Completeness and Unbiased Estimation for Sum—Quota Sampling

Abstract

Abstract Most statistical methodology in use today is designed for sampling schemes or experimental designs in which the sample size or number of data points is regarded as a fixed, preset parameter. This is in part for mathematical convenience but is also suggested by the intuition that sampling is restricted by a total cost of data collection and that the cost of sampling remains the same from unit to unit. The cost of sampling, however, may vary from unit to unit and the costs may be unknown before the sample is taken. If so, the objective of taking a sample of preset cost may force one to sample sequentially until achieving the preset cost and then stop. That is, the objective of taking a sample of fixed size may be turned against itself if the cost of sampling varies from unit to unit. A cost may be monetary or general in character. Motivated by the possibility of a quota for costs, consider the sampling scheme in which random variables, Xi are sampled sequentially until the sum of the associated nonnegative random variables, Ci , is greater than or equal to a predetermined quota, Q. The objective of this article is to show that the minimal sufficient statistic described by Pathak (1976) is complete when sampling is from a finite population and also to obtain uniformly minimum variance unbiased estimators (UMVUE) of population parameters by use of sufficiency and completeness. These estimators are similar to their fixed sample size counterparts, except that they have an adjustment for the sampling bias of the terminal observation, or the observation that causes the quota to be achieved. When sampling is from an infinite population these estimators continue to be unbiased, and in the absence of distributional assumptions they continue to be UMVUE.