Two seminal budget studies by David Davies (1795) and Frederick Eden (1797) are employed below to investigate place of bread in diets of English rural laborers at end of eighteenth century.' Because of considerable geographic and temporal dispersion in prices of foodstuffs found in these budgets, they afford a unique opportunity to study influences of both prices and income on individual household consumption decisions. In particular a test is made of famous hypothesis, attributed by Marshall to Robert Giffen,2 that a rise in price of bread, ceteris paribus, increases its consumption among lower classes. Wheaten bread was, in Middle Ages, a luxury food of landed classes in Europe. Its gradual introduction into laboring class diets in modern period prompted David Landes (1969, p. 47) to conclude, ... one of best signs of comfort in Europe is consumption of white bread. The transition in England to wheat as the almost universal bread corn of whole people took place, according to Sir William Ashley (1928, pp. 1-2) primarily in eighteenth century and was virtually complete by 1795. The rise of wheat occurred most rapidly in southern and eastern counties and was favored by a more capitalistic agriculture since it often required special liming, other fertilization, and tilling. To contemporaries, and some modern historians, this change to wheaten bread seemed, for purely psychological reasons, to be irreversible. Radcliffe Salaman (1949, pp. 480-481) writes:

There is an innate human tendency, one might call it the “league table mentality,” to construct rankings. Schools, hospitals, sports teams, movies, and myriad other objects are ranked even though their inherent multi‐dimensionality would suggest that—at best—only partial orderings were possible. We consider a large class of elementary ranking problems in which we observe noisy, scalar measurements of merit for n objects of potentially heterogeneous precision and are asked to select a group of the objects that are “most meritorious.” The problem is naturally formulated in the compound decision framework of Robbins's (1956) empirical Bayes theory, but it also exhibits close connections to the recent literature on multiple testing. The nonparametric maximum likelihood estimator for mixture models (Kiefer and Wolfowitz (1956)) is employed to construct optimal ranking and selection rules. Performance of the rules is evaluated in simulations and an application to ranking U.S. kidney dialysis centers.

In this article, we analyze the impact of uncertainty on the hiring process. We show the connection between models of statistical discrimination where uncertainty can work against groups that have less reliable indicators of future productivity and models of option value where uncertainty about future productivity can be beneficial for these groups. These models generate hypotheses about the relationship between ex ante hiring patterns and ex post productivity. This is applied to the market for NFL football players. We provide various estimates of NFL success, which suggest that statistical discrimination and option value influence choices in this market.

Tests based on the quantile regression process can be formulated like the classical Kolmogorov-Smirnov and Cramer-von-Mises tests of goodness-of-t employing the theory of Bessel processes as in Kiefer (1959). However, it is fre-quently desirable to formulate hypotheses involving unknown nuisance parameters, thereby jeopardizing the distribution free character of these tests. We characterize this situation as \ he Durbin problem " since it was posed in Durbin (1973), for parametric empirical processes. In this paper we consider an approach to the Durbin problem involving a mar-tingale transformation of the parametric empirical process suggested by Khmaladze (1981) and show that it can be adapted to a wide variety of inference problems involving the quantile regression process. In particular, we suggest new tests of the location shift and location-scale shift models that underlie much of classical econometric inference. The methods are illustrated with a reanalysis of data on unemployment durations from the Pennsylvania Reemployment Bonus Experiments. The Pennsylvania ex-periments, conducted in 1988-89, were designed to test the ecacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells. 1.

statistics of a linear hypothesis in the standard linear model. These test statistics, which correspond to Wald, likelihood ratio, and Lagrange multiplier tests, are shown to have the same limiting chi-square behavior under mild regularity conditions on design and the distribution of errors. The asymptotic theory of the tests is derived for a large class of error distributions; thus in Huber's [10] terminology we investigate the behavior of the likelihood ratio test under non-standard conditions. The asymptotic efficiency of the 11 tests involves a modest sacrifice of power compared to classical tests in cases of strictly Gaussian errors but may yield large efficiency gains in non-Gaussian situations. The Lagrange multiplier test seems particularly attractive from a computational standpoint. We derive the asymptotic distribution of the three alternative 11 test statistics for a simple linear exclusion hypothesis. Extension of these results to hypotheses of the form R,8 = r is a straightforward exercise. When the density of the error distribution is strictly positive at the median, all three test statistics have the same limiting central x2 behavior at the null and noncentral x2 behavior for local alternatives to the null. When the variance of the error distribution is bounded, analogous results are well known for classical forms of the Wald, likelihood ratio, and Lagrange multipler tests based on least-squares methods. See, for example, Silvey [18] and the discussion in Section 4 below.

[A new class of tests for heteroscedasticity in linear models based on the regression quantile statistics of Koenker and Bassett [17] is introduced. In contrast to classical methods based on least-squares residuals, the new tests are robust to departures from Gaussian hypotheses on the underlying error process of the model.]

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Following Jureckova (1981) we introduce a finite-sample measure of performance of regression estimators based on tail behavior.The least squares estimator is studied in detail, and we find that it may achieve good tail performance under strictly Gaussian conditions.However, the tail performance of the least-squares estimator is found to be extremely poor in the case of heavy-tailed error distributions or when leverage points are present.Further analysis of the least-squares estimator with light-tailed errors indicates the strong influence of the design matrix in determining tail performance.Turning to the tail behavior of various robust estimators of the parameters of the linear model, we focus on tail performance under heavy (algebraic) tailed errors.The /^estimator is seen to be a leading case: we find a simple characterization of its tail behavior in terms of the design configuration and show that a broad class of M-estimators have the same performance.Perhaps most significantly, it is shown that our finite-sample measure of tail performance is, for heavy tailed error distributions, essentially the same as the finite sample concept of breakdown point introduced by Donoho and Huber(1983).This finding provides an important probabilistic interpretation of the breakdown point and clarifies the role of tail behavior as a quantitative measure of robustness.This link is further explored for high-breakdown regression estimators including Rousseeuw's (1982) least-median-of-squares estimator.