70 RANDOM NUMBERS 3.3.2 Sl. [Initialize.] Set A[j] (Web site)
70 RANDOM NUMBERS 3.3.2 Sl. [Initialize.] Set A[j] e 0 for 0 2 j 5 n; then set A[l] t 1 and j, e ji t 1. Then do step S2 exactly n - 1 times and go on to step S3. S2. [Update probabilities.] (Each time we do this step it corresponds to tossing a ball into an urn; A[j] represents the probability that exactly j of the urns are occupied.) Set jr +ji+l. Thenforjcjl,ji-l,…, jh(inthis order), set A[j] +- (j/mkUl + ((1 + l/m) -(jlm))Ab -11. If ALI has become very small as a result of this calculation, say A[j] < 10m2 , set A[j] + 0; and in such a case, if j = j, decrease ji by 1, or if j = jo increase jo by 1. S3. [Compute the answers.] In this step we make use of an auxiliary table (Tl I r2 f . . * , Ttmax ) = (.Ol, .05, .25, .50, .75, .95, .99, 1.00) containing the specified percentage points of interest. Set p t 0, t t 1, and j c j. -1. Do the following iteration until t = tmax: Increase j by 1, and set p e p + A[j]; then if p > Tt, output n -j -1 and 1 - p (meaning that with probability 1 -p there are at most n -j -1 collisions) and repeatedly increase t by 1 until p 5 Tt. I J. Serial correlation test. We may also compute the following statistic: n(U0Ul + GU2 + . . . + un-2un–l + un-1Uo) -(VII + Ul + . . . + un-1y = n(U$ + UT + .*a+ q-,)-(uo + u1+…+ un-ly . (23) This is the serial correlation coefficient, which is a measure of the amount lJ+l depends on Uj. Correlation coefficients appear frequently in statistics; if we have n quantities uo, Ul, f U,-I and n others Vi, VI, . . . , V,-I , the correlation coefficient between them is defined to be C= n m4w -cc WC vj) (24 J(nCu3 -Euj)2)(nCV~ -(,IEV,)2) All summations in this formula are to be taken over the range 0 5 j < n; Eq. (23) is the special case V, = U(j+l) modn. (Note: The denominator of (24) is zero when UO = VI = ... = Un-l or VO = VI = ... = V,-1; we exclude this case from discussion.) A correlation coefficient always lies between -1 and +l. When it is zero or very small, it indicates that the quantities Uj and I+ are (relatively speaking) independent of each other, but when the correlation coefficient is fl it indicates total linear dependence; in fact Vj = cx f PiIJj for all j in such a case, for some constants cr and p. (See exercise 17.) Therefore it is desirable to have C in Eq. (23) close to zero. In actual fact, since U&J1 is not completely independent of UIU2, the serial correlation
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