  We consider estimating the proportion of random variables for two types of
composite null hypotheses: (i) the means or medians of the random variables
belonging to a non-empty, bounded interval; (ii) the means or medians of the
random variables belonging to an unbounded interval that is not the whole real
line. For each type of composite null hypotheses, uniform consistent estimators
of the proportion of false null hypotheses are constructed respectively for
random variables whose distributions are members of a Type I location-shift
family or are members of the Gamma family. Further, uniformly consistent
estimators of certain functions of a bounded null on the means or medians are
provided for the two types of random variables mentioned earlier. These
functions are continuous and of bounded variation. The estimators are
constructed via solutions to Lebesgue-Stieltjes integral equations and harmonic
analysis, do not rely on a concept of p-value, can be used to construct
adaptive false discovery rate procedures and adaptive false nondiscovery rate
procedures for multiple hypothesis testing, can be used in Bayesian inference
via mixture models, and may be used to estimate the sparsity level in
high-dimensional Gaussian linear models.
