Percentile Approximation Via Orthogonal Polynomials Hyung-Tae Ha Supervisor : Prof. Serge B. Provost

2 u Orthogonal Polynomial Approximants u Application to Statistics on (a, b) O U T L I N E u Introduction u Application to Statistics on (0, ) u Computation and Mathematica Codes u Conclusions

Introduction

4 Research Domain: Density Approximation Density Estimate Density Approximant X i 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)

5 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.

6 Literature Review Characteristic Paper Pearson CurveSaddlepoint Concept Daniels, H.E. (1954), “Saddlepoint Approximations in Statistics”, Annals of Mathematical Statistics 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 function using a few moments Approximating density function using cumulant generating function

7 Literature Review Characteristic Papers Cornish-Fisher ExpansionOrthogonal Series Expansion Concept Tiku (1965) Laguerre series forms 1. Expressible in terms of Hermite Polynomial 2. Gram-Charlier series 3. Edgeworth’s Expansion Cornish and Fisher (1938) Fisher and Cornish (1960) Hill and Davis (1968) Based on cumulants of a distribution Approximating the density function of noncentral Chi-squared and F random variables

Orthogonal Polynomial Approximants

9 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 positive.

Then, there exists a sequence of polynomials p 0 (x), p 1 (x),…, p n (x),… that is uniquely determined by the following conditions: 1) is a polynomial of degree n and the coefficient of x n in this polynomial is positive. 2) The polynomials p 0 (x), p 1 (x),…, p n (x), … are orthogonal w.r.t. the weight function w(x) if We say that the polynomials p n (x) constitute a system of orthogonal polynomials on the interval (a, b) with the weight function w(x) and orthogonal factor. If, p n (x) is called orthonormal polynomials.

11 Orthogonal Polynomial Approximation Approximant Base Density Orthogonal Polynomial Coefficients

12 Jacobi Polynomials Base Density Jacobi Polynomial

13 Jacobi Polynomial Approximant Transformation Approximant X (-1, 1) Y (a, b)

14 Jacobi Polynomial Approximant Distribution Approximant where

15 Laguerre Polynomials Base Density Laguerre Polynomial

16 Laguerre Polynomial Approximant Transformation Approximant Y X = Given the moments of Y

Application to Statistics on Compact Support ~

18 The L vc Test Statistic * Hypothesis : All the variances and covariances are equal. * Test Statistic : by Wilks (1946) * Moments : where

19 In the case of P=3, N=11 * 4 th degree Jacobi Polynomial Density Approximant * Wilks (1946) determined that 1 st and 5 th percentiles are and , respectively. F 4 [0.1682]= F 4 [0.2802]=

20 The V test statistic * Hypothesis : Equality of variances in independent normal populations * Test Statistic : * Moments :

21 p=5, N=12 * 4 th degree Jacobi Polynomial Density Approximant * Mathai (1979) determined that 1 st and 5 th percentiles are and , respectively. F 4 [ ]= F 4 [ ]=

Application to Statistics on the Positive Half Line ~

23 Test statistic for a single covariance matrix * Hypothesis : Covariance matrix of multivariate normal population is equal to a given matrix * Test Statistic : * MGF :

24 p=5 and N=10 * 4 th degree Laguerre Polynomial Density Approximant * Korin (1968) determined that 95 st and 99 th percentiles are and 38.60, respectively. F 4 [31.40]= F 4 [ ]=

25 Generalized Test of Homoscedasticity * Hypothesis : The constancy of variance and covariance in k sets of p-variate normal samples * Test Statistic : * MGF :

26 p=2, k=5 and N=45

Computation and Mathematica Codes

28 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.

29 Mathematica Code : Jacobi Polynomial Approximant

30 Mathematica Code : Laguerre Polynomial Approximant

Conclusion

32 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.