Finding Critical Values Using Numerical Integration - CiteSeerX

Finding Critical Values Using Numerical Integration  Alan Genz Department of Mathematics Washington State University Pullman, WA 99164-3113 [email protected]

Abstract

 and the Non-central MVT (NCMVT), de ned by

The problem of nding critical values for multivariate distributions is discussed and analyzed. The solution for this problem requires the combination of a numerical optimization method with a numerical integration method. Key issues are a) how to balance the error from the optimization method with the estimated error from the numerical integration method, and b) how to make ecient use of time consuming numerical integration computations for multivariate distributions. An algorithm for the problem will be described. The use of the algorithm will be illustrated with examples that use multivariate Normal and multivariate t distributions.

T(a; b; ; ; )

1 Introduction

Assume that x = (x1 ; x2; :::; xm)t , ?1  ai < bi  1, for all i, and  is positive semi-de nite symmetric m  m covariance matrix. The following distributions (see Tong, 1990) will be considered:  the multivariate Normal (MVN), de ned by

1

Z 1?  2 2 = 2?(  ) s ?1 e? s2 ( psa ?  ; psb ?  ; )ds: 2 0

Ecient and robust numerical software is now available (see Genz, 1992 and Genz and Bretz, 1999, and also Fortran software available from the website with url http://www.sci.wsu.edu/math/faculty/genz/homepage) for these MVN and MVT problems for 1  m  20. The focus of this paper is the use of this type of software for several standard con dence interval (CI) problems. A variety of these problems are described in references given at the end of this paper (Edwards and Berry (1987), Genz and Bretz (1999), Genz and Kwong (1999), Hsu (1996), Somerville (1997, 1998, 1999)). These problems all require the determination of a number t that solves the equation P(t) = 1 ? , for a given , where P(t) takes one of the following forms:

Zb1 Zb2 Zbm 1 t ?1 1 ::: e? 2 x  x dx; (a; b; ) = pjj(2)m a a a

 P(t) = (?1; t; ) (one-sided MVN CI's),

 the multivariate t (MVT), de ned by

 P(t) = T(?1; t; ; ) (one-sided MVT CI's),

1 2

m

T(a; b; ; )

Zb1 Zb2 Zbm  +m ) ?( 1 2 dx ::: = p t m ?( 2 ) jj() a a a (1 + x ?1 x ) +2m 1

1 2

m

Z 1?  2 2 2  ?(  ) s ?1 e? s2 ( psa ; psb ; )ds; 2 0

 To

appear in

Computing Science and Statistics 31, 2000.

 P(t) = ( ?t; t; ) (two-sided MVN CI's),  P(t) = T( ?t; t; ; ) (for two-sided MVT CI's),  P(t) = T(?1; t; ; ; ) (one-sided NCMVT CI's),  P(t) = T( ?t; t; ; ; ) (two-sided NCMVT CI's), with t = t(d1; :::; dm)t (for di > 0, often di = 1), and 1 = (1; :::; 1)t.

2 General Methods A simulation method that is often used for CI problems is the %-rejection method (Edwards and Berry, 1987). This method consists of the following steps: 1. generate N random vectors xj from the chosen distribution; 2. for each xj , let vj = 1max (x ) (for the one-sided im i;j cases), or 1max (jx j) (for the two-sided cases); im i;j 3. order the v's and set t = v((N +1)(1?)), rejecting the smallest (N + 1)(1 ? )% of v's. This method can sometimes quickly provide low accuracy approximations to t, but it suers from the slow convergence typical of Monte Carlo algorithms, when higher accuracy is desired. Other popular methods use various types of approximations to the chosen distribution in order to simplify the integration, but the result is often an approximate solution to a problem which is already an approximation to the original problem. The book by Hsu (1996) reviews many of these methods. The numerical integration-optimization method considered in this paper uses the function h(t) = P(t) ? (1 ? ): The basic method involves nding t, the point where h(t ) = 0, using a numerical optimization method. But h(t) is often expensive to compute using numerical integration, particularly for large m, so a numerical optimization method that requires only a few iterations is needed. This can be accomplished by combining a method for getting good starting points for the optimization method, with an optimization method that converges rapidly. An additional complication that arises with the combined numerical integration-optimization method is the presence of numerical integration errors, which must be controlled along with with the numerical optimization errors. These issues are discussed in detail in the next section.

3 Numerical Optimization

3.1 Choice of Optimization Method

A primary goal for the selection of an optimization method is to nd a method the requires only a few iterations for a large class of problems, so it would be

desirable to use a second (or even higher) order method like Newton's method to nd t , but Newton's method and higher order methods all use h0 (t). The computation of h0 (t) requires m (m-1)-dimensional distribution integrals, so the numerical computation of h0 (t) is probably infeasible for many problems. Therefore, Secantlike methods were considered. Some preliminary tests showed that the simple Secant method is not suitable for many problems because jh0(t)j is sometimes very small near t (particularly when  is small), and this can result in divergence unless a very very good starting value is available. The more robust bisection method could be used, but the convergence is only linear. Therefore, various bisection-Secant hybrid methods were considered. After some experiments, the \Pegasus" method (see Ralston and Rabinowitz, 1978) was selected. This method has asymptotic order of convergence similar to that of the Secant method and, at each iteration it provides a bracketing interval for t . The Pegasus method (PM) consists of the following steps: 1. given an initial error tolerance , start with an interval [ta ; tb] where (tb ? ta ) > 2 and t 2 [ta ; tb]; let ha = h(ta ) and hb = h(tb), so ha < 0 < hb ; 2. compute the Secant tc = tb ? hb (tb ? ta )=(hb ? ha ); (a) if hc = h(tc ) < 0, swap (tc ; hc) and (ta ; ha), (b) if hc  0, set ha ha hb=(hb + hc ) and swap (tc ; hc) and (tb ; hb); 3. if (tb ? ta ) > 2 repeat step 2; 4. output t  (ta + tb)=2. The Pegasus method is the same as the linearly convergent False-Position method except for the ha modi cation at step 2(b), which improves the approximate order of convergence to 1:64.

3.2 Starting Interval Selection

Let Aj (t) = fXj : Xj  t for the one-sided cases, or Aj = jXj j  t for the two-sided cases. The Bonferroni bound for 1 ? P(t) (see Hsu, 1996) is 1 ? P(t) = Prob([mj=1Acj (t)) 

m X j =1

Prob(Acj (t)) = U(t);

where Acj (t) is the compliment of the set Aj (t). A simple lower bound for 1 ? P(t) is L(t) = min Prob(Acj (t))  1 ? P(t) j

Both of these bounds require only 1-dimensional distribution values. If ta and tb are determined by solving L(t) =  and U(t) = , respectively, then t 2 [ta; tb]. In both the MVN and the MVT cases, this bounding interval for t can be found directly using the appropriate 1-dimensional inverse distribution function. For example, with the two-sided MVN case,  )]: [ta; tb] = [?1(1 ? 2 ); ?1(1 ? 2m Shorter intervals can be found using bivariate distribution values if a modi ed Bonferroni bound is combined with the Hunter-Worsley bound. These bounds are described the book by Hsu (1966, Appendix A). The practical use of these bounds requires the assumption that the bivariate distribution values can be eciently determined. If S(t) is de ned by S(t) =

X j