Mle-equivariance, data transformations and invariant tests of fit

We define data transformations that leave certain classes of distributions invariant, while acting in a specific manner upon the parameters of the said distributions. It is shown that under such transformations the maximum likelihood estimators behave in exactly the same way as the parameters being estimated. As a consequence goodness--of--fit tests based on standardized data obtained through the inverse of this invariant data--transformation reduce to the case of testing a standard member of the family with fixed parameter values. While presenting our results, we also provide a selective review of the subject of equivariant estimators always in connection to invariant goodness--of--fit tests. A small Monte Carlo study is presented for the special case of testing for the Weibull distribution, along with real--data illustrations.Comment: 12 pages, 1 figur

Robust estimators of ar-models : a comparison

Many regression-estimation techniques have been extended to cover the case of dependent observations. The majority of such techniques are developed from the classical least squares, M and GM approaches and their properties have been investigated both on theoretical and empirical grounds. However, the behavior of some alternative methods- with satisfactory performance in the regression case- has not received equal attention in the context of time series. A simulation study of four robust estimators for autoregressive models containing innovation or additive outliers is presented. The robustness and efficiency properties of the methods are exhibited, some finite-sample results are discussed in combination with theoretical properties and the relative merits of the estimators are viewed in connection with the outlier-generating scheme.peer-reviewe

A unified approach to goodness-of-fit testing for spherical and hyperspherical data

We propose a general and relatively simple method for the construction of goodness-of-fit tests on the sphere and the hypersphere. The method is based on the characterization of probability distributions via their characteristic function, and it leads to test criteria that are convenient regarding applications and consistent against arbitrary deviations from the model under test. We emphasize goodness-of-fit tests for spherical distributions due to their importance in applications and the relative scarcity of available methods.Comment: 29 pages, 2 figures, 6 table

Fourier-type monitoring procedures for strict stationarity

We consider model-free monitoring procedures for strict stationarity of a given time series. The new criteria are formulated as L2-type statistics incorporating the empirical characteristic function. Asymptotic as well as Monte Carlo results are presented. The new methods are also employed in order to test for possible stationarity breaks in time-series data from the financial sector

Fourier--type estimation of the power garch model with stable--paretian innovations

We consider estimation for general power GARCH models under stable--Paretian innovations. Exploiting the simple structure of the conditional characteristic function of the observations driven by these models we propose minimum distance estimation based on the empirical characteristic function of corresponding residuals. Consistency of the estimators is proved, and we obtain a singular asymptotic distribution which is concentrated on a hyperplane. Efficiency issues are explored and finite--sample results are presented as well as applications of the proposed procedures to real data from the financial markets. A multivariate extension is also considered

Fourier--type estimation of the power garch model with stable--paretian innovations

We consider estimation for general power GARCH models under stable--Paretian innovations. Exploiting the simple structure of the conditional characteristic function of the observations driven by these models we propose minimum distance estimation based on the empirical characteristic function of corresponding residuals. Consistency of the estimators is proved, and we obtain a singular asymptotic distribution which is concentrated on a hyperplane. Efficiency issues are explored and finite--sample results are presented as well as applications of the proposed procedures to real data from the financial markets. A multivariate extension is also considered

Fourier methods for analysing piecewise constant volatilities

We develop procedures for testing the hypothesis that a parameter of a distribution is constant throughout a sequence of independent random variables. Our proposals are illustrated considering the variance and the kurtosis. Under the null hypothesis of constant variance, the modulus of a Fourier type transformation of the volatility process is identically equal to one. The approach proposed utilizes this property considering a canonical estimator for this modulus under the assumption of indepen- dent and piecewise identically distributed observations with zero mean. Using blockwise estimators we introduce several test statistics resulting from different weight functions which are all given by simple explicit for- mulae. The methods are compared to other tests for constant volatility in extensive Monte Carlo experiments. Our proposals offer comparatively good power particularly in the case of multiple structural breaks and allow adequate estimation of the positions of the structural breaks. An appli- cation to process control data is given, and it is shown how the methods can be adapted to test for constancy of other quantities like the kurtosis

Characterizations of multinormality and corresponding tests of fit, including for Garch models

We provide novel characterizations of multivariate normality that incorporate both the characteristic function and the moment generating function, and we employ these results to construct a class of affine invariant, consistent and easy-to-use goodness-of-fit tests for normality. The test statistics are suitably weighted L2-statistics, and we provide their asymptotic behavior both for i.i.d. observations as well as in the context of testing that the innovation distribution of a multivariate GARCH model is Gaussian. We also study the finite-sample behavior of the new tests and compare the new criteria with alternative existing tests