A common statistical problem is that of estimating the parameters It and C in a population having distribution function of the form F[(x -u l)/o-] from a complete or censored random sample taken from the population. More generally an estimate of a linear combination kl It + k2 omay be required, for example, for an estimate of a percentile of the distribution. One method of dealing with this situation is to arrange the sample values in order of magnitude and to obtain an estimate by the method of least squares from this ordered sample, xfn " x .n . < X(n) The theory of this method, where the complete sample is available, was given by Lloyd (1952) and this theory mayreadilybe extended to the case where either one or both ends of the sample are censored. Using the method of least squares provides an estimate which is linear in these (ordered) observations and has minimum variance among all such linear estimates. Except in certain special cases, however, using the method involves, for uncensored samples, the evaluation of l n(n -1) double integrals and 2n single integrals, followed by the inversion of an (n x n) matrix. This is not in general practicable, so that to overcome this, various methods which approximate to this least-squares process have been proposed. Notable among these proposals is that by Blom (1958), who obtains 'nearly best' estimates based on an approximation to the moments of ordered observations described by Mosteller (1946). Blom's estimates are in many cases extremely efficient for quite small sample sizes in spite of the fact that the approximations on which they are based are in general very poor unless sample sizes are quite large. One corollary of this is that although an efficient estimate may be obtainable, information about its standard error may be very poor. Another is that in many cases where Blom's method provides efficient estimates for small sample sizes, the efficiency of the linear estimates does not seem to be particularly sensitive to changes in the coefficients. In the present paper, therefore, no attempt has been made to obtain estimates which in any sense approximate to the best (least-squares) estimates. Estimates are proposed in which the general structure of the coefficients is chosen for mathematical tractability, both in the determination of these coefficients and also in the computation of the standard error of the estimates so obtained. It is shown that these estimates have variances and covariances which in general, behave asymptotically as 1/n, and the class of distributions for which they may be suitable is discussed. Two examples (the normal distribution and the extreme value distribution) are used to illustrate the method. In the case of the normal distribution an extremely simple unbiased estimate of Cis obtained, which is uncorrelated with the mean. It has asymptotic efficiency of 98 % and has high efficiency even for small values of n. This estimate is slightly more difficult to compute than the range, but its higher efficiency suggests that there may be situations in which it could be preferred to that statistic.