Generalised smooth tests of goodness of fit

Smooth tests of goodness of fit may be constructed by defining an order k alternative to the hypothesised probability density function and deriving the score test to assess whether or not the data are consistent with the hypothesised probability density function. For many important distributions the form of the score test statistic is the sum of squares of components that are asymptotically independent and asymptotically standard normal. Moreover each component has a moment interpretation that assists with interpreting rejection of the null hypothesis. Here a sufficient condition is given for the score test statistic to have this form and for the components to have this simple and convenient moment interpretation. Alternative approaches, using generalised score tests, are given for when the sufficient condition is not satisfied. This enables the construction of convenient tests of fit for distributions not from exponential families of distributions, such as the logistic and extreme value distributions.