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A new approach to the BHEP tests for multivariate normality. (English) Zbl 0874.62043

Summary: Let \(X_1,\dots,X_n\) be i.i.d. random \(d\)-vectors, \(d\geq 1\), with sample mean \(\overline{X}\) and sample covariance matrix \(S\). For testing the hypothesis \(H_d\) that the law of \(X_1\) is some non-degenerate normal distribution, there is a whole class of practicable affine invariant and universally consistent tests. These procedures are based on weighted integrals of the squared modulus of the difference between the empirical characteristic function of the scaled residuals \(Y_j=S^{-1/2}(X_j-\overline{X})\) and its almost sure pointwise limit \(\text{exp}(-|t|^2/2)\) under \(H_d\). The test statistics have an alternative interpretation in terms of \(L^2\)-distances between a nonparametric kernel density estimator and the parametric density estimator under \(H_d\), applied to \(Y_1,\dots,Y_n\).
By working in the Fréchet space of continuous functions on \(\mathbb{R}^d\), we obtain a new representation of the limiting null distributions of the test statistics and show that the tests have asymptotic power against sequences of contiguous alternatives converging to \(H_d\) at the rate \(n^{-1/2}\), independent of \(d\).

MSC:

62G10 Nonparametric hypothesis testing
62H15 Hypothesis testing in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics
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