Percentile ApproximationVia

Orthogonal Polynomials

Hyung-Tae Ha

Supervisor : Prof. Serge B. Provost

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e-business 적용 능력 및 사례 Orthogonal Polynomial ApproximantsOrthogonal Polynomial Approximants

e-business 적용 능력 및 사례 Application to Statistics on (a, b)Application to Statistics on (a, b)

O U T L I N E

회사의 성과 및 업적 분석자료 IntroductionIntroduction

e-business 적용 능력 및 사례 Application to Statistics on (0, )Application to Statistics on (0, )

e-business 적용 능력 및 사례 Computation and Mathematica CodesComputation and Mathematica Codes

e-business 적용 능력 및 사례 ConclusionsConclusions

Introduction

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Research Domain: Density Approximation

Density Estimate

Density Approximant

Xi

Sample

Theoretical Moments

Focus : Continuous distributions

Unknown

* We are considering the problem of approximating a density function from the theoretical moments (or cumulants) of a given distribution (for example, that of the sphericity test statistic)

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Moment Problem 1. While it is usually possible to determine the moments of various random quantities used in statistical inference, their exact density functions are often times analytically intractable or difficult to obtain in closed forms.

2. Suppose a density function admits moments of all orders. A given moment sequence doesn’t define a density function uniquely in general. But it does when the random variable is on compact support.

3. The sufficient condition for uniqueness is that

4. The moments can be obtained from the derivatives of its moment generating function (MGF) or by making use of the recursive relationship to express moments in terms of cumulants.

is absolutely convergent for some t > 0.

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Literature Review

Characteristic

Paper

Pearson Curve Saddlepoint

Concept

Daniels, H.E. (1954), “Saddlepoint Approximations in Statistics”, Annals of MathematicalStatistics

• Adequate Approximation• A Variety of Applications• Unimodal• Difficult to implement• Tail Approximation is good.

A Variety of Applications• Unimodal• Using up to 4 moments

Solomon and Stephens (1978), “Approximations to density functions using Pearson curves”, JASA

Approximating density functionusing a few moments

Approximating density functionusing cumulant generating function

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Literature Review

Characteristic

Papers

Cornish-Fisher ExpansionOrthogonal Series Expansion

Concept

Tiku (1965)

• Laguerre series forms

1. Expressible in terms of Hermite Polynomial2. Gram-Charlier series3. Edgeworth’s Expansion

Cornish and Fisher (1938)Fisher and Cornish (1960)Hill and Davis (1968)

Based on cumulants of a distribution

Approximating the density functionof noncentral Chi-squared and F random variables

Orthogonal Polynomial

Approximants

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Brief Review of Orthogonal Polynomials

Suppose that w(x) is a nonnegative real function of a real variable x. Let (a, b) be a fixed interval on the x-axis and suppose further that, for n=0,1,…, the integral

exists and that the integral

is

positi

ve.

Then, there exists a sequence of polynomials p0(x), p1(x),…, pn(x),… that is uniquely determined by the following conditions:

1) is a polynomial of degree n and the coefficient of xn in this polynomial is positive.

2) The polynomials p0(x), p1(x),…, pn(x), … are orthogonal w.r.t. the weight function w(x) if

We say that the polynomials pn (x) constitute a system of orthogonal polynomials

on the interval (a, b) with the weight function w(x) and orthogonal factor .

If , pn (x) is called orthonormal polynomials.

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Orthogonal Polynomial Approximation

Approximant

Base Density

Orthogonal Polynomial

Coefficients

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Jacobi Polynomials

Base Density

Jacobi Polynomial

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Jacobi Polynomial Approximant

Transformation

Approximant

X(-1, 1)

Y(a, b)

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Jacobi Polynomial Approximant

DistributionApproximant

where

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Laguerre Polynomials

Base Density

LaguerrePolynomial

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Laguerre Polynomial Approximant

Transformation

Approximant

Y X = Given the moments of Y

Application to Statistics on Compact

Support~

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The Lvc Test Statistic

* Hypothesis : All the variances and covariances are equal.

* Test Statistic : by Wilks (1946)

* Moments :

where

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In the case of P=3, N=11

* 4th degree Jacobi Polynomial Density Approximant

* Wilks (1946) determined that 1st and 5th percentiles are 0.1682 and 0.2802, respectively.

F4 [0.1682]=0.0100071

F4 [0.2802]=0.050019

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The V test statistic

* Hypothesis : Equality of variances in independent normal populations

* Test Statistic :

* Moments :

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p=5, N=12

* 4th degree Jacobi Polynomial Density Approximant

* Mathai (1979) determined that 1st and 5th percentiles are 0.27336 and 0.38595, respectively. F4 [0.27336]=0.00999801

F4 [0.38595]=0.0049923

Application to Statistics on the Positive Half Line

~

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Test statistic for a single covariance matrix

* Hypothesis : Covariance matrix of multivariate normal population is equal to a given matrix

* Test Statistic :

* MGF :

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p=5 and N=10

* 4th degree Laguerre Polynomial Density Approximant

* Korin (1968) determined that 95st and 99th percentiles are 31.40 and 38.60, respectively. F4 [31.40]=0.950368

F4 [0.38595]=0.990075

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Generalized Test of Homoscedasticity

* Hypothesis : The constancy of variance and covariance in k sets of p-variate normal samples

* Test Statistic :

* MGF :

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p=2, k=5 and N=45

Computation and Mathematica Codes

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Computational consideration

1. The symbolic computational package Mathematic was used for evaluating the approximants and plotting the graphs.

2. The code is short and simple.

3. The formula will be easier to program when orthogonal polynomials are built-in functions in the computing packages.

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Mathematica Code : Jacobi Polynomial Approximant

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Mathematica Code : Laguerre Polynomial Approximant

Conclusion

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Concluding Remarks

1. The proposed density approximation methodology yields remarkably accurate percentage points while being relatively easy to program.

2. The proposed approximants can also accommodate a large number of moments, if need be.

3. For a vast array of statistics that are not widely utilized, statistical tables, when at all available or accessible, are likely to be incomplete; the proposed methodology could then prove particularly helpful in evaluating certain p-values.

4. When a table is needed for a specific combination of parameters, the proposed methodology could readily generate it.