“Hedging” on statetical assumptions

It is pointed out that in many cases statistical assumptions, such as the assumption of a normal population, really cannot be known to be true with the absolute certainty that the word “assumption” has come to imply. In fact, in some instances data collected for purposes other than testing the assumptions cast serious doubt on the validity of the assumptions. A simple safety or hedging device is proposed, and illustrated for the case where the problem is to estimate the qth quantile (m, say) of a distribution assumed to be normal with given standard deviation σ. Let X, R denote the sample mean and the qth sample quantile, respectively. Under the assumption, the best estimate of m is X + t σ, where t is a properly chosen constant, while R is a reasonable estimate of m even if the assumption is not true. The hedging device is to use X + t σ as the estimate if |X + t σ – R| ≤ d, and to use R as the estimate if || + t σ – R| > d, where d is a preassigned value. In order to choose d to reflect our faith in the validity of the assumption, and to obtain the operating characteristics of our estimation rule, it is necessary to know the joint distribution of X and R. This is shown to approach a bivariate normal distribution as the sample size increases, for any population satisfying mild regularity conditions.