Kurtosis can be measured in more than one way. A modification of Geary's measure of kurtosis is shown to be more sensitive to kurtosis in the center of the distribution while Pearson's measure of kurtosis is more sensitive to kurtosis in the tails of the distribution. The modified Geary measure and the Pearson measure are used to define a joint test of kurtosis that has high uniform power across a very wide range of symmetric nonnormal distributions.

Goodness-of-fit and correlation tests are considered for dependent univariate data that arises when multivariate data is projected to the real line with a data-suggested linear transformation. Specifically, tests for multivariate normality are investigated. Let { } i Y be a sequence of independent k-variate normal random vectors, and let 0 d be a fixed linear transform from Rk to R . For a sequence of linear transforms { ( )} 1 , , n d Y Y converging almost surely to 0 d , the weak convergence of the empirical process of the standardized projections from d to a tight Gaussian process is established. This tight Gaussian process is identical to that which arises in the univariate case where the mean and standard deviation are estimated by the sample mean and sample standard deviation (Wood, 1975). The tight Gaussian process determines the limiting null distribution of E.D.F. goodness-of-fit statistics applied to the process of the projections. A class of tests for multivariate normality, which are based on the Shapiro-Wilk statistic and the related correlation statistics applied to the dependent univariate data that arises with a data-suggested linear transformation, is also considered. The asymptotic properties for these statistics are established. In both cases, the statistics based on random linear transformations are shown to be asymptotically equivalent to the statistics using the fixed linear transformation. The statistics based on the fixed linear transformation have same critical points as the corresponding tests of univariate normality; this allows an easy implementation of these tests for multivariate normality. Of particular interest are two classes of transforms that have been previously considered for testing multivariate normality and are special cases of the projections considered here. The first transformation, originally considered by Wood (1981), is based on a symmetric decomposition of the inverse sample covariance matrix. The asymptotic properties of these transformed empirical processes were fully developed using classical results. The second class of transforms is the principal components that arise in principal component analysis. Peterson and Stromberg (1998) suggested using these transforms with the univariate Shapiro-Wilk statistic. Using these suggested projections, the limiting distribution of the E.D.F. goodness-of-fit and correlation statistics are developed.

When data is collected sample size and precision in measurements are often limited. In what sense this impacts the size, unadjusted and adjusted power of different normality tests is a relatively unexplored field. Therefore this paper is dedicated to perform a simulation study where these three properties of the normality tests Anderson-Darling, Jarque-Bera and Shapiro-Wilk are examined. The study is based on different combinations of sample sizes and roundings where repeated samples are drawn from both normally and asymmetrically distributed populations. The results from the study indicate that coarser roundings results in increased size and unadjusted power of Anderson-Darling and Shapiro-Wilk, while Jarque-Bera is seemingly unaffected by roundnings. The three tests have in common that a larger sample size leads to an increase in the size, unadjusted and adjusted power of the tests and that roundings have no substantial impact on adjusted power. / När data samlas in är ofta stickprovsstorlek och precision i mätningarna begränsad i olika grad. Vilken betydelse detta får för sannolikheten för typ I-fel, ojusterad samt justerad styrka hos olika normalitetstest är ett förhållandevis outforskat område. Därför dedikeras denna uppsats till att genomföra en simuleringsstudie där dessa tre egenskaper hos normalitetstesten Anderson-Darling, Jarque-Bera samt Shapiro-Wilk undersöks. Studien baseras på olika kombinationer av stickprovsstorlekar samt avrundningar där upprepade stickprov dras från både normalfördelade och asymmetriskt fördelade populationer. Resultaten från studien indikerar att grövre avrundningar leder till ökad sannolikhet för typ I-fel och ojusterad styrka hos Anderson-Darling och Shapiro-Wilk, medan Jarque-Bera inte påverkas nämnvärt av avrundningar. Gemensamt för samtliga test är att en större stickprovsstorlek leder till ökad sannolikhet för typ I-fel, ojusterad styrka och justerad styrka samt att avrundningar inte nämnvärt påverkar justerad styrka.

In the globally collaborative effort of maintaining an accurate space catalog, it is of utmost importance for ground tracking stations to provide observations which can be used to update and improve the catalog. However, each tracking station is responsible for viewing thousands of objects in a limited window of time. Limitations in sensor capabilities, human error, and other circumstances inevitably result in erroneous, or unusable, data, but when receiving information from a tracking station, it may be difficult for the end-user to determine a data set's usability. Variables in equipment, environment, and processing create uncertainties when computing the positions and orbits of the satellites. Firstly, this research provides a reference frame for what degrees of errors or biases in equipment translate to different levels of orbital errors after a least squares orbit determination. Secondly, using just an incoming data set's angle error distribution compared to the newly determined orbit, statistical distribution testing is used to determine the validity and usability of the newly received data set. In the context of orbit position uncertainty, users are then able to communicate and relay the uncertainties in the data they share while assessing incoming data for potential sources of error.

This paper presents a stability analysis of the sensor data which is collected by QTAGG from alarge ocean going ship and using the stability results, introduces some information about howthe measurements that come from the sensors can be improved and how reliable they are. In thetheoretical part, some background information is given mainly based on British Standard(BS)ISO 19030 which was published in November, 2016. This source basically includes someinformation about the measurement of changes in hull and propeller performance of a vessel.Using the theoretical information, in the implementation part, the necessary methods areimplemented in python programming language on a real life data set of a vessel which is givenfrom QTAGG company. To measure the stability of the parameters in the data, we loosenthe filters of the parameters and observe how they respond to the technical changes. In orderto understand how loosen the filters can be made, a reference speed-power curve is createdby using a curve fitting method, and after creating a performance indicator by utilizing thereference curve, Anderson-Darling and Shapiro-Wilk tests are used to measure the stability ofthe performance indicator. Besides these numerical tests, some visual methods such as Q-Qplot and histogram plot are also used in this process. Finally, we could provide stability resultsby using both our theoretical knowledge and the practical implementation.

The importance of checking the normality assumption in most statistical procedures especially parametric tests cannot be over emphasized as the validity of the inferences drawn from such procedures usually depend on the validity of this assumption. Numerous methods have been proposed by different authors over the years, some popular and frequently used, others, not so much. This study addresses the performance of eighteen of the available tests for different sample sizes, significance levels, and for a number of symmetric and asymmetric distributions by conducting a Monte-Carlo simulation. The results showed that considerable power is not achieved for symmetric distributions when sample size is less than one hundred and for such distributions, the kurtosis test is most powerful provided the distribution is leptokurtic or platykurtic. The Shapiro-Wilk test remains the most powerful test for asymmetric distributions. We conclude that different tests are suitable under different characteristics of alternative distributions.