Rational Arithmetic Mathematica Functions to Evaluate the One-sided One-sample K-S Cumulative Sample Distribution

One of the most widely used goodness-of-fit tests is the Kolmogorov-Smirnov (KS) family of tests which have been implemented by many computer statistical software packages. To calculate a p value (evaluate the cumulative sampling distribution), these packages use various methods including recursion formulae, limiting distributions, and approximations of unknown accuracy developed over thirty years ago. Based on an extensive literature search for the one-sided one-sample K-S test, this paper identifies two direct formulae and five recursion formulae that can be used to calculate a p value and then develops two additional direct formulae and four iterative versions of the direct formulae for a total of thirteen formulae. To ensure accurate calculation by avoiding catastrophic cancelation and eliminating rounding error, each formula is implemented in rational arithmetic. Linear search is used to calculate the inverse of the cumulative sampling distribution (find the confidence interval bandwidth). Extensive tables of bandwidths are presented for sample sizes up to 2, 000. The results confirm the hypothesis that as the number of digits in the numerator and denominator integers of the rational number test statistic increases, the computation time also increases. In comparing the computational times of the thirteen formulae, the direct formulae are slightly faster than their iterative versions and much faster than all the recursion formulae. Computational times for the fastest formula are given for sample sizes up to fifty thousand.

Rational Arithmetic Mathematica Functions to Evaluate the Two-Sided One Sample K-S Cumulative Sampling Distribution

One of the most widely used goodness-of-fit tests is the two-sided one sample Kolmogorov-Smirnov (K-S) test which has been implemented by many computer statistical software packages. To calculate a two-sided p value (evaluate the cumulative sampling distribution), these packages use various methods including recursion formulae, limiting distributions, and approximations of unknown accuracy developed over thirty years ago. Based on an extensive literature search for the two-sided one sample K-S test, this paper identifies an exact formula for sample sizes up to 31, six recursion formulae, and one matrix formula that can be used to calculate a p value. To ensure accurate calculation by avoiding catastrophic cancelation and eliminating rounding error, each of these formulae is implemented in rational arithmetic. For the six recursion formulae and the matrix formula, computational experience for sample sizes up to 500 shows that computational times are increasing functions of both the sample size and the number of digits in the numerator and denominator integers of the rational number test statistic. The computational times of the seven formulae vary immensely but the Durbin recursion formula is almost always the fastest. Linear search is used to calculate the inverse of the cumulative sampling distribution (find the confidence interval half-width) and tables of calculated half-widths are presented for sample sizes up to 500. Using calculated half-widths as input, computational times for the fastest formula, the Durbin recursion formula, are given for sample sizes up to two thousand.

Arbitrary Precision Mathematica Functions to Evaluate the One-Sided One Sample K-S Cumulative Sampling Distribution

Efficient rational arithmetic methods that can exactly evaluate the cumulative sampling distribution of the one-sided one sample Kolmogorov-Smirnov (K-S) test have been developed by Brown and Harvey (2007) for sample sizes n up to fifty thousand. This paper implements in arbitrary precision the same 13 formulae to evaluate the one-sided one sample K-S cumulative sampling distribution. Computational experience identifies the fastest implementation which is then used to calculate confidence interval bandwidths and p values for sample sizes up to ten million.

Tension is Dimension

We propose a simple universal formula for the tension of a D-brane in terms of a regularized dimension of the associated conformal field theory statespace.Comment: 18 pages, harvmac (b), one ref added, one typo fixe

Scale-factor duality in string Bianchi cosmologies

We apply the scale factor duality transformations introduced in the context of the effective string theory to the anisotropic Bianchi-type models. We find dual models for all the Bianchi-types [except for types $VIII$ and $IX$] and construct for each of them its explicit form starting from the exact original solution of the field equations. It is emphasized that the dual Bianchi class $B$ models require the loss of the initial homogeneity symmetry of the dilatonic scalar field.Comment: 18 pages, no figure