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This chapter chronicles the period from 1980 to 2000. Progress on the development of permutation methods continued unabated during this period, paralleling advancements in high-speed computing and the subsequent wide-spread availability of both university mainframes and, later in the period, personal desktop computers. Also, a number of books were published that introduced permutation methods to a wide variety of audiences and there was a decided shift in the literature away from computer science journals into discipline journals. These progressions were accompanied by an increasing emphasis on statistical applications of permutation methods, both exact and resampling, since efficient permutation generators were readily available.

This chapter chronicles the development of permutation statistical methods from 1940 to 1959. This period may be considered a bridge between the early years of 1920–1939 where permutation tests were first conceptualized and the next period, 1960–1979, in which gains in computer technology provided the necessary tools to successfully employ permutation tests. The recognition of permutation methods as the gold standard against which conventional statistical methods were to be evaluated, while often implicit in the 1920s and 1930s, is manifest in many of the publications on permutation methods that appeared between 1940 and 1959. Also, a number of researchers turned their attention during this time period to rank tests, which simplified the calculation of exact probability values; other researchers continued work on calculating exact probability values, creating tables for small samples; and still others continued the theoretical work begun in the 1920s.

This chapter chronicles the development of permutation statistical methods from 1920 to 1939, when the earliest discussions of permutation methods appeared in the literature. In this period J. Spława-Neyman, R.A. Fisher, R.C. Geary, T. Eden, F. Yates, and E.J.G. Pitman laid the foundations of permutation methods as we know them today. As is evident in this period, permutation methods had their roots in agriculture and, from the beginning, were widely recognized as the gold standard against which conventional methods could be verified and confirmed.

This chapter chronicles the period from 2001 to 2010. By the beginning of this period, permutation statistical methods had come of age and advances were comprised more of application and expansion into new fields and disciplines than the development of new permutation methods that characterized earlier years. In this period computing power was sufficient to accommodate the needs of computational statisticians utilizing permutation statistical methods, including both exact and Monte Carlo permutation tests.

This chapter chronicles the development of permutation statistical methods from 1960 to 1979. This was a period that witnessed dramatic improvements in computer technology, a process that was integral to the development of permutation tests. Prior to 1960, computers were based on vacuum tubes and were large, slow, expensive, and availability was severely limited. Between 1960 and 1979 computers became based on transistors and were smaller, faster, more affordable, and more readily available to researchers. During this period, work on permutation tests fell primarily into three categories: writing algorithms that efficiently generated permutation sequences; designing exact permutation probability values for known statistics; and, for the first time, the development of statistics specifically designed for permutation methods.

Airport runways, radio spectrum, and hospital beds are resources with capacity limits used to provide multiple services with specific capacity requirements in separate markets, which contribute to recover capacity investment costs. A welfare-maximizing and (possibly) budget-constrained firm, whose operating costs significantly increase as total capacity use presses against capacity, chooses prices and capacity. When the equilibrium capacity is reached, second-best Ramsey prices must be adjusted, and mark-ups on marginal costs may be higher for services with higher demand elasticities, if they intensively use capacity. Moreover, for a given output vector, the firm invests more than in first best. Instead, the equilibrium capacity may be first best when there is excess capacity to reduce operating costs and thus improve welfare. Our model can be used as a benchmark to evaluate the efficiency of market mechanisms for resource allocation and pricing, or when market mechanisms are not adopted.