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SummaryA comprehensive approach to goodness of fit testing is possible using the smooth tests described in detail in Rayner & Best (1989). Here we give an overview of this area and demonstrate the power and flexibility of the smooth tests. Our emphasis is on the use of orthonormal functions, tests for composite hypotheses, and tests of categorised data. We have developed tests for families of distributions, such as the univariate and multivariate normal, exponential and Poisson. The tests are essentially omnibus tests but the components provide useful and powerful directional tests. The history of the smooth tests of goodness of fit is reviewed from Neyman (1937), through to Lancaster, to Thomas, Kopecky and Pierce, and to Rayner and Best. The formulation of categorised smooth models leads to X2 tests and their components. A generalisation of the smooth categorised model, when allied with Hall's (1985) idea of overlapping, leads to focused tests, and to an alternative to pooling. Examples are taken from D'Agostino & Stephens (1986), who have several different contributors and therefore approaches, none of which is recommended above the others. Our resolution is simple: don't use those other methods-use a smooth test! Neyman (1937) introduced the smooth tests for goodness of fit. He considered only the case of no nuisance parameters. Application of the probability integral transformation means that the distribution for which we test can be taken to be uniform. His tests use the Legrendre polynomials. Components of the smooth test statistic are such that the first detects a mean shift, the second a variance shift, the third a change in skewness and the fourth a change in kurtosis. If the orthonormal set was chosen differently, different alternatives would be detected by the components. So the orthonormal set should be chosen with the alternatives one wishes to most powerfully detect in mind.Neyman (1937) did not use score tests, but instead required that the tests derived for the smooth model be locally uniformly most powerful symmetric, unbiased and of size a. Unfortunately, only asymptotically are those constraints realised for the tests suggested. Barton (1953Barton ( , 1955Barton ( , 1956) extended Neyman's work, and other contributions were made by Watson (1959), and Hamdan (1962, 1963, 1964) (who worked with H. O.