We derive optimality conditions for the optimum sample allocation problem in stratified sampling, formulated as the determination of the fixed strata sample sizes that minimize the total cost of the survey, under the assumed level of variance of the stratified $\pi$ estimator of the population total (or mean) and one-sided upper bounds imposed on sample sizes in strata. In this context, we presume that the variance function is of some generic form that, in particular, covers the case of the simple random sampling without replacement design in strata. The optimality conditions mentioned above will be derived from the Karush-Kuhn-Tucker conditions. Based on the established optimality conditions, we provide a formal proof of the optimality of the existing procedure, termed here as LRNA, which solves the allocation problem considered. We formulate the LRNA in such a way that it also provides the solution to the classical optimum allocation problem (i.e. minimization of the estimator's variance under a fixed total cost) under one-sided lower bounds imposed on sample sizes in strata. In this context, the LRNA can be considered as a counterparty to the popular recursive Neyman allocation procedure that is used to solve the classical problem of an optimum sample allocation with added one-sided upper bounds. Ready-to-use R-implementation of the LRNA is available through our stratallo package, which is published on the Comprehensive R Archive Network (CRAN) package repository.

翻译：我们推导了分层抽样中最优样本分配问题的最优性条件，该问题被表述为：在假设总体总量（或均值）的分层$\pi$估计量方差水平下，在每层样本量受单侧上界约束时，确定使调查总成本最小化的固定层样本量。在此背景下，我们假设方差函数具有某种通用形式，该形式特别涵盖了分层内无放回简单随机抽样设计的情况。上述最优性条件将从卡鲁什-库恩-塔克条件推导得出。基于建立的最优性条件，我们提供了现有程序（此处称为LRNA）的最优性正式证明，该程序解决了所考虑的最优分配问题。我们对LRNA进行公式化，使其也能在每层样本量受单侧下界约束的情况下，提供经典最优分配问题（即在固定总成本下最小化估计量方差）的解。在此背景下，LRNA可被视为流行的递归内曼分配程序的对应方法，后者用于解决添加了单侧上界约束的经典最优样本分配问题。LRNA的即用型R实现可通过我们的stratallo包获取，该包已发布在综合R档案网络（CRAN）包库中。