This paper proposes a class of origin-smooth approximators of indicators underlying the sum-of-negative-part statistic for testing multiple inequalities. The need for simulation or bootstrap to obtain test critical values is thereby obviated. A simple procedure is enabled using fixed critical...

Let X be a random variate whose distribution conditional on an unobservable variate Y is chi-square with s degrees of freedom and non-centrality parameter equal to Y. Upper and lower bounds for the unconditional d.f. of X are derived when only the values of location and randomness indices for Y...