Abstract
This paper presents some results dealing with statistical inferences in presence of isotonic conditions. Isotonic regression problem is to minimize Σ__<xεX>{g(x)-f(x)}^2 w(x) subject to the restriction of partial ordering of f(x) where w(x)>0 and g(x) are given for any x ε X. The solution is called the isotonic regression on g(x). A generalization of this regression and algorithms to obtain the solution are given. The isotonic regression also solves many other problems such as mulidimensional scaling, inventory problems, production plannings, taut string and so on. Statistical tests under ordered hypothesis have been studied by many authors, some of whom are Bartholomew, Barlow, Brunk, Bremner [1]. In this paper, Bartholomew's X^^-^2 and E^^-^2 tests are considered under normal assumption. It is shown that these tests are more powerful than X^2 test.
