On confidence intervals for nonmonotone parametric functions and an application to the squared mean of the normal distribution

Consider the problem of obtaining a confidence interval for some function g(θ) of an unknown parameter 9, for which a (1-α)-confidence interval is given. If g(θ) is one-to-one the solution is immediate. However, if g is not one-to-one the problem is more complex and depends on the structure of g. In this note the situation where g is a nonmonotone convex function is considered. Based on some inequality, a confidence interval for g(θ) with confidence level at least 1-α is obtained from the given (1-α) confidence interval on θ. Such a result is then applied to the N(μ,σ²) distribution with σ known. It is shown that the coverage probability of the resulting confidence interval, while being greater than 1-α, has in addition an upper bound which does not exceed Φ(3z_1-α/2)-α/2.