3. Approximants Based on Laguerre Polynomials

As pointed out in the Introduction, the density functions of numerous statistics distributed on the positive half-line can be approximated from their exact moments by means of sums involving Laguerre polynomials. It should be pointed out that such approximants should only be used when the underlying distribution possesses the tail behaviour of a gamma random variable. Fortunately, this is often the case for test statistics whose support is semi-infinite. Note that for other types of distributions defined on the positive half-line, such as the lognormal which is considered in Example 3, the moments may not uniquely determine the distribution; see [15, 106] for conditions ensuring that they do.

Consider a random variable defined on the interval , whose th moment is denoted by , and let

and

As explained in Remark 3.1, when the parameters and are so chosen, the leading term of the approximant is a shifted gamma density function whose mean and variance agree with those of . Although can be any finite real number, it is in most cases of interest equal to zero. By definition, belongs to , the set of positive real numbers. Denoting the th moment of by

that is,

the density function of the random variable defined on the interval can be expressed as

where

is a Laguerre polynomial of order in with parameter , that is, in Mathematica notation and

which also can be represented by times , wherein is replaced with , [13, 17]. Then, on truncating the series given in equation (24) and making the change of variable , we obtain the following density approximant for :

that is,

where denotes the gamma function or, observing that with replaced by as defined earlier, is equivalent to with replaced by , we obtain

where and are defined in equations (19) and (20), respectively. It should be noted that the representation of the approximant appearing in equation (29) does not require the evaluation of , .

Remark 3.1 Note that is a shifted gamma density function with parameters and whose mean, , and variance, , match those of and that, in light of equation (27), we can express as the product of an initial shifted gamma density approximation specified by times a polynomial adjustment. That is,

where .

Example 3: The Case of the Standard Lognormal Distribution

As pointed out at the beginning of this section, the proposed methodology is contraindicated when a distribution is not uniquely defined by its moments or when its tail behaviour is not that of a gamma random variable. A case in point is the lognormal distribution. As shown in Figure 5, if we employ the methodology outlined in this section, a very crude approximation of the CDF of the standard lognormal distribution is obtained on the basis of its first three moments. When additional moments are being used, the resulting density approximants turn out to be unusable.

Figure 5. Exact and approximate (dashed line) CDFs. [CDFLN in the Appendix]

The following example is relevant as nonnegative definite quadratic forms in normal variables--which happen to be ubiquitous in statistics--can be expressed as mixtures of chi-square random variables, [18, Chapters 2, 7].

Example 4: A Mixture of Gamma Random Variables

Let the random variable be a mixture of three equally weighted, shifted gamma random variables defined on the interval with parameters , , and . The density and moment-generating functions of are given in the Appendix by and , respectively. The th moment of this distribution, denoted by , is determined by evaluating the hth derivative of with respect to at .

Figure 6 shows the exact density function of the mixture as well as the initial gamma density approximation given by . Clearly, traditional approximants which make use of three or four moments could not capture adequately all the distinctive features of this particular distribution.

Figure 6. Exact density function and initial gamma approximant. [PGE in the Appendix]

The exact density function, , and its approximant, , evaluated from equation (29), are plotted in Figure 7. As pointed out in Remark 3.1, this density approximant results from a polynomial adjustment applied to the shifted gamma density specified by . (Once such an approximant is obtained, a spline could be fitted in order to reduce the degree of precision that would be required in subsequent calculations.)

Figure 7. Exact and approximate (dashed line) PDFs. [PDEA in the Appendix]

This example illustrates that the proposed approximation formulae can also accommodate multimodal distributions and that calculations involving high-order Laguerre polynomials will readily produce remarkably accurate approximations when performed in an advanced computing environment such as that provided by Mathematica.