Pearson Curves Research Articles

Representation of ion implantation profiles by Pearson frequency distribution curves

A method is described whereby ion implantation profiles in amorphous materials may be accurately represented by Pearson frequency distribution curves. The type of Pearson distribution to be used depends upon the implantation conditions but it is shown that the three main types, I, IV and VI, together with the transition types II, III, V, VII and the Gaussian are all suitable representations. Previous attempts to fit Pearson curves to implantation profiles have often failed due to semi-infinite moments being used instead of infinite moments. This approximation is only valid when the surface concentration of implanted ions is negligibly small.

The normal approximation to the Bartlett-Nanda-Pillai trace test in multivariate analysis

SUMMARY The distribution of the trace criterion in multivariate analysis of variance can be approximated by a normal distribution with correct first two moments. Results show that the usual significance points agree very closely with values earlier tabulated which are based on fitting Pearson curves to the correct first four moments.

Exact-and approximate bayes credibility intervals for the one-way analysis of variance model

In most hierarchical Bayes cases the posterior distributions are difficult to derive and cannot be obtained in closed form. In some special cases, however, it is possible to obtain the exact moments of the posterior distributions. By applying these moments and Pearson curves or Cornish-Fisher expansions to real problems, good approximations of the exact posterior distributions of individual parameter values as well as linear combinations of parameter values could easily be obtained.

Comparison Of Some Mrpp And Standard Rank Tests For Three Equal Sized Samples

MRPP rank tests for three equal sized samples are studied. Based on the Pearson criteria, appropriate Pearson curves are suggested for their approximate distributions. Empirical powers of these tests are compared among themselves and with some comparable standard nonparametric rank tests for several underlying populations.

Comparison of Some Mrpp and Standard Rank Tests for Two Unequal Samples

MRPP rank tests for two subgroups of unequal size are studied in detail. Based on the Pearson criteria, appropriate Pearson curves are suggested for their approximate distributions. On the basis ot extensive simulations, empirical powers of these tests are compared among themselves and with some standard nonparametric tests for several underlying populations.

Evaluation of Pearson Curves as an Approximation of the Maximum Probable Annual Aggregate Loss

Evaluation of Pearson Curves as an Approximation of the Maximum Probable Annual Aggregate Loss

A Credibility Interval for the Probability that a New Observation Belongs to One of Two Multivariate Normal Populations

Summary The moments of the logarithm of the likelihood ratio of a new observation, drawn from one of two multivariate normal populations, are obtained from the posterior moment generating function. The distribution of the logarithm of the likelihood ratio is approximated using Pearson curves. The distribution of the probability p that a new observation belongs to one of the two populations can be obtained by transformation. The relationship between the expected value of this distribution and the “predictive” estimate of p is shown.

Equation of the ferromagnetic resonance line in polycrystalline ferrites

Ferromagnetic resonance in polycrystalline ferrites is considered in the framework of the independent-grain model. It is shown that the ferromagnetic resonance absorption line can be obtained analytically from its moments. When a limited number of moments of the line are known explicitly, the method of Pearson curves is used to obtain the equation of the line. The obtained expression is analyzed and the resonance field and absorption line width deduced from the equation are compared with results obtained earlier.

Further Percentage Points for Greenwood's Statistic

Abstract : Percentage points for Greenwood's statistic obtained by fitting Pearson curves to the first four moments are given. Comparisons are given with the exact points for n=10, recently given by Burrows (1979) and these suggest that the approximate points will be accurate for practical purposes. (Author)

Approximations to Density Functions Using Pearson Curves

Abstract This article is an expository paper to demonstrate the usefulness of Pearson curves in density estimation especially for those unaware of this early development in statistics. It is shown how to fit the curves and how very good approximate percentage points can be obtained for intractable distributions when the first four moments (or three moments and one endpoint) are known exactly (not estimated from sample data). The effectiveness of this method in density estimation is illustrated in three somewhat disparate contexts and reference is given to others. In general, the Pearson curves give an excellent approximation to the long tail of a distribution, the tail most often needed in practical work.

Approximations to Density Functions Using Pearson Curves

Abstract This article is an expository paper to demonstrate the usefulness of Pearson curves in density estimation especially for those unaware of this early development in statistics. It is shown how to fit the curves and how very good approximate percentage points can be obtained for intractable distributions when the first four moments (or three moments and one endpoint) are known exactly (not estimated from sample data). The effectiveness of this method in density estimation is illustrated in three somewhat disparate contexts and reference is given to others. In general, the Pearson curves give an excellent approximation to the long tail of a distribution, the tail most often needed in practical work.

Mantel's space-time clustering statistic: computing higher moments and a comparison of various data transforms

Mantel proposed a statistic for assessing space-time clustering. He showed how to compute its theoretical expectation and variance and pointed out its asymptotic normality, noting that a simulation approach might be used in insufficiently asymptotic situations. Another approach to this distribution problem is presented by which graduation curves (e.g. Pearson curves) are fitted to the theoretical distribution. This requires the computation of the exact first four moments and the formulae are presented herein. In terms of computer time, this compares favourably with a simulation approach. The analysis of two sets of data confirmed that normality of the test statistic cannot be taken for granted and that a variety of inverse transformations of the time and space data and Knox's method were equally robust.

A test for regression function adequacy

A generalization of the mean-square-successive-difference statistic is used to detect lack of fit in the multiple linear regression model. The distribution of the test statistic is approximated with Pearson Curves. A numerical example is used to illustrate the procedure and compare the test with the usual F test and a nonparametric test.

Generalized pearson distributions and nonlinear programing

A generalization of the Pearson curves is obtained as the solution to the differential equation which best fits a histogram in the mean square and satisfies certain statistical constraints, e.g., the mean and variance may be prescribed. Φ is the theoretical distribution defined on the intervel is a rational function with numerator and denomenator orders of mand n, respevtively. The values of the coefficients in are obtained from a Powell minimization of the mean-square value plus sum-of-squares constraints. Excellent fits are obtained effciently which, furthermore, are capable of providing an analytical representation of an infinite tail. An accurate initial approximation of for starting the Powell minimization is obtained from a symbol manipulation code in PL/1—Formac and based on the visual decomposition of a histogram into sums of normal curves.

Corrigenda: Table of Percentage Points of Pearson Curves, for given | surd | beta 1 and | beta 2 , Expressed in Standard Measure

Corrigenda: Table of Percentage Points of Pearson Curves, for given | surd | beta 1 and | beta 2 , Expressed in Standard Measure

Table of Percentage Points of Pearson Curves, for Given √β 1 and β 2 , Expressed in Standard Measure

Table of Percentage Points of Pearson Curves, for Given √β 1 and β 2 , Expressed in Standard Measure

Table of percentage points of Pearson curves, for given √β1 and β2, expressed in standard measure

Table of percentage points of Pearson curves, for given √β₁ and β₂, expressed in standard measure

Tables of the 5% and 0.5% points of Pearson curves (with argument beta1 and beta2) expressed in standard measure.

TABLES OF THE 5% AND 0.5% POINTS OF PEARSON CURVES (WITH ARGUMENT β1 AND β2) EXPRESSED IN STANDARD MEASURE Get access E. S. PEARSON, E. S. PEARSON Search for other works by this author on: Oxford Academic Google Scholar MAXINE MERRINGTON MAXINE MERRINGTON Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 38, Issue 1-2, June 1951, Pages 4–10, https://doi.org/10.1093/biomet/38.1-2.4 Published: 01 June 1951

On Multinomial Distributions with Limited Freedom: A Stochastic Genesis of Pareto's and Pearson's Curves

On Multinomial Distributions with Limited Freedom: A Stochastic Genesis of Pareto's and Pearson's Curves

Molecular Weight and Polydispersivity of Rubber by Diffusion Measurements

Abstract The study of diffusion in solutions of natural rubber (light crepe) by Lamm's method showed that even with a concentration of 0.1 per cent the normalized experimental diffusion curves diverge from the ideal Gaussian curve (Figure 1), in that they are characterized by a marked asymmetry and an excess of the maximal ordinate. It follows from an analysis of the experimental curves by the method of moments (up to moments of the fourth order) that they belong to Type IV Pearson curves, that is, to asymmetrical distribution curves with asymptotic branches. The determination of the perturbation multiplier enables us to calculate the course of the experimental curves with a fair degree of accuracy. The physical cause of asymmetry of the diffusion curves is the difference in the rate of diffusion to both sides of the interface (of the polymer into the solvent and back) due to a marked intermolecular interaction in the solution of the polymer at a given concentration. With a decrease of the concentration or of the molecular weight of the dissolved substance, the asymmetry of the diffusion curves becomes less pronounced. However, this asymmetry does not preclude the computation of the average diffusion coefficient D from the standard deviation of the curve. It can, indeed, be shown that the probable error does not exceed 1 per cent. The average value found for natural rubber in carbon tetrachloride is D20°=0.71×10−7 sq. cm. per sec.