Modified sampling scheme in inverse sampling without replacement

Inverse sampling with unequal probabilities without replacement is modified from the Midzuno´s scheme. An unbiased estimator of the population total of a study variable is presented. The variance and the corresponding unbiased estimator of this variance are given. An unbiased estimator of the number of units in a given class, its variance and an unbiased estimate of the variance are also given. These estimators are obtained via the Murthy´s estimator using the Rao-Blackwell approach conditioned on a minimum sufficient statistic. The estimates do not depend on the order of draws and are easy to compute. An example is given to illustrate the idea.