Akihiro Nishi

Tests for the marginal probabilities in the two-way contingency table under restricted alternatives

Testing hypotheses on the marginal probabilities of a two-way contingency table is discussed. Three statistics are considered for testing the hypothesis of specified probabilities in the margins against alternatives with certain kind of order restriction. The properties of these statistics are discussed and their asymptotic behaviors are compared in depth. An appliction which motivated the consideration of the original testing problem is illustrated with a practical data.

A generalization of the Binet-Minc formula for the evaluation of permanents

Abstract A formula for the sum of the coefficients of monomials of the form xl1j1⋯xlpjP is given, where j1,…,jp are given positive integers, in the polynomial ∏ i=1 n ∑ j=1 m a ij x j . When p=n and j1 = ⋯jn = 1, this formula coincides with the Binet-Minc formula for the evaluation of the permanent of the matrix (aij).

A method of obtaining Pythagorean triples

The discriminant of fb(X) is the square of an integer and it follows that b2 4a2 = y2, say. Since b = u + v and a2 = uv, y = u v. It follows that y = u v and z = b = u + v is a positive integral solution of (2). Conversely, suppose that y, z is a positive integral solution of (2). Then y and z have the same parity and if we let u = (z + y)/2 and v = (z y)/2 then u > v > 0 where u and v are integers. Also, UV = (z2 _ y2)/4 = a2, and if b = u + v then u and v are solutions of X2 bX + a2 = 0. Thus, there is a one-to-one correspondence between the positive integral solutions of (2) and the integral solutions u, v of fb(X) = 0 such that u > v > O. It is easy to see that the number of pairs u, v in (3) such that u > v is given by {(2eo + 1) ... (2e,1 + 1) 1}/2 = N, say. Thus, the equation (2) has exactly N solutions for a given value of a.