Variability in Modified Estimators of VaR and ES

Thesis (Master's)--University of Washington, 2016-06

Copulas in QTL mapping

The standard variance components method for mapping quantitative trait loci is derived on the assumption of normality. Unsurprisingly, statistical tests based on this method do not perform so well if this assumption is not satisfied. We use the statistical concept of copulas to relax the assumption of normality and derive a test that can perform well under any distribution of the continuous trait. In particular, we discuss bivariate normal copulas in the context of sib-pair studies. Our approach is illustrated by a linkage analysis of lipoprotein(a) levels, whose distribution is highly skewed. We demonstrate that the asymptotic critical levels of the test can still be calculated using the interval mapping approach. The new method can be extended to more general pedigrees and multivariate phenotypes in a similar way as the original variance components method.

Nutrient partitioning during treatment of tuberculosis: gain in body fat mass but not in protein mass

Background: Tuberculosis is an important cause of wasting. The functional consequences of wasting and recovery may depend on the distribution of lost and gained nutrient stores between protein and fat masses. Objective: The goal was to study nutrient partitioning, ie, the proportion of weight change attributable to changes in fat mass (FM) versus protein mass (PM), during anti mycobacterial treatment. Design: Body-composition measures were made of 21 men and 9 women with pulmonary tuberculosis at baseline and after 1 and 6 mo of treatment. All subjects underwent dual-energy X-ray absorptiometry and deuterium bromide dilution tests, and a four-compartment model of FM, total body water (TBW), bone minerals (BM), and PM was derived. The ratio of PM to FM at any time was expressed as the energy content (p-ratio). Changes in the p-ratio were related to disease severity as measured by radiologic criteria. Results: Patients gained 10% in body weight (P < 0.001) from baseline to month 6. This was mainly due to a 44% gain in FM (P < 0.001); PM, BM, and TBW did not change significantly. Results were similar in men and women. The p-ratio decreased from baseline to month 1 and then fell further by month 6. Radiologic disease severity was not correlated with changes in the p-ratio. Conclusions: Microbiological cure of tuberculosis does not restore PM within 6 mo, despite a strong anabolic response. Change in the p-ratio is a suitable parameter for use in studying the effect of disease on body composition because it allows transformation of such effects into a normal distribution across a wide range of baseline proportion between fat and protein mass.

Towards a framework for combining stochastic and deterministic descriptions of nonstationary financial time series

We present in this paper ideas to tackle the problem of analysing and forecasting nonstationary time series within the financial domain. Accepting the stochastic nature of the underlying data generator we assume that the evolution of the generator's parameters is restricted on a deterministic manifold. Therefore we propose methods for determining the characteristics of the time-localised distribution. Starting with the assumption of a static normal distribution we refine this hypothesis according to the empirical results obtained with the methods anc conclude with the indication of a dynamic non-Gaussian behaviour with varying dependency for the time series under consideration.

Non-normally distributed data and non-parametric statistics

Different types of numerical data can be collected in a scientific investigation and the choice of statistical analysis will often depend on the distribution of the data. A basic distinction between variables is whether they are ‘parametric’ or ‘non-parametric’. When a variable is parametric, the data come from a symmetrically shaped distribution known as the ‘Gaussian’ or ‘normal distribution’ whereas non-parametric variables may have a distribution which deviates markedly in shape from normal. This article describes several aspects of the problem of non-normality including: (1) how to test for two common types of deviation from a normal distribution, viz., ‘skew’ and ‘kurtosis’, (2) how to fit the normal distribution to a sample of data, (3) the transformation of non-normally distributed data and scores, and (4) commonly used ‘non-parametric’ statistics which can be used in a variety of circumstances.

Statnote 14: the correlation of two variables (Pearson's 'r')

Pearson's correlation coefficient (‘r’) is one of the most widely used of all statistics. Nevertheless, care needs to be used in interpreting the results because with large numbers of observations, quite small values of ‘r’ become significant and the X variable may only account for a small proportion of the variance in Y. Hence, ‘r squared’ should always be calculated and included in a discussion of the significance of ‘r’. The use of ‘r’ also assumes that the data follow a bivariate normal distribution (see Statnote 17) and this assumption should be examined prior to the study. If the data do not conform to such a distribution, the use of a non-parametric correlation coefficient should be considered. A significant correlation should not be interpreted as indicating ‘causation’ especially in observational studies, in which the two variables may be correlated because of their mutual correlations with other confounding variables.

The use of data analysis methods: basic concepts for optometrists

This article explains first, the reasons why a knowledge of statistics is necessary and describes the role that statistics plays in an experimental investigation. Second, the normal distribution is introduced which describes the natural variability shown by many measurements in optometry and vision sciences. Third, the application of the normal distribution to some common statistical problems including how to determine whether an individual observation is a typical member of a population and how to determine the confidence interval for a sample mean is described.

Data analysis methods in optometry: is there a difference between two samples? Part 2.

In this second article, statistical ideas are extended to the problem of testing whether there is a true difference between two samples of measurements. First, it will be shown that the difference between the means of two samples comes from a population of such differences which is normally distributed. Second, the 't' distribution, one of the most important in statistics, will be applied to a test of the difference between two means using a simple data set drawn from a clinical experiment in optometry. Third, in making a t-test, a statistical judgement is made as to whether there is a significant difference between the means of two samples. Before the widespread use of statistical software, this judgement was made with reference to a statistical table. Even if such tables are not used, it is useful to understand their logical structure and how to use them. Finally, the analysis of data, which are known to depart significantly from the normal distribution, will be described.

Data analysis methods in optometry. Part 5: correlation

1. Pearson's correlation coefficient only tests whether the data fit a linear model. With large numbers of observations, quite small values of r become significant and the X variable may only account for a minute proportion of the variance in Y. Hence, the value of r squared should always be calculated and included in a discussion of the significance of r. 2. The use of r assumes that a bivariate normal distribution is present and this assumption should be examined prior to the study. If Pearson's r is not appropriate, then a non-parametric correlation coefficient such as Spearman's rs may be used. 3. A significant correlation should not be interpreted as indicating causation especially in observational studies in which there is a high probability that the two variables are correlated because of their mutual correlations with other variables. 4. In studies of measurement error, there are problems in using r as a test of reliability and the ‘intra-class correlation coefficient’ should be used as an alternative. A correlation test provides only limited information as to the relationship between two variables. Fitting a regression line to the data using the method known as ‘least square’ provides much more information and the methods of regression and their application in optometry will be discussed in the next article.

The choice of the model in inventory control and the cost of sophistication

This thesis is concerned with the inventory control of items that can be considered independent of one another. The decisions when to order and in what quantity, are the controllable or independent variables in cost expressions which are minimised. The four systems considered are referred to as (Q, R), (nQ,R,T), (M,T) and (M,R,T). Wiith ((Q,R) a fixed quantity Q is ordered each time the order cover (i.e. stock in hand plus on order ) equals or falls below R, the re-order level. With the other three systems reviews are made only at intervals of T. With (nQ,R,T) an order for nQ is placed if on review the inventory cover is less than or equal to R, where n, which is an integer, is chosen at the time so that the new order cover just exceeds R. In (M, T) each order increases the order cover to M. Fnally in (M, R, T) when on review, order cover does not exceed R, enough is ordered to increase it to M. The (Q, R) system is examined at several levels of complexity, so that the theoretical savings in inventory costs obtained with more exact models could be compared with the increases in computational costs. Since the exact model was preferable for the (Q,R) system only exact models were derived for theoretical systems for the other three. Several methods of optimization were tried, but most were found inappropriate for the exact models because of non-convergence. However one method did work for each of the exact models. Demand is considered continuous, and with one exception, the distribution assumed is the normal distribution truncated so that demand is never less than zero. Shortages are assumed to result in backorders, not lost sales. However, the shortage cost is a function of three items, one of which, the backorder cost, may be either a linear, quadratic or an exponential function of the length of time of a backorder, with or without period of grace. Lead times are assumed constant or gamma distributed. Lastly, the actual supply quantity is allowed to be distributed. All the sets of equations were programmed for a KDF 9 computer and the computed performances of the four inventory control procedures are compared under each assurnption.

Within-site variation in lichen growth rates and its implications for direct lichenometry

Variation in lichen growth rates poses a significant challenge for the application of direct lichenometry, i.e. the construction of lichen dating curves from direct measurement of growth rates. To examine the magnitude and possible causes of within-site growth variation, radial growth rates (RaGRs) of thalli of the fast-growing foliose lichen Melanelia fuliginosa ssp. fuliginosa (Fr. ex Duby) Essl. and the slow-growing crustose lichen Rhizocarpon geographicum (L.) DC. were studied on two S-facing slate rock surfaces in north Wales, UK using digital photography and an image analysis system (Image-J). RaGRs of M. fuliginosa ssp. fuliginosa varied from 0.44 to 2.63 mmyr-1 and R. geographicum from 0.10 to 1.50 mmyr-1.5. Analysis of variance suggested no significant variation in RaGRs with vertical or horizontal location on the rock, thallus diameter, aspect, slope, light intensity, rock porosity, rock surface texture, distance to nearest lichen neighbour or distance to vegetation on the rock surface. The frequency distribution of RaGR did not deviate from a normal distribution. It was concluded that despite considerable growth rate variation in both species studied, growth curves could be constructed with sufficient precision to be useful for direct lichenometry. © 2014 Swedish Society for Anthropology and Geography.

Individual variability analysis of fluorescence parameters measured in skin with different levels of nutritive blood flow

Fluorescence spectroscopy has recently become more common in clinical medicine. However, there are still many unresolved issues related to the methodology and implementation of instruments with this technology. In this study, we aimed to assess individual variability of fluorescence parameters of endogenous markers (NADH, FAD, etc.) measured by fluorescent spectroscopy (FS) in situ and to analyse the factors that lead to a significant scatter of results. Most studied fluorophores have an acceptable scatter of values (mostly up to 30%) for diagnostic purposes. Here we provide evidence that the level of blood volume in tissue impacts FS data with a significant inverse correlation. The distribution function of the fluorescence intensity and the fluorescent contrast coefficient values are a function of the normal distribution for most of the studied fluorophores and the redox ratio. The effects of various physiological (different content of skin melanin) and technical (characteristics of optical filters) factors on the measurement results were additionally studied.The data on the variability of the measurement results in FS should be considered when interpreting the diagnostic parameters, as well as when developing new algorithms for data processing and FS devices.

A laser induced breakdown spectroscopy quantitative analysis method based on the robust least squares support vector machine regression model

Data fluctuation in multiple measurements of Laser Induced Breakdown Spectroscopy (LIBS) greatly affects the accuracy of quantitative analysis. A new LIBS quantitative analysis method based on the Robust Least Squares Support Vector Machine (RLS-SVM) regression model is proposed. The usual way to enhance the analysis accuracy is to improve the quality and consistency of the emission signal, such as by averaging the spectral signals or spectrum standardization over a number of laser shots. The proposed method focuses more on how to enhance the robustness of the quantitative analysis regression model. The proposed RLS-SVM regression model originates from the Weighted Least Squares Support Vector Machine (WLS-SVM) but has an improved segmented weighting function and residual error calculation according to the statistical distribution of measured spectral data. Through the improved segmented weighting function, the information on the spectral data in the normal distribution will be retained in the regression model while the information on the outliers will be restrained or removed. Copper elemental concentration analysis experiments of 16 certified standard brass samples were carried out. The average value of relative standard deviation obtained from the RLS-SVM model was 3.06% and the root mean square error was 1.537%. The experimental results showed that the proposed method achieved better prediction accuracy and better modeling robustness compared with the quantitative analysis methods based on Partial Least Squares (PLS) regression, standard Support Vector Machine (SVM) and WLS-SVM. It was also demonstrated that the improved weighting function had better comprehensive performance in model robustness and convergence speed, compared with the four known weighting functions.

An Estimate of the Probability Pr(X

2000 Mathematics Subject Classification: 33C90, 62E99

Minimum Description Length Principle in Discriminating Marginal Distributions

2010 Mathematics Subject Classification: 94A17, 62B10, 62F03.