Simple and highly efficient estimators for a type I censored normal sample

SUMMARY In a type I left censored normal sample the information consists of the observations xl, ..., which fell above the 'observation limit' c and of the number, n - k, of those observations which fell below c. This paper considers the estimation of the parameters of a normal population given a sample which has been censored at the known point c. When the maximum likelihood method is used to produce estimates of It and oC one has to resort to numerical solution of the resulting equations. In this paper simple estimators ,t* and o-* are presented. They are shown to be almost as good as the maximum likelihood estimators both for small and large samples. In the 'fixed ki' case the censored sample is generated by a sequential procedure: independent observations are made, one by one, until a predetermined number k of observations above c is obtained. The distribution of n - k will then be negative binomial. The 'fixed n' situation occurs when in a random sample of fixed size n all observations, if any, below c are deleted. In this case the number of remaining observations, k, will have a binomial distribution. However, it may then happen that the censored sample is void, and then no reasonable estimates of the population parameters can be produced. When we investigate small-sample properties of estimators we prefer to avoid that complication by excluding all such samples from our considerations: if all n observations fall below c a new sample of size n is taken, and so on, until a sample with at least one observation above c is obtained; furthermore we assume that, whenever this happens, we completely ignore how many times a useless sample was obtained. This situation arises naturally in practice: the client will not consult the stati- stician unless k exceeds zero, and the statistician will never know how many times the client did not consult him. When n is large this change in the meaning of 'fixed n'is of little importance. In particular, it does not affect any asymptotic results. Throughout the paper we assume that the population from which the original observations are taken is normal with unknown mean ,u and standard deviation cr, while the 'observation