Approximation of the Test-Statistic PDF by Orthogonal Polynomial Expansion

The exact analytical expression (12) of the moments can be also used for the determination of the PDF of the test-statistic $V$. The PDF $W_V(x)$ of the random variable $V$ can be represented as a series from a full system of orthogonal polynomials $Q_n(x)$

$$W_V (x)=\sum\limits_{n=0}^{\infty}c_n f(x)Q_n(x) ,$$ (13)

where $c_n$ are the coefficients of this expansion and $f(x)$ is a given weight function which determines the set of orthogonal polynomials $Q_n(x)$.

The basic problem that arises by using the expansion (13) consists in the choice of the weight function $f(x)$ for which this series has the most rapid convergence rate [5]. For this purpose as weight function $f(x)$, the beta density function was chosen

$$f(x)=\frac{\Gamma(p'+q')}{\Gamma(p')\Gamma(q')}x^{p'-1}(1-x)^{q'-1},$$ (14)

$$0 < x < 1, \quad p' > 0, \quad q' > 0$$ (15)

The investigation carried out has shown that the function (14) is a very successful zero approximation of the analytically unknown PDF of the test-statistic $V$. For this weight function, the set of orthogonal polynomials $Q_n(x)$ is a system of biased orthogonal Jacobi polynomials defined on the interval [0,1]. These polynomials are written as follows

$$\begin{array}{l} Q_n(x)=\frac{\Gamma(q'+n)}{\Gamma(p'+2n)} \times... ...frac{n!}{k!(n-k)!}\frac{\Gamma(p'+2n-k)}{\Gamma(q'+n-k)} x^{(n-k)}, \end{array}$$ (16)

From the condition of equalities for the first two moments of the beta probability distribution (14) and the test-statistic $V$ (12) it is possible to evaluate two parameters $p'$ and $q'$ which do not depend upon the beta density function $f(x)$. Therefore, the moments of the beta distribution are equal to

$$M[\beta^k]=\frac{\Gamma(p'+q')\Gamma(p'+k)}{\Gamma(p')\Gamma(p'+q'+k)},$$ (17)

From (12) and (17) the system of two equations for the unknown parameters $p'$ and $q'$ can be obtained

$$\begin{array}{l} \frac{\Gamma(p'+q')\Gamma(p'+1)}{\Gamma(p')\Gamm... ... \frac{\Gamma(p'+q')\Gamma(p'+2)}{\Gamma(p')\Gamma(p'+q'+2)}=<V^2>, \end{array}$$ (18)

that can be simplified to

$$\begin{array}{l} \frac{p'}{p'+q'}=<V> \\ \frac{p'(p'+1)}{(p'+q')\Gamma(p'+q'+1)}=<V^2>, \end {array}$$ (19)

Because the parameters $p'$ and $q'$ are positive and the values of all moments $M[V^h]$ are in the interval [0,1] it can be easily shown that for any values of the first two moments of the random variable $V$ and for any $p$ and $N$ there is unique solution of the system (19)

$$\begin{array}{l} p'=\frac{<V>(<V>-<V^2>)}{<V^2>-<V>^2} \\ q'=\frac{(<V>-<V^2>)(<V>-1)}{<V>^2-<V^2>}, \end {array}$$ (20)

Substituting (20) into expression (14), we can obtain the zero approximation for the PDF $W_V(x)$ (the first term in the series expansion (13)). It can be derived that the following two coefficients in the expansion (13) are identically equal to zero ( $c_1 = 0, c_2 = 0$) for any $N$ and $p$.

Higher order coefficients $c_n$ ($n \ge 3$) in the expansion can be expressed as function of the known test-statistic $V$ moments (of order $k < n$) by a standard procedure which involves the condition of mutual orthogonality of the polynomials $Q_n(x)$. The common expression for $c_n$ is given by

$$\begin{array}{l} c_n = \frac{(2n+p)\Gamma^2(2n+p)}{n!\Gamma(n+q)\... ...n}(-1)^k \frac{n!\Gamma(p+2n-k)} {k!(n-k)!\Gamma(q+n-k)} M[V^{n-k}] \end{array}$$ (21)

As it can be seen from the simulation results shown in the next section, the series (13) with coefficients (21) has a very high rate of convergence to $W_V(x)$ in the whole definition interval [0,1].