Johnson System Of Distributions Research Articles

Dynamic prediction of over-excavation gap due to posture adjustment of shield machine in soft soil

The probability analysis of ground deformation is becoming a trend to estimate and control the risk brought by shield tunnelling. The gap parameter is regarded as an effective tool to estimate the ground loss of tunnelling in soft soil. More specifically, ω, which is a gap parameter component defined as the over (or insufficient) excavation due to the change in the posture of the shield machine, may contribute more to the uncertainty of the ground loss. However, the existing uncertainty characterization methods for ω have several limitations and cannot explain the uncertain correlations between the relevant parameters. Along these lines, to better characterize the uncertainty of ω, the multivariate probability distribution was developed in this work and a dynamic prediction was proposed for it. To attain this goal, 1 523 rings of the field data coming from the shield tunnel between Longqing Road and Baiyun Road in Kunming Metro Line 5 were utilized and 44 parameters including the construction, stratigraphic, and posture parameters were collected to form the database. According to the variance filter method, the mutual information method, and the value of the correlation coefficients, the original 44 parameters were reduced to 10 main parameters, which were unit weight, the stoke of the jacks (A, B, C, and D groups), the pressure of the pushing jacks (A, C groups), the chamber pressure, the rotation speed, and the total force. The multivariate probability distribution was constructed based on the Johnson system of distributions. Moreover, the distribution was satisfactorily verified in explaining the pairwise correlation between ω and other parameters through 2 million simulation cases. At last, the distribution was used as a prior distribution to update the marginal distribution of ω with any group of the relevant parameters known. The performance of the dynamic prediction was further validated by the field data of 3 shield tunnel cases.

Robust methods for controlling casting processes and the quality of castings

The paper considers the ways of controlling the processes of producing good-quality castings by using robust methods based on the identification of objects under uncertainty, adaptive control methods, stabilization methods for automatic control systems and their mathematical description. The purpose of the paper is to practically elaborate on the task of creating robust technologies for casting based on the study of the stochastic dispersion models of the casting process parameters and to develop a method for stabilizing the casting processes and a method for stabilizing the quality parameters of the castings. The regular patterns of the stochastic dispersion of the casting process parameters were investigated based on the mechanical performance of the casting alloys, their chemical composition, the physical and mechanical properties of the molding mixtures, the dimensions of the castings, and the features of the casting production process such as pouring temperature, element loss during melting and mold hardness. The results of the conducted investigations have shown that to describe the stochastic dispersion of the casting process parameters, it is appropriate to use the Johnson system of distributions. A method for stabilizing the casting quality parameters was developed. It has been shown that it is more efficient to stabilize the strength and improve the dimensional accuracy of the casting simultaneously. With the existing production process, the dimensional accuracy of the casting corresponds to Grade 11 according to GOST 26645-85. With the increase of the dimensional accuracy grade, the metal content of the casting can be reduced by 16.3% after reaching Grade 10, by 20.4% after reaching Grade 9, and by 24.2% after reaching Grade 8.

How accurate are modern Value-at-Risk estimators derived from extreme value theory?

In this study, we compare the out-of-sample forecasting performance of several modern Value-at-Risk (VaR) estimators derived from extreme value theory (EVT). Specifically, in a multi-asset study covering 30 years of stock, bond, commodity and currency market data, we analyse the accuracy of the classic generalised Pareto peak over threshold approach and three recently proposed methods based on the Box–Cox transformation, L-moment estimation and the Johnson system of distributions. We find that, in their unconditional form, some of the estimators may be acceptable under current regulatory assessment rules but none of them can continuously pass more advanced tests of forecasting accuracy. In their conditional forms, forecasting power is significantly increased and the Box–Cox method proves to be the most promising estimator. However, it is also important to stress that the traditional historical simulation approach, which is currently the most frequently used VaR estimator in commercial banks, can not only keep up with the EVT-based methods but occasionally even outperforms them (depending on the setting: unconditional versus conditional). Thus, recent claims to generally replace this simple method by theoretically more advanced EVT-based methods may be premature.

Moment Problem and Its Applications to Risk Assessment

ABSTRACTThis article discusses how to assess risk by computing the best upper and lower bounds on the expected value E[φ(X)], subject to the constraints E[Xi] = μi for i = 0, 1, 2, …, n. φ(x) can take the form of the indicator function in which the bounds on are calculated and the form φ(x) = (ϕ(x) − K)+ in which the bounds on financial payments are found. We solve the moment bounds on through three methods: the semidefinite programming method, the moment-matching method, and the linear approximation method. We show that for practical purposes, these methods provide numerically equivalent results. We explore the accuracy of bounds in terms of the number of moments considered. We investigate the usefulness of the moment method by comparing the moment bounds with the “point” estimate provided by the Johnson system of distributions. In addition, we propose a simpler formulation for the unimodal bounds on compared to the existing formulations in the literature. For those problems that could be solved both analytically and numerically given the first few moments, our comparisons between the numerical and analytical results call attention to the potential differences between these two methodologies. Our analysis indicates the numerical bounds could deviate from their corresponding analytical counterparts. The accuracy of numerical bounds is sensitive to the volatility of X. The more volatile the random variable X is, the looser the numerical bounds are, compared to their closed-form solutions.

A practical method to calculate probabilities: illustrative example from the electronic industry business

ABSTRACTThe real-life environment is made of probabilistic data by nature and the ability to make decisions based on probabilities is crucial in the business world. It is common to have a set of data and the need of calculating the probability of taking a value greater or less than a specific value. It is also common in many companies the unavailability of a statistical software or a specialized professional in statistics. The purpose of this paper is to present a practical and simple method to calculate probabilities from normal or non-normal distributed data set and illustrate it with an application from the electronic industry. The method does not demand statistical knowledge from the user; there is no need of normality assumptions, goodness test or transformations. The proposed method is easy to implement, robust and the experiments have evidenced its quality. The technique is validated with a large variety of instances and compared with the well-known Johnson system of distributions.

Empirical distributions of daily equity index returns: A comparison

The normality assumption concerning the distribution of equity returns has long been challenged both empirically and theoretically. Alternative distributions have been proposed to better capture the characteristics of equity return data. This paper investigates the ability of five alternative distributions to represent the behavior of daily equity index returns over the period 1979–2014: the skewed Student-t distribution, the generalized lambda distribution, the Johnson system of distributions, the normal inverse Gaussian distribution, and the g-and-h distribution. We find that the generalized lambda distribution is a prominent alternative for modeling the behavior of daily equity index returns.

Empirical Distributions of Stock Returns: A Comparison

The normality assumption concerning the distribution of equity returns has long been challenged both empirically and theoretically. Alternative distributions have been proposed to better capture the characteristics of equity return data. This paper investigates the ability of five alternative distributions to represent the behavior of daily equity index returns over the period 1979–2014: the skewed Student-t distribution, the generalized lambda distribution, Johnson system of distributions, the normal inverse Gaussian distribution, and the g-and-h distribution. We find that the generalized lambda distribution is a prominent alternative for modeling the behavior of daily equity index returns.

Modeling of Human Exposure to Benzene in Urban Environments

Urban areas characterized by high spatial and temporal variability in air pollution levels require implementation of comprehensive approaches to address exposure of individuals. The main objective of this study was to implement a quantitative assessment of individual exposure to benzene in urban environments. For this purpose, ExPOSITION model based on a global positioning system (GPS) tracking approach was applied to estimate individual exposure in different microenvironments. The current investigation provides an application example and validation of the modeling approach against personal and biological exposure measurements collected during the measurements campaign. The probabilistic approach using the Johnson system of distributions was implemented to characterize variability of indoor concentrations. The results obtained for daily average individual exposure to benzene corresponded to mean levels of 1.6 and 0.8–2.7 μg/m3 in terms of 5th–95th percentiles. Validation of the model results against several personal exposure samples collected for the selected individuals revealed a Pearson’s correlation coefficient of .66. This modeling approach explicitly addressed the temporal and spatial variability in the exposure and established a source–receptor relationship.

Johnson binomial trees

Rubinstein developed a binomial lattice technique for pricing European and American derivatives in the context of skewed and leptokurtic asset return distributions. A drawback of this approach is the limited set of skewness and kurtosis pairs for which valid stock return distributions are possible. A solution to this problem is proposed here by extending Rubinstein's Edgeworth tree idea to the case of the Johnson system of distributions which is capable of accommodating all possible skewness and kurtosis pairs. Numerical examples showing the performance of the Johnson tree to approximate the prices of European and American options in Merton's jump diffusion framework and Duan's GARCH framework are examined.

Computational Issues in Fitting Statistical Distributions to Data

SYNOPTIC ABSTRACTIn fitting statistical distributions to data the practitioner almost always needs to compute the parameters of a family of distributions. Such computations are frequently difficult to implement and may require considerable effort and computation time. This paper discusses the computing difficulties that can arise in fitting statistical distributions to data and illustrates these problems through examples that use the Weibull distribution, the generalized lambda distribution, and the Johnson system of distributions. In all cases fits are obtained by the method of moments using either the Maple computing environment or the R statistical system.

An asymptotically optimal detector for Gaussianity testing

The aim of this paper is to provide a novel approach to the classical problem of testing the Gaussianity of a stationary process where the data can be assumed to be independent and identically distributed. Some results of Large Deviations theory are used to select a set of asymptotically optimal test statistics so that their combination results in a suitable detection procedure. The Johnson system of distributions has been used as the basis for the design of the detector. The fact that the good asymptotical properties can be extended to finite samples with small sizes provides the motivation for the rest of the work. In this respect, the performance of the proposed procedure is also compared with that of some known Gaussianity tests. The possibility to directly calculate a reliable measure of support such as the Bayesian posterior probabilities, as opposed to P-values, is an additional advantage. Furthermore, it will be shown how the proposed procedure can provide some extra information concerning the type of deviation from Gaussianity, which is present in the data.

An Experimental Comparison of Four Methods for Assessing Judgmental Distributions

For a number of statistical applications, experts' subjective estimates are needed for distributional parameters, like mean and variance; sometimes even the whole density is wanted. The literature agrees that it is wise behaviour to ask the expert only quantile estimates; from these, the desired parameter or density is derived. Many methods, using different sets of quantiles, have been suggested. A new method is proposed here. Experts are asked the relatively simple task of estimating the seven values that divide the total probability mass into eight equal parts. These seven quantiles uniquely determine one distribution within both the Pearson and the Johnson system of distributions. Comparisons between different methods were, in the past, generally based on theoretical arguments. A more practice-oriented approach is followed here: together with three often-used methods, our new method is tested experimentally: 56 students with some expert knowledge estimated quantiles of four distributions. The accuracy of the four methods in estimating both the quantiles and the distributional parameters is calculated and discussed.

MODELING FINANCIAL SERIES DISTRIBUTIONS: A VERSATILE DATA FITTING APPROACH

The empirical distribution of common stock returns is a subject of interest to many researchers, as it often determines the validity of theoretical models proposed in the economics and finance studies. This paper brings to the attention the availability of two flexible systems of distributions for fitting data: the Johnson system of distributions and the Tadikamalla–Johnson system of distributions. We explore the feasibility of fitting the empirical distributions of several financial series to these two systems of distributions. Both systems of distributions are highly flexible and capable of accommodating all possible skewness and kurtosis values. The probability density function and the cumulative distribution function take on simple closed forms and appropriate transformations of the data lead to normal/logistic distributions. In addition, the parameter estimation procedures are easy to implement. When the results are compared with those of other data fitting models, in all cases tested, the proposed distributions provide a good fit to the empirical distribution of data.

Transforming Non-Normal Data to Normality in Statistical Process Control

Quality characteristics analyzed in statistical process control (SPC) often are required to be normally distributed. This is true in many types of control charts and acceptance sampling plans, as well as in process capability studies. If a characteristic is not normally distributed, but normal-based techniques are used, serious errors can result. One approach to solving this problem is to transform the non-normal data to normality using the Johnson system of distributions. In this paper, we use the sample quantile ratio, in conjunction with the Shapiro-Wilk test of normality, to find a suitable transformation for non-normal data. Examples of fitting non-normal SPC data are presented and discussed. The effect of the Johnson transformation on an SPC procedure involving an estimator for the population standard deviation is studied using non-normal data and Johnson-transformed data.

Comparison and evaluation of five methods of estimation of the Johnson system parameters

The Johnson system of distributions was used to fit both diameter and height data collected from 226 sample plots located in the ponderosa pine (Pinusponderosa P. Laws, ex C. Laws.) and mixed-conifer forest types in New Mexico and Arizona. To estimate the Johnson system parameters, five methods (namely the four-percentile method, Knoebel–Burkhart method, mode method, maximum likelihood estimation method, and a new method developed in the study, linear regression method) were compared and evaluated. For these sample plots, the linear regression method was superior for estimating parameters of SB distributions for both diameter and height.

On weighting the studentized empirical characteristic function for testing normality

The aim in this paper is to propose a test of the composite hypothesis of ρ-variate normality. The test is based on the weighted integral of the squared modulus of the studentized empirical characteristic function. In the present weighting method, it is easy to investigate the asymptotic performance of the proposed test through its asymptotic normality. The performances of the proposed test through Monte Carlo experiments are also discussed. Especially the estimated power of the test against the Johnson system of distributions are illustrated by plots of isodynes.

On using percentiles to fit data by a johnson distribution

The well-known Johnson system of distributions was developed by N. L. Johnson (1949). Slifker and Shapiro (1980) presented a criterion for choosing a member from the three distributional classes (SB,SL, and Sv) in the Johnson system to fit a set of data. The criterion is based on the value of a quantile ratio which depends on a specified positive z value and the parameters of the distribution. In this paper, we present some properties of the quantile ratio for various distributions and for some selected z values. Some comments are made on using the criterion for selecting a Johnson distribution to fit empirical data.

The modelling and simulation of a rough surface

In this paper, new simulation methods of the actual surface are presented. A simulated surface which has similar statistical characteristics to those of the actual surface can be obtained through measurement of several profiles along different directions. The method is as follows. 1. (1) The rough surface is classified and described by the sample autocorrelation function (ACF) and the first four moments of the probability density functions of profiles along different directions. 2. (2) The optimum separable approximation of a given ACF matrix in the least-squares sense is found through singular value decomposition, so that the two-dimensional autoregressive modelling problem can be reduced to a onedimensional problem. 3. (3) Two simple two-dimensional autoregressive models, with the assumption that the given ACF matrix is separable or of exponential form, are used in surface modelling and simulation. 4. (4) The non-gaussian random surface with the desired skew and kurtosis is obtained by transforming the gaussian surface with desired ACFs using the Johnson system of distributions. The results of modelling and simulation of the actual engineering surfaces showed that the simulated surfaces agree well with the actual surfaces in the statistical sense.

Recent and classical tests for normality - a comparative study

We give a critical synopsis of classical and recent tests for univariate normality, our emphasis being on procedures which are consistent against all alternatives. The power performance of some selected tests (Anderson-Darling, Shapiro-Wilk, Shapiro-Francia, Epps-Pulley) is assessed in a simulation study. Numerical results are illuminated by plots of isodynes, i.e., lines of constant estimated power, for the Johnson-system of distributions.