C. Schmegner and M. Baron. Sequential Plans and Risk Evaluation. Sequential Analysis, 26(4), 335–354, 2007. Sequential Plans and Risk Evaluation Claudia Schmegner Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA Michael Baron Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas, USA Abstract: Introduced in 1990s, sequential plans generalize and optimize the classical sequential procedures allowing sampling in groups of variables sizes. Sequential plans are preferred to pure sequential procedures when the cost function is non-linear so that each observed group implies an additional fixed cost. Optimal plans minimize the risk, a weighted sum of the expected loss, cost of observations, and cost of groups. Although risk evaluation is a challenging problem in sequential planning, risk functions of sequentially planned probability ratio tests (SPPRT) can be obtained as roots of certain functional equations. Explicit solutions are derived for SPPRT on a lattice, allowing practitioners to compare exact risks and choose an optimal procedure. Keywords: Lattice, Markov chain, Sequential plan, Sequentially planned probability ratio test, Stopping boundaries Subject classification: 62H30, 62-07, 65U05, 68T05 1 Introduction Abraham Wald [11] introduced the sequential probability ratio test (SPRT) that marked the beginning of modern sequential analysis ([7, 12]). In SPRT, observations are sampled one by one until the gained infor- mation guarantees desired error probabilities. At that moment data collection is terminated and a decision is made. The total sample size is not predetermined, it depends on data. After each observation, one decides whether to stop sampling and make an inference based on the data already collected, or to postpone the 1