this paper. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks phenomenon is unveiled. We demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow

. 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 bears 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. 1. Introduction In testing whether or not a parametric statistical model is correctly specified, there are a number of apparently distinct approaches one might take. T...

With modern technology, massive data can easily be collected in a form of multiple sets of curves. New statistical challenge includes testing whether there is any statistically significant difference among these sets of curves. In this paper, we propose some new tests for comparing two groups of curves based on the adaptive Neyman test and the wavelet thresholding techniques introduced in Fan (1996). We demonstrate that these tests inherit the properties outlined in Fan (1996) and they are simple and powerful for detecting di erences between two sets of curves. We then further generalize the idea to compare multiple sets of curves, resulting in an adaptive high-dimensional analysis of variance, called HANOVA. These newly developed techniques are illustrated by using a dataset on pizza commercial where observations are curves and an analysis of cornea topography in ophthalmology where images of individuals are observed. A simulation example is also presented to illustrate the power of the adaptive Neyman test.

Several new tests are proposed for examining the adequacy of a family of parametric models against large nonparametric alternatives. These tests formally check if the bias vector of residuals from parametric ts is negligible by using the adaptive Neyman test and other methods. The testing procedures formalize the traditional model diagnostic tools based on residual plots. We examine the rates of contiguous alternatives that can be detected consistently by the adaptive Neyman test. Applications of the procedures to the partially linear models are thoroughly discussed. Our simulation studies show that the new testing procedures are indeed powerful and omnibus. The power of the proposed tests is comparable to the F-test statistic even in the situations where F -test is known to be suitable and can be far more powerful than the F-test statistic in other situations. An application to testing linear models versus additive models are discussed.

Generalized likelihood ratio statistics have been proposed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153–193] as a generally applicable method for testing nonparametric hypotheses about nonparametric functions. The likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family. We extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio (SELR) tests. The approach is also applied to test conditional estimating equations on the distributions of stochastic errors. It is shown that the proposed SELR statistics follow asymptotically rescaled χ 2-distributions, with the scale constants and the degrees of freedom being independent of the nuisance parameters. This demonstrates that the Wilks phenomenon observed in Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153–193] continues to hold under more relaxed models and a larger class of techniques. The asymptotic power of the proposed test is also derived, which achieves the optimal rate for nonparametric hypothesis testing. The proposed approach has two advantages over the generalized likelihood ratio method: it requires one only to specify some conditional estimating equations rather than the entire distribution of the stochastic error, and the procedure adapts automatically to the unknown error distribution including heteroscedasticity. A simulation study is conducted to evaluate our proposed procedure empirically.

The classical problem of testing goodness-of-t of a parametric family is reconsidered. A new test for this problem is proposed and investigated. The new test statistic is a combination of the smooth test statistic and Schwarz's selection rule. More precisely, as the sample size increases, an increasing family of exponential models describing departures from the null model is introduced and Schwarz's selection rule is presented to se-lect among them. Schwarz's rule provides the "right" dimension given by the data, while the smooth test in the "right" dimension nishes the job. Theoretical properties of the selection rules are derived under null and al-ternative hypotheses. They imply consistency of data driven smooth tests for composite hypotheses at essentially any alternative.

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Abstract. To test if a density f is equal to a specified f0, one knows by the Neyman–Pearson lemma the form of the optimal test at a specified alternative f1. Any nonparametric density estimation scheme allows an estimate of f. This leads to estimated likelihood ratios. Properties are studied of tests which for the density estimation ingredient use log-linear expansions. Such expansions are either coupled with subset selectors like the AIC and the BIC regimes, or use order growing with sample size. Our tests are generalised to testing adequacy of general parametric models, and work also in higher dimensions. The tests are related to but different from the ‘smooth tests ’ which go back to Neyman (1937) and which have been studied extensively in recent literature. Our tests are large-sample equivalent to such smooth tests under local alternative conditions, but different and often better under non-local conditions.

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We propose novel misspeci…cation tests of semiparametric and fully parametric univariate di¤usion models based on the estimators developed in Kristensen (Journal of Econometrics, 2010). We …rst demonstrate that given a preliminary estimator of either the drift or the di¤usion term in a di¤usion model, nonparametric kernel estimators of the remaining term can be obtained. We then propose misspeci…cation tests of semparametric and fully para-metric di¤usion models that compare estimators of the transition density under the relevant null and alternative. The asymptotic distribution of the estimators and tests under the null are derived, and the power properties are analyzed by considering contiguous alternatives. Test directly comparing the drift and di¤usion estimators under the relevant null and alter-native are also analyzed. Markov Bootstrap versions of the test statistics are proposed to improve on the …nite-sample approximations. The …nite sample properties of the estimators are examined in a simulation study.