3.17  Quantiles of Quadratic Polynomials of Joint-Normal Random Vectors

Consider a random variable Y that is a quadratic polynomial of a joint-normal random vector X. We can approximate its quantiles using the Cornish-Fisher expansion. Alternatively, if exact quantiles are required, we may employ the inversion theorem in a manner described by Imhof (1961) and Davies (1973). This provides the CDF of Y. From this, we can calculate quantiles.

3.17.1 The CDF of a quadratic polynomial of a joint-normal random vector

As described in Section 3.12, express Y as a linear polynomial of independent random variables,

[3.223]

where Q0 ~ N(0,1) and Qk ~ χ2(1,) for k > 0. Based upon this representation, the characteristic function of Y is calculated from [3.217], [3.219], and [3.220] as

[3.224]

Inversion theorem [3.221] provides an expression for the CDF of Y in terms of this characteristic function

[3.225]

but it involves an integral that is not amenable to standard techniques of numerical integration, such as the trapezoidal rule or Simpson’s rule. Employing the algebra of complex numbers, the theorem can be reformulated as

[3.226]

Substituting characteristic function [3.224] into [3.226] and simplifying yields

[3.227]

where

[3.228]

[3.229]

[3.230]

[3.231]

and tan–1 denotes the inverse tangent function with output in radians. The integral in [3.227] appears cumbersome, but it entails no imaginary numbers and its integrand is easily evaluated by a computer. To solve the integral numerically, two problems must be addressed:

• The integrand has form 0/0 as w approaches 0.
• The interval of integration is unbounded.

To solve the first problem, we apply l’Hopital’s rule to obtain14

[3.232]

We solve the second problem with the approximation

[3.233]

where u < ∞ is chosen sufficiently large. Valuing this integral is one instance where the trapezoidal rule provides consistently superior results to Simpson’s rule. By selecting an appropriate value for u, we can make the error in approximation [3.233] arbitrarily small. The solution is essentially exact.

3.17.2 Quantiles of a quadratic polynomial of a joint-normal random vector

Since we can evaluate the CDF Φ(y) of Y, we can now calculate any q-quantile of Y. Consider a specific value q. We seek that value y such that Φ(y) = q. We might find this by evaluating Φ at a range of values for y and finding which one yields a probability closest to q. A faster and more systematic approach is to use the secant method of Section 2.12.

The secant method requires two seed values y[1] and y[2]. Subsequent values y[3], y[4], y[5], … are obtained with the recursive equation

[3.234]

The resulting sequence of values should converge to the q-quantile y of Y.

3.17.3 Example

Consider again the random variable Y that is a quadratic polynomial of a joint-normal random vector X as defined by [3.168]. We have considered this random variable in several examples. Let’s find its .10-quantile.

Based upon representation [3.180] and formula [3.227], we can use the trapezoidal rule to evaluate the Φ(y) at any point y. For this purpose, we use approximation [3.233] with u = 1. We partition [0, u] into 500 subintervals to apply the trapezoidal rule. Consider seed values y[1] = 0 and  y[2] = 1. Applying the trapezoidal rule to each, we obtain

• Φ(0) = 0.21752,
• Φ(1) = 0.24546.

Applying the secant method, we obtain the results indicated in Exhibit 3.32.

Exhibit 3.32: Results of applying the secant method to evaluate the .10-quantile of Y. Each iteration requires the evaluation of Φ using the trapezoidal rule.

The .10-quantile of Y is –5.004, which is exact to the number of decimal places shown. Compare this with the –5.029 approximation we obtained for the same quantile using the Cornish-Fisher expansion in Section 3.14.

Exercises
3.50

Independently reproduce the results of the example of this section.