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The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bear closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banach-valued random elements and applying the Banach central limit theorem and law of iterated logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted.

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Consumers' average value for information goods, websites, weather forecasts, music, and news declines with the number consumed. This paper provides simple guidelines to optimal bundling marketing strategies in this case. If consumers' values do not decrease too quickly, we show that bundling is approximately optimal. If consumers' values to subsequent goods decrease quickly, we show by example that one should expect bundling to be suboptimal.bundling, electronic commerce, price discrimination, digital products

Recently Barron (1993) has given rates for hidden layer feedforward networks with sigmoid activation functions approximating a class of functions satisfying a certain smoothness condition. These rates do not depend on the dimension of the input space. We extend Barron's results to feedforward networks with possibly nonsigmoid activation functions approximating mappings and their derivatives simultaneously. Our conditions are similar but not identical to Barron's, but we obtain the same rates of approximation, showing that the approximation error decreases at rates as fast as n-'I2, where n is the number of hidden units. The dimension of the input space appears only in the constants of our bounds.

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