Suppose that a sample of size n from a continuous and symmetric population with an unknown parameter is given. We consider a fictitious random subsample of sizekdrawn from the original sample and construct the best linear estimator based on the subsample. Applying the Rao-Blackwell type argument, we get an estimator which uses the information contained in the whole sample and is supposed to be uniformly efficient for a wide class of distributions. Monte Carlo experiments established that this estimator is highly efficient for small samples of size 10 to 20.