Analysis of Cumulant Coefficients of Two-Component Mixtures of Shifted Non-Gaussian Distributions

The dependence of the cumulant coefficients of two-component mixtures of shifted non- Gaussian distributions on the weight coefficient is analyzed and conditions are determined under which the cumulant coefficients of any orders are equal to zero. The dependence of the cumulant coefficients of two-component mixtures on the shear parameter is investigated and the parameter values are determined at which the cumulant coefficients of any orders have extrema and zeros. The dependence of the skewness and excess kurtosis of a two-component mixture of shifted Gumbel distributions of type 1 on the weight coefficient and the shear parameter is investigated and their values are obtained at which the skewness and excess kurtosis of the mixture are equal to zero. The features of computer modeling of random variables, the probability density of which is a two-component mixture of shifted distributions, are considered.

OPTION IMPLIED VIX, SKEW AND KURTOSIS TERM STRUCTURES

Comparisons are made of the Chicago Board of Options Exchange (CBOE) skew index with those derived from parametric skews of bilateral gamma models and from the differentiation of option implied characteristic exponents. Discrepancies can be due to strike discretization in evaluating prices of powered returns. The remedy suggested employs a finer and wider set of strikes obtaining additional option prices by interpolation and extrapolation of implied volatilities. Procedures of replicating powered return claims are applied to the fourth power and the derivation of kurtosis term structures. Regressions of log skewness and log excess kurtosis on log maturity confirm the positivity of decay in these higher moments. The decay rates are below those required by processes of independent and identically distributed increments.

Two component modified Lilliefors test for normality

Research background: Commonly known and used parametric tests e.g. Student, Behrens? Fisher, Snedecor, Bartlett, Cochran, Hartley tests are applicable when there is an evidence that samples come from the Normal general population. What makes things worse is that testers are not fully aware in what degree of abnormality distorts results of parametric tests listed above and suchlike. So, it is no exaggeration to say that testing for normality (goodness-of-fit testing, GoFT) is a gate to proper parametric statistical reasoning. It seems that the gate opens too easily. In other words, most popular goodness-of-fit tests are weaker than statisticians want them to be. Purpose of the article: The main purpose of this paper is to put forward the GoFT that is, in particular circumstances, more powerful than GoFTs used until now. The other goals are to define a similarity measure between an alternative distribution and the normal one and to calculate the power of normality tests for a big set of alternatives. And, of course, to interest statisticians in using the GoFTs in their practice. Method: There are two ways to make GoFT more powerful: extensive and intensive one. The extensive method consists in drawing large samples. The intensive method consists in extracting more information from mall samples. In order to make the test method intensive, the test statistics, as distinct from all existing GoFTs, has two components. The first component (denoted by ?) is a classic Kolmogorov / Lilliefors test statistics i.e. the greatest absolute difference between theoretical and empirical cumulative distribution functions. The second component is the order statistics (r) at which the ?_max^((r) ) locate itself. Of course ?_max^((r) ) is the conditional random variable with (r) being the condition. Large scale Monte Carlo simulations provided data sufficient to in-depth study of properties of distributions of ?_max^((r) ) random variable. Findings & value-added: Simulation study shows that the Two Component Modified Lilliefors test for normality is the most powerful for some type of alternatives, especially for the symmetrical, unimodal and bimodal distributions with positive excess kurtosis, for symmetrical and unimodal distributions with negative excess kurtosis and small sample sizes. Due to the values of skewness and excess kurtosis, and the defined similarity measure between the ND and an alternative, alternative distributions are close to the normal distribution. Numerous examples of real data show the usefulness of the proposed GoFT.

A Numerical Study on the Evolution of Random Seas With the Occurrence of Rogue Waves

Numerous numerical and experimental investigations show that rogue waves present much larger probabilities of occurrence than predicted by the linear random wave model, i.e., Gaussian distributed waves. The deviation from normal statistical events excites a continuous concern about rogue wave research. In this study, rogue waves under long-crested and narrow-banded wave trains are checked using the high-order spectral (HOS)-NST model. The JONSWAP wave spectra with random phases are selected as the initial state of the incoming wave trains. Different values of spectral parameters are chosen to reproduce different random sea states with different Benjamin–Feir index (BFI). Numerical results are compared with the classical experimental study and show good agreements. Statistical properties of rogue waves are recounted again within the analysis of exceedance distribution function (EDF) of wave heights and wave crests. Spectral changes are examined, and the monotonic increases with BFI are stressed. However, no bifurcations are observed for BFI near 1. For large BFI, quasi-resonance interactions dominate the wave nonlinearities, and the resulted dynamic excess kurtosis involves initially monotonic enhancement along with space, peaking at around 20–30 wavelengths, but stays at stably high-level values. The quasi-steady-state of dynamic excess kurtosis after full interaction of wave nonlinearities in time and space demonstrates a continuous emergence of rogue waves much more frequent than normality. The changes of excess kurtosis along x are complicated where BFI near 1 and the occurrence of rogue waves might be enhanced even for BFI slightly inferior to 1.

Forecasting VIX Using Filtered Historical Simulation*

We propose a new VIX forecast method using Generalized Autoregressive Conditional Heteroscedasticity models based on the filtered historical simulation put forward in Barone-Adesi, Engle, and Mancini (2008). The flexible change of measure accommodates for non-normalities such as negative skewness and positive excess kurtosis. We present an application for four well-established volatility indices (VIX9D, VIX, VIX3M, and VIX6M). Our results show that our proposed estimation method outperforms the Normal-VIX model of Hao and Zhang (2013) both in-sample and out-of-sample. Furthermore, the use of volatility indices reduces the computational burden significantly compared to the options-based pricing method.

Near-Gaussian distributions for modelling discrete stellar velocity data with heteroskedastic uncertainties

ABSTRACT The velocity distributions of stellar tracers in general exhibit weak non-Gaussianity encoding information on the orbital composition of a galaxy and the underlying potential. The standard solution for measuring non-Gaussianity involves constructing a series expansion (e.g. the Gauss–Hermite series) that can produce regions of negative probability density. This is a significant issue for the modelling of discrete data with heteroskedastic uncertainties. Here, we introduce a method to construct positive-definite probability distributions by the convolution of a given kernel with a Gaussian distribution. Further convolutions by observational uncertainties are trivial. The statistics (moments and cumulants) of the resulting distributions are governed by the kernel distribution. Two kernels (uniform and Laplace) offer simple drop-in replacements for a Gauss–Hermite series for negative and positive excess kurtosis distributions with the option of skewness. We demonstrate the power of our method by an application to real and mock line-of-sight velocity data sets on dwarf spheroidal galaxies, where kurtosis is indicative of orbital anisotropy and hence a route to breaking the mass–anisotropy degeneracy for the identification of cusped versus cored dark matter profiles. Data on the Fornax dwarf spheroidal galaxy indicate positive excess kurtosis and hence favour a cored dark matter profile. Although designed for discrete data, the analytic Fourier transforms of the new models also make them appropriate for spectral fitting, which could improve the fits of high-quality data by avoiding unphysical negative wings in the line-of-sight velocity distribution.

Conceptual Framework for Invariant Protein Fragment Library

Proteins are essential and are present in all life forms and determining its structure is cumbersome, laborious and time consuming. Hence, over 3-4 decades, researchers have been using computational techniques such as template and template free based protein structure prediction from its sequence. This research focuses on developing a conceptual basis for establishing an invariant fragment library which can be used for protein structure prediction. Based on 20 amino acids, fragments can be classified into lengths of 3 to 41 size. Further, they can be classified based on the identical number of amino acids present in the fragment. This encompasses theoretically the number of fragments that can exist and in no way represent the actual possible fragments that can exist in nature. Invariant fragments are ones which are rigid in structure 3-dimensionally and do not change. A formula was arrived at to determine all possible permutations that can exist for length 3 to 41 based on the 20 amino acids. 100 proteins from the Protein Data Bank were downloaded, broken into fragments of 3 to 41 resulting in a total of 6102,102 fragments using Asynchronous Distributed Processing. Then identical fragments in sequence were superimposed and Root Mean Square Deviation (RMSD) values were obtained resulting in roughly 3.2% of the original framgnets.. t-score and z-scores were obtained from which Skewness, Kurtosis and Excess Kurtosis were determined. For invariance, skewness cutoff was set at + 0.1 and using the excess kurtosis, fragments whose distribution were either leptokurtic or platykurtic and were within + 1 standard deviation of the mean value were considered as invariant i.e., if there were no outliers in the distribution and if most of the t-score or z-score values were centered around its average value. Using these cutoff values, fragments were classified and deposited into an invariant fragment library. Roughly 3,81,799 invariant fragments were obtained which is roughly 6.3% of the total number of initial fragments. This would be way less than the number of fragments that one has to either use in homology or de-novo modelling thereby reducing the design space. Further work is underway to set up the entire invariant fragment library which can then be used to predict protein structure by template-based approach.