Bounds for non-central chi-square distributions having unobservable random non-centrality parameters

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 are known. The numerical quality of the bounds is illustrated for a range of parameter values. An application to testing hypotheses in random coefficients regression models is presented.