fitted parameters a1Γ a2Γ and C of the Rayleigh/generalized log-normal distribution when sorted according to wind speed and mean wave slope. At mid-range incidence angles, the Rayleigh/generalized log-normal distribution is found to adequately characterize both low and high amplitude portions of YSCAT94 backscatter distributions. However, at higher incidence angels (50°and 60°) the more general Weibull/generalized log-normal distributions is found to better characterized the low amplitude portion of the backscatter distributions.

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<p>When researchers work on time series or sequence, certain fundamental questions will naturally arise. One of them will be whether the series or sequence exhibits a gradual trend over time. In this thesis, we propose a test statistic based on moving order statistics and establish an exact procedure to test for the presence of monotone trends. We show that the test statistic under the null hypothesis that there is no trend follows the closed skew normal distribution. An efficient algorithm is then developed to generate realizations from this null distribution. A simulation study is conducted to evaluate the proposed test under the alternative hypotheses with linear, logarithmic and quadratic trend functions. Finally, a practical example is provided to illustrate the proposed test procedure.</p> / Master of Science (MSc)

To extract the best possible information from geodetic and geophysical observations, it is necessary to select a model of the observation errors, mostly the family of Gaussian normal distributions. However, there are alternatives, typically chosen in the framework of robust M-estimation. We give a synopsis of well-known and less well-known models for observation errors and propose to select a model based on information criteria. In this contribution we compare the Akaike information criterion (AIC) and the Anderson Darling (AD) test and apply them to the test problem of fitting a straight line. The comparison is facilitated by a Monte Carlo approach. It turns out that the model selection by AIC has some advantages over the AD test.

To extract the best possible information from geodetic and geophysical observations, it is necessary to select a model of the observation errors, mostly the family of Gaussian normal distributions. However, there are alternatives, typically chosen in the framework of robust M-estimation. We give a synopsis of well-known and less well-known models for observation errors and propose to select a model based on information criteria. In this contribution we compare the Akaike information criterion (AIC) and the Anderson Darling (AD) test and apply them to the test problem of fitting a straight line. The comparison is facilitated by a Monte Carlo approach. It turns out that the model selection by AIC has some advantages over the AD test.

Stein's method is a powerful technique that can be used to obtain bounds for approximation errors in a weak convergence setting. The method has been used to obtain approximation results for a number of distributions, such as the normal, Poisson and Gamma distributions. A major strength of the method is that it is often relatively straightforward to apply it to problems involving dependent random variables. In this thesis, we consider the adaptation of Stein's method to the class of Variance-Gamma distributions. We obtain a Stein equation for the Variance-Gamma distributions. Uniform bounds for the solution of the Symmetric Variance-Gamma Stein equation and its first four derivatives are given in terms of the supremum norms of derivatives of the test function. New formulas and inequalities for modified Bessel functions are obtained, which allow us to obtain these bounds. We then use local approach couplings to obtain bounds on the error in approximating two asymptotically Variance-Gamma distributed statistics by their limiting distribution. In both cases, we obtain a convergence rate of order n^-1 for suitably smooth test functions. The product of two normal random variables has a Variance-Gamma distribution and this leads us to consider the development of Stein's method to the product of r independent mean-zero normal random variables. An elegant Stein equation is obtained, which motivates a generalisation of the zero bias transformation. This new transformation has a number of interesting properties, which we exploit to prove some limit theorems for statistics that are asymptotically distributed as the product of two central normal distributions. The Variance-Gamma and Product Normal distributions arise as functions of the multivariate normal distribution. We end this thesis by demonstrating how the multivariate normal Stein equation can be used to prove limit theorems for statistics that are asymptotically distributed as a function of the multivariate normal distribution. We establish some sufficient conditions for convergence rates to be of order n^-1 for smooth test functions, and thus faster than the O(n^-1/2) rate that would arise from the Berry-Esseen Theorem. We apply the multivariate normal Stein equation approach to prove Variance-Gamma and Product Normal limit theorems, and we also consider an application to Friedman's X² statistic.

In this thesis we introduce, implement and compare several multivariate goodness-of-fit tests. First of all, we will focus on universal mul- tivariate tests that do not place any assumptions on parametric families of null distributions. Thereafter, we will be concerned with testing of multi- variate normality and, by using Monte Carlo simulations, we will compare power of five different tests of bivariate normality against several alternati- ves. Then we describe multivariate skew-normal distribution and propose a new test of multivariate skew-normality based on empirical moment genera- ting functions. In the final analysis, we compare its power with other tests of multivariate skew-normality. 1

In the present work we study families of skew-probability distributions. We will gradually build concept of families of more and more general distributions. For us the most important ones are skew normal distribution, elliptical distri- bution and skew elliptical distribution. On the each of them we will present basic properties and visualize particular examples. At the end we will generate realizations of variates and propose how to estimate the original distribution.

No presente trabalho, são estudadas propriedades de uma distribuição pertencente à classe de distribuições Kumaraswamy generalizadas, denominada Kumaraswamy normal, formulada a partir da distribuição Kumaraswamy e da distribuição normal. Algumas propriedades estudadas são: expansão da função densidade de probabilidade em série de potências, função geradora de momentos, momentos, função quantílica, entropia de Shannon e de Rényi e estatísticas de ordem. São construídos dois modelos de regressão lineares do tipo localização-escala para a distribuição Kumaraswamy normal, um para dados sem censura e o outro com a presença de observações censuradas. Os parâmetros dos modelos são estimados pelo método de máxima verossimilhança e algumas medidas de diagnóstico, como influência global, influência local e resíduos são desenvolvidos. Para cada modelo de regressão é realizada uma aplicação a um conjunto de dados reais. / In this work, properties of a distribution belonging to the class of generalized Kumaraswamy distributions, called Kumaraswamy normal, are studied. The Kumaraswamy normal distribution is formulated from the Kumaraswamy distribution and from the normal distribution. Some properties studied are: expansion of the probability density function in power series, moment generating function, moments, quantile function, Shannon and Rényi entropy, and order statistics. Two location-scale linear regression models are constructed for the Kumaraswamy-normal distribution, one for datas uncensored and the other with the presence of censoreds observations. The parameters of these models are estimated by the maximum likelihood method and some diagnostic measures such as global influence, local influence and residuals are developed. For each regression model an application is made to a real data set.