In this paper we extend Varian's (1984) nonparametric production analysis to situations when the set of observed output, input, and price data is not consistent with profit maximization for at least one firm. In such cases, Varian's results imply that no production possibility set containing all observations can rationalize the observed data. We identify each firm whose performance, given the prices faced by it, may be found consistent with profit maximization relative to some production possibility set containing all observed output-input vectors. We show that the set @e of all such firms can itself be weakly rationalized in the sense that there exists a (closed, convex, and "monotone") production possibility set that contains all the observations, and relative to which the performance of all the firms in the set @e is consistent with profit maximization given their respective prices. By definition, firms not included in this largest set @e of efficient observations unambiguously deviate from profit maximizing behavior for any production possibility set containing all observations. We follow Farrell (1957) and analyze these deviations into technical and allocative efficiency measures, considering as admissible all closed, convex, and "monotone" production possibility sets relative to which the performance of each firm in the set @e remains consistent with profit maximization. We then describe nonparametric methods for determining the tightest upper and lower bounds on the technical, allocative, and aggregate efficiency measures evaluated relative to all such admissible production possibility sets. It is seen that the tightest upper bound on the technical efficiency measure is the same as the value computed by the nonparametric efficiency evaluation technique known as data envelopment analysis, thus establishing a link between this literature in management science/operations research and the nonparametric production analysis in economics.