Upper bounds for coverage probabilities of confidence intervals for nonmonotone parametric functions

Consider a statistical model F∈F and let θ=θ(F) be a structural parameter which admits a (1-α)-level two-sided confidence interval based on a random sample taken from F. Let g(θ) be some parametric function of interest. The problem of deriving a confidence interval for g(θ) directly from that given on θ is considered. If g is one-to-one then a (1-α)-level two-sided confidence interval is immediately available. If however, g is not one-to-one the problem becomes more complex. In this paper the situation where g is a nonmonotone function is considered. Under the assumption that g has a unique minimum γ at x=δ and that g(x) is strictly decreasing (increasing) for x<δ (x>δ) a two-sided confidence interval for g(θ) can be obtained from the (1-α)-level confidence interval on θ whose confidence level, while being at least 1-α, is not greater than 1-α/2. Moreover, if in addition g is symmetric then an improved upper bound, smaller than 1-α/2, can be achieved when F is a location or location and scale distribution.