For real-valued x1, x2 , ..., xn with real-valued “responses” y1, y2 , ...,yn and “scores” s1, s2, ..., sn we solve the problem of computing the maximum of C(k) = Σh=1,...,n sh I[yh ≥ k(xh)] over all convex functions k on R1. The article describes a recursive relation and an algorithm based on it to compute this value and an optimal k in O(n3) steps. For a special choice of scores, max C(k) can...

We prove the following conjecture of Narayana: there are no nontrivial dominance refinements of the Smirnov two-sample test if and only if the two sample sizes are relatively prime. We also count the number of natural significance levels of the Smirnov two-sample test in terms of the sample sizes and relate this to the Narayana conjecture. In particular, Smirnov tests with relatively prime samp...

We prove the following conjecture of Narayana: there are no nontrivial dominance reene-ments of the Smirnov two-sample test if and only if the two sample sizes are relatively prime. We also count the number of natural signiicance levels of the Smirnov two-sample test in terms of the sample sizes and relate this to the Narayana conjecture. In particular, Smirnov tests with relatively prime sampl...

In this paper we propose an improvement of the Kolmogorov-Smirnov test for normality. In the current implementation of the Kolmogorov-Smirnov test, a sample is compared with a normal distribution where the sample mean and the sample variance are used as parameters of the distribution. We propose to select the mean and variance of the normal distribution that provide the closest fit to the data....