Pearson Curves Research Articles

On approximation of linear regression disturbance distribution

Inference on the parameters of a linear regression usually requires the assumption that the disturbances are normally distributed. But sometimes this assumption does not correspond to the residual distribution. Thus, there is a need to infer the disturbances of the regression model from its residuals. In this article, the analysis is mainly focused on estimation of moments of the disturbances based on residuals. The main goal of this article is derivation of the unbiased estimators of the third and forth central moments of the linear regression disturbances. They are applied to estimation parameters of the random component distribution based on empirical data. In addition to the normal distribution of the random component, the gamma distribution and the class of Pearson curves are considered. An example of skewness and kurtosis coefficient estimation is shown. An example of estimating the dominant of the gamma distribution is shown. Finally, different types of point and interval forecasts of the examined variable are presented.

A Sensitivity Analysis of a Certain Mathematical Model of Expert Evaluation of Investment Projects

The paper presents a sensitivity analysis of the developed mathematical model of expert evaluation of investment projects. Processes of investment projects implementations are analyzed as decision-making processes under uncertainty. The mathematical model under study evaluates the effectiveness of an investment project using the NPV index. This index is considered a random variable and can be estimated by an investor as a segment [NPV1; NPV2J. The proposed mathematical model utilizes the probability density function of NPV in the form of Pearson curves of the first type. Another peculiar feature of the mathematical model is utilization of the subjective utility function in decision making of whether to invest or not to invest in some project. The subjective utility function considers individual characteristics of investors and their decision making under risks and uncertainty. Perception of additional information by investors changes depending on their attitude toward risk and missed opportunities and thus reduces uncertainty in decision making. The paper studies this step in detail and provides some conclusions proving the correctness of the developed mathematical model.

Computational Methods in Problems of Subjective Assessment of Investment Decisions Effectiveness

The paper describes the implementation of investment projects under conditions of uncertainty. In the developed mathematical model, the effectiveness of an investment project is evaluated by the NPV index. This index is considered a random variable that can be estimated by an investor to within a segment [NPV1;NPV2]. The main difficulties of the decision-making process arise when the segment [NPV1;NPV2] includes zero value. In the developed model, we use the probability density function of NPV value in the form of Pearson curves of the first type. This paper discusses in detail some particular moments, which are to be taken into consideration while choosing a specific type of probability density function of NPV.
 The main element of the proposed model is the subjective utility function. Many questions regarding the usage of this function in practice are also extensively reviewed in this article. The main requirement for successful usage of the subjective utility function in real life is a well-calculated chapter of scenario analysis of an investment project. This chapter is present in almost all modern business plans. The practical application of the developed mathematical model improves the quality of decisions concerning the investment of funds into projects.

Michael Arthur Stephens, 1927–2019

Michael Arthur Stephens, 1927–2019

Computing tail areas for a high-dimensional Gaussian mixture

We consider the problem of computing tail probabilities - that is, probabilities of regions with low density - for high-dimensional Gaussian mixtures. We consider three approaches: the first is a bound based on the central and non-central ?2 distributions; the second uses Pearson curves with the first three moments of the criterion random variable U; the third embeds the distribution of U in an exponential family, and uses exponential tilting, which in turn suggests an importance sampling distribution. We illustrate each method with examples and assess their relative merits.

Mathematical modeling of the multimodal distributions of elastomer relaxation spectra using the family of Pearson curves

An approach to the mathematical modeling of elastomer relaxation spectra obtained via acoustic spectroscopy is presented. The solving of Pearson differential equations is the basis for the calculated dependences. The solutions to the equations describe the frequency and temperature distribution of the mechanical loss tangent. The form of the distribution is estimated from selected statistical moments of the experimental relaxation spectra and the mechanical loss tangent.

A New Approach to the Construction of the APF Algorithm by Applying the Pearson Curves Technique

A New Approach to the Construction of the APF Algorithm by Applying the Pearson Curves Technique

A method of statistical modeling of electromagnetic interference in automatics and telemechanics channels in railway transport

Data sets in the form of time variation of traction current are obtained as a result of an experimental investigation of interference in the channels of automatics and telemechanics systems in railways and subways. Visual assessment of the obtained records suggests the presence of transients in the traction current that are diverse in both amplitude and pulse width. In order to analyze and synthesize noise-resistant receivers of rail track circuits and locomotive devices of the automatic locomotive signaling, it is necessary to construct mathematical models that describe this class of interference. The experimentally obtained histogram should be approximated by an analytical expression that represents a theoretical probability density function that should meet two important conditions, i.e., nonnegativity and normalization. The approximating probability density is selected based on the condition of the conservation of the first four statistical distribution moments for the Pearson family. Twelve types of Pearson curves are described in this paper that depend on the values of individual parameters. Calculation formulas of unbiased estimates of the central moments are given in accordance with statistical data, based on which the appropriate type of the distribution of impulse noise from the traction current is selected. A generalized method of approximation using Pearson curves and a chi-squared test (χ2) for determining the divergence of theoretical and statistical distributions are formulated.

Influence of higher semiinvariants on the fitting parameters of implanted ions distribution

The influence of higher semiinvariants of impurity distribution on fitting parameters of the Pearson curves is investigated. An exact distribution and a package for its parameter calculations, suggested by the authors, are used. Calculations are made for implantation of the ions Li, C, Ar and Ge into diamond, graphite and fullerite C 60 over the energy range 0.1 keV to 1 GeV. It is found that (1) at the low energies, when nuclear scattering dominates, the higher semiinvariants underestimate the fitting skewness and excesses on 20–50% concerning their true values, if their modules are more than 0.1, (2) at the high energies, when electronic stopping dominates, the fitting skewness is always underestimated, and the fitting excesses are always overestimated in the same limits, (3) if module of the skewness or excesses are smaller than 0.1 and at the medial energies, when the nuclear and electronic energy losses are commensurable, the higher semiinvariants can change these fitting parameters several times with respect to true values.

Generalized pearson distributions and related characterization problems

This paper is concerned with a Rodrigues' formula for generating new classes of polynomials. The associated density functions belong to the family of generalized Pearson curves, which extends the classical Pearson family. Various properties of these polynomials (degree, orthogonality ...) are investigated and then exploited to derive several related results, especially characterizations, in probability.

Information processing in a gravitational-wave experiment subject to pulse noise

We present a new algorithm for processing information in a gravitational-wave experiment; the algorithm is based on using Pearson curves to approximate the unknown probability density of the output noise of a gravitational antenna subject to chaotic pulse noise.

Generating ion implantation profiles in one and two dimensions. 1: density functions

In this, the first of two papers, the problem of constructing ion implantation profiles in one and two dimensions from depth-independent spatial moments is discussed. Comparisons are made between Pearson and Johnson curves, constructed from moments produced by a transport equation solver, and profiles obtained directly from Monte Carlo simulations. A set of such comparisons, using consistent input quantities, is performed over a range of ion - target mass ratios and energies. For projected range distributions of the ions B, P and As into a-Si, a single Johnson type describes the implants over the energy range 1 keV to 1 MeV. The description using Pearson curves requires two types (I and VI). Also, taking the Monte Carlo data as a reference, the Johnson curves are equivalent, if not superior, to the Pearson curves in terms of fit accuracy. For lateral distributions of the same ion types over the same energy range it is shown that if the depth-dependent lateral kurtosis is less than 3.0, then the Pearson type II (bounded), Johnson type (bounded) and the modified Gaussian (unbounded) curves prove acceptable representations. If the depth-dependent lateral kurtosis is greater than 3.0 then the Pearson type VII (unbounded) and Johnson type (unbounded) curves are good representations.

Representation of two-dimensional ion implantation rest distributions by Pearson distribution curves for silicon technology

This is the second of two papers concerned with fitting Pearson curves to Monte Carlo simulations of implants into amorphous targets. In the first paper [ Solid-St. Electron. 35, 1151 (1992)] it was shown that accurate Pearson curve fitting to projected range profiles is possible when implant profiles are available for which optimised moments can be generated. In the present paper we extend the fitting to simulations of two-dimensional rest distributions. Comparisons are made between Pearson curve fits and the original high-resolution implant profiles, in two-dimensions, for the ions B and As implanted into amorphous silicon. The profiles were derived from Monte Carlo simulations, each of one million ion trajectories. Fit coefficients are provided that allow the regeneration of the moment surfaces for the depth and implantation energy dependent lateral straggle and lateral kurtosis for the ions B, P, As and Sb implanted, with energies in the range 25–300 keV, into targets of amorphous silicon, silicon dioxide and silicon nitride. The depth-dependent lateral distribution is then constructed using symmetrical Pearson curves driven by analytical formulae for the moment surfaces. The two-dimensional rest distribution is then reconstructed from the product of this depth-dependent lateral distribution and the projected range distribution derived in the first paper.

Bayesian Analysis of Two Overdispersed Poisson Models

In this paper, we consider the Bayesian analysis of two overdispersed Poisson models. The first is an overdispersed generalized Poisson model. The second is an ordinary Poisson and overdispersed generalized Poisson mixture model. Shoukri and Consul (1989, Comimintiicationis inz Statistics: Simnllationi anzd Comiiptutationi 18, 1465-1480) have previously considered a limited form of approximate Bayesian analysis for the first of these two models requiring the use of Pearson curves and the assumption that a certain model parameter has support on a finite number of values. By way of comparison, this paper demonstrates how a full Bayesian analysis of either model may proceed by making use of the Gibbs sampler and adaptive rejection sampling methods for log-concave densities. The methodology is illustrated with an application to a biological data set.

Structural reliability estimation using Monte Carlo simulation and Pearson's curves

At present Level 2 and importance sampling methods are the main tools used to estimate reliability of structural systems. But sometimes application of these techniques to realistic problems involves certain difficulties. In order to overcome the difficulties it is suggested to use Monte Carlo simulation in combination with two other techniques-extreme value and tail entropy approximations; an appropriate Pearson's curve is fit to represent simulation results. On the basis of this approach an algorithm and computer program for structural reliability estimation are developed. A number of specially chosen numerical examples are considered with the aim of checking the accuracy of the approach and comparing it with the Level 2 and importance sampling methods. The field of application of the approach is revealed.

The Sampling Distribution of the Serial Correlation Coefficient

SYNOPTIC ABSTRACTA methodology for the derivation of the cumulants of the lag- k serial covariance is proposed for the case of a Gaussian white-noise process. Explicit representations of the first five moments of the serial correlation coefficient are then given. Alternate representations which are expressed in terms of trigonometric functions are also derived. The first four moments are used to approximate the density of the serial correlation by means of Pearson curves. A computable representation of the exact distribution function is also given. The exact, the approximate and the simulated distributions are compared in a numerical example.

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the n X s random variable Z = N/D where N = }-~.1 rn~ i + }-~.n+l rniX~, D = 7" ~-~.~+1 liX~ + ~-~-s+l niXi, and the Xi's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution of Z by means of Pearson curves. The exact density function of Z is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of the h-th moment of the ratio N/D where h is a complex number. The case n = 1, s = 2 and r = 3 is discussed in detail and a general technique which applies to any ratio having the structure of Z is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quadratic forms is evaluated at various points and then compared with simulated values.

Representation of ion implantation projected range profiles by Pearson distribution curves for silicon technology

Several authors have reported difficulties in fitting Pearson curves to implants into amorphous targets. However, we show that accurate Pearson curve fitting to simulations of projected range profiles in amorphous targets is possible when implant profiles are available for which optimised moments can be generated. In previous literature few comparisons can be found between Pearson curve fits and the original implant profiles whereas, in the current paper, comparisons are made with high-resolution simulated profiles for B, P and As implanted into amorphous Si. These profiles were derived using Monte-Carlo simulations each of 1,000,000 ion trajectories. Fit coefficients that allow the regeneration of Pearson curves, optimised to fit Monte-Carlo implant profiles, are provided for the ions, B, P, As and Sb implanted, with energies in the range 25–300 keV, into targets of amorphous Si, silicon dioxide and silicon nitride. Comparisons of infinite and optimised moments, for B and P implants, show that the best fit moments (and corresponding Pearson type) can only be determined as part of an optimisation process, and not from infinite moments directly (such as moments from a backward Boltzmann transport equation solver).

Representation of ion implantation distributions in two and three dimensions

The representation of two- and three-dimensional point response functions by Pearson frequency distribution curves is discussed for the case of energetic ions implanted into homogeneous, amorphous material. Comparisons are made between the Pearson curves and high-resolution Monte Carlo simulations. It is shown that no extra information is required to derive the three-dimensional representation once the two-dimensional representation is known.

Approximation of structureless bands in the electron spectra of molecules using Pearson curves

Approximation of structureless bands in the electron spectra of molecules using Pearson curves