TVT Algorithm Tests

A method similar to the TVN method based on equation (10) is also considered here. In this case the TVT distribution can be written in the form

\begin{displaymath} T_\nu({\bf b}, R)= \frac{\Gamma(\frac{\nu+1}{2})}{\Gamma(\fr... ...} \int_{-\infty}^{b_1} (1+x^2/\nu)^{-\frac{\nu+1}{2}} G(x)dx, \end{displaymath}

where

\begin{displaymath} G(x) = T_\nu\bigg(\Big( \frac{(\nu+1)^\frac{1}{2}(b_2-\rho_{... ...o_{21}}{((1-\rho^2_{21})(1-\rho^2_{31}))^\frac{1}{2}} \bigg) . \end{displaymath}

The transformation $x = T_\nu^{-1}(t)$ produces the formula
\begin{displaymath} T_\nu({\bf b}, R) = \int_{0}^{T_\nu(b_1)} G(T_\nu^{-1}(t)) dt, \end{displaymath} (28)

which can be used with numerical integration to compute TVT probabilities.

Algorithms that combine numerical integration with equations (25), (27) and (28) were implemented and tested in a manner similar to that used for the TVN algorithms where the correlation matrices depend on the $C$ (15) matrices with $\theta_1 = 1/258, 17/258, ..., 257/258$, $\theta_2 = 1/258, 17/258, ..., 257/258$, $\theta_3 = 1/258, 17/258, ..., 257/258$, and with limits $b_1 = -5, -4, ..., 5$, $b_2 = -5, -4, ..., 5$, $b_3 = b_2, b_2+1, ..., 5$. Some tests (indicated with $\epsilon > 0$) were also completed with nearly equal $b$ values. The adaptive integration algorithm used for the three methods had an absolute error tolerance level set at $10^{-14}$. A quadruple precision implementation of the adaptive integration algorithm applied to equation (27) with absolute error tolerance level set at $10^{-18}$ was used to provide high accuracy TVT values for comparisons. This algorithm included a quadruple precision Dunnett and Sobel (1954) algorithm implementation for special case BVT probabilities.


Table 3: Maximum (Average) Errors for TVT Methods for Grid of $\theta $'s, $\nu = 1$
  $\epsilon$ $T_\nu^{-1}$ Transformed Singular Generalized Combined Generalized
6-Point Rule 0 $1\cdot10^{-3}$ ($1\cdot10^{-5}$) $6\cdot10^{-5}$ ($7\cdot10^{-7}$) $4\cdot10^{-5}$ ($7\cdot10^{-7}$)
12-Point Rule 0 $2\cdot10^{-4}$ ($1\cdot10^{-6}$) $2\cdot10^{-5}$ ($1\cdot10^{-7}$) $1\cdot10^{-5}$ ($1\cdot10^{-7}$)
24-Point Rule 0 $3\cdot10^{-5}$ ($1\cdot10^{-7}$) $5\cdot10^{-6}$ ($2\cdot10^{-8}$) $2\cdot10^{-6}$ ($1\cdot10^{-8}$)
24-Point Rule .01 $3\cdot10^{-5}$ ($1\cdot10^{-7}$) $2\cdot10^{-5}$ ($2\cdot10^{-7}$) $2\cdot10^{-5}$ ($3\cdot10^{-7}$)
48-Point Rule 0 $1\cdot10^{-5}$ ($3\cdot10^{-8}$) $6\cdot10^{-7}$ ($3\cdot10^{-9}$) $1\cdot10^{-7}$ ($1\cdot10^{-9}$)
48-Point Rule .01 $7\cdot10^{-6}$ ($2\cdot10^{-8}$) $1\cdot10^{-6}$ ($7\cdot10^{-9}$) $4\cdot10^{-6}$ ($7\cdot10^{-9}$)
Adaptive 0 $4\cdot10^{-15}$ ( $2\cdot10^{-17}$) $2\cdot10^{-11}$ ( $7\cdot10^{-16}$) $8\cdot10^{-14}$ ( $3\cdot 10^{-17}$)
Adaptive .01 $2\cdot10^{-14}$ ( $2\cdot10^{-17}$) $1\cdot 10^{-12}$ ( $6\cdot10^{-16}$) $2\cdot10^{-13}$ ( $3\cdot 10^{-17}$)
Adaptive Times 0 $2\cdot10^{-4}$ s $9\cdot10^{-4}$ s $3\cdot10^{-4}$ s

Tables 3, 4 and 5 provide results using Gauss rule and adaptive integration for methods using equations (28), (25) and (27) (respectively referred to as the $T_\nu^{-1}$ Transformed, Singular Generalized and Combined Generalized methods) for $\nu = 1, 5, 25$. Average errors are given in parenthesis for each table entry. The last line of each Table provides average times in seconds for the adaptive algorithms using an 800 MHZ Pentium III computer. These times are significantly smaller than typical times for general-purpose algorithms developed for multivariate normal and multivariate t probability computations (see Genz and Bretz, 2002, and Genz, 1993), where high accuracy computations are often infeasible. Some testing was also done using adaptive integration for a method based on equation (23). The outer infinite integration interval was transformed to the adaptive integration interval $[0,1]$ using the $\chi_\nu^{-1}(s)$ function, and a Drezner-Plackett method with a fixed integration rule was used for the inner TVN integral. This method cannot achieve accuracy comparable to the other TVT methods discussed in this section and average computation times were more than one hundred times larger than average computation times for the generalized Plackett formula TVT method implementations.


Table 4: Maximum (Average) Errors for TVT Methods for Grid of $\theta $'s, $\nu = 5$
  $\epsilon$ $T_\nu^{-1}$ Transformed Singular Generalized Combined Generalized
6-Point Rule 0 $1\cdot10^{-3}$ ($8\cdot10^{-6}$) $6\cdot10^{-5}$ ($1\cdot10^{-7}$) $2\cdot10^{-5}$ ($6\cdot10^{-8}$)
12-Point Rule 0 $1\cdot10^{-4}$ ($8\cdot10^{-7}$) $2\cdot10^{-5}$ ($2\cdot10^{-8}$) $1\cdot10^{-6}$ ($3\cdot10^{-9}$)
24-Point Rule 0 $3\cdot10^{-5}$ ($1\cdot10^{-7}$) $8\cdot10^{-7}$ ($1\cdot10^{-9}$) $1\cdot10^{-7}$ ( $2\cdot10^{-10}$)
24-Point Rule .01 $3\cdot10^{-5}$ ($1\cdot10^{-7}$) $8\cdot10^{-5}$ ($2\cdot10^{-7}$) $8\cdot10^{-5}$ ($4\cdot10^{-7}$)
48-Point Rule 0 $1\cdot10^{-5}$ ($3\cdot10^{-8}$) $2\cdot10^{-7}$ ( $2\cdot10^{-10}$) $3\cdot10^{-9}$ ( $4\cdot10^{-12}$)
48-Point Rule .01 $3\cdot10^{-6}$ ($2\cdot10^{-8}$) $1\cdot10^{-6}$ ($4\cdot10^{-9}$) $1\cdot10^{-6}$ ($7\cdot10^{-9}$)
Adaptive 0 $5\cdot10^{-7}$ ( $2\cdot10^{-12}$) $1\cdot10^{-11}$ ( $7\cdot10^{-16}$) $1\cdot10^{-13}$ ( $2\cdot10^{-17}$)
Adaptive .01 $5\cdot10^{-7}$ ( $4\cdot10^{-12}$) $1\cdot 10^{-12}$ ( $5\cdot10^{-16}$) $2\cdot10^{-13}$ ( $3\cdot 10^{-17}$)
Adaptive Times 0 $5\cdot10^{-3}$ s $2\cdot10^{-4}$ s $4\cdot10^{-4}$ s


Table 5: Maximum (Average) Errors for TVT Methods for Grid of $\theta $'s, $\nu = 25$
  $\epsilon$ $T_\nu^{-1}$ Transformed Singular Generalized Combined Generalized
6-Point Rule 0 $1\cdot10^{-3}$ ($6\cdot10^{-6}$) $6\cdot10^{-5}$ ($1\cdot10^{-7}$) $1\cdot10^{-5}$ ($1\cdot10^{-7}$)
12-Point Rule 0 $1\cdot10^{-4}$ ($5\cdot10^{-7}$) $2\cdot10^{-5}$ ($2\cdot10^{-8}$) $4\cdot10^{-7}$ ($1\cdot10^{-9}$)
24-Point Rule 0 $2\cdot10^{-5}$ ($6\cdot10^{-8}$) $1\cdot10^{-6}$ ($1\cdot10^{-9}$) $9\cdot10^{-9}$ ( $2\cdot10^{-11}$)
24-Point Rule .01 $2\cdot10^{-5}$ ($5\cdot10^{-8}$) $1\cdot10^{-4}$ ($3\cdot10^{-7}$) $1\cdot10^{-4}$ ($7\cdot10^{-7}$)
48-Point Rule 0 $5\cdot10^{-6}$ ($6\cdot10^{-6}$) $3\cdot10^{-8}$ ( $5\cdot10^{-11}$) $4\cdot10^{-10}$ ( $5\cdot10^{-13}$)
48-Point Rule .01 $3\cdot10^{-6}$ ($1\cdot10^{-8}$) $6\cdot10^{-6}$ ($2\cdot10^{-8}$) $6\cdot10^{-6}$ ($4\cdot10^{-8}$)
Adaptive 0 $1\cdot10^{-6}$ ( $6\cdot10^{-12}$) $1\cdot10^{-11}$ ( $7\cdot10^{-16}$) $1\cdot10^{-13}$ ( $5\cdot10^{-17}$)
Adaptive .01 $1\cdot10^{-6}$ ( $1\cdot10^{-11}$) $1\cdot 10^{-12}$ ( $6\cdot10^{-16}$) $2\cdot10^{-13}$ ( $5\cdot10^{-17}$)
Adaptive Times 0 $6\cdot10^{-3}$ s $2\cdot10^{-4}$ s $6\cdot10^{-4}$ s

The combined generalized Plackett formula method (equation (27)) provides the highest level of overall accuracy. An algorithm based on this method with only a 6-point Gauss rule can provide single precision accuracy for most TVT problems. An adaptive algorithm (with Fortran implementation TVTL in TVPACK, available from the author's website) can provide double precision accuracy for most TVT problems. The algorithm that uses adaptive integration with the singular generalized Plackett formula is often faster than the combined generalized Plackett formula algorithm, and this difference is more significant for the larger $\nu$ values, but this algorithm (like the related TVN algorithm) appears to be more sensitive to loss of accuracy from rounding errors.



2004-04-13