On the calculation of standard error for quotation in confidence statements

Abstract

The variability of a parameter estimator θ is often expressed by quoting, within parentheses, a value of standard error, say σ . Typically the latter is calculated via an asymptotic formula, or by application of the bootstrap or jackknife. A common interpretation of the quotation “ θ ( σ )” is that the confidence interval I = ( θ − 2 σ , θ covers the true value of θ with probability approximately 0.95. However, in problems where the allowable range of θ is sharply restricted by the context it is sometimes the case that one or other of the endpoints of I lies outside the allowable range. This happens because information about the allowable range is not adequately taken into account when computing the standard error. Examples include confidence statements about ratios of means. In the present paper we suggest a remedy for this problem. We propose a new, symmetrized version of Owen's empirical likelihood method, and use it to construct range-respecting, symmetric confidence intervals. These intervals suggest new formulae for standard error. An alternative approach may be based on a version of the bootstrap, although the latter is more expensive in computing time.