A family of estimators for multivariate kurtosis in a nonnormal linear regression model
In this paper, we propose a new estimator for a kurtosis in a multivariate nonnormal linear regression model. Usually, an estimator is constructed from an arithmetic mean of the second power of the squared sample Mahalanobis distances between observations and their estimated values. The estimator gives an underestimation and has a large bias, even if the sample size is not small. We replace this squared distance with a transformed squared norm of the Studentized residual using a monotonic increasing function. Our proposed estimator is defined by an arithmetic mean of the second power of these squared transformed squared norms with a correction term and a tuning parameter. The correction term adjusts our estimator to an unbiased estimator under normality, and the tuning parameter controls the sizes of the squared norms of the residuals. The family of our estimators includes estimators based on ordinary least squares and predicted residuals. We verify that the bias of our new estimator is smaller than usual by constructing numerical experiments.
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Volume (Year): 98 (2007)
Issue (Month): 1 (January)
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- Yanagihara, Hirokazu, 2003. "Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 222-246, February.
- Kano, Yutaka, 1992. "Robust statistics for test-of-independence and related structural models," Statistics & Probability Letters, Elsevier, vol. 15(1), pages 21-26, September.
- Klar, Bernhard, 2002. "A Treatment of Multivariate Skewness, Kurtosis, and Related Statistics," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 141-165, October.
- Fujikoshi, Yasunori, 2000. "Transformations with Improved Chi-Squared Approximations," Journal of Multivariate Analysis, Elsevier, vol. 72(2), pages 249-263, February.
- Yasunori Fujikoshi & Takafumi Noguchi & Megu Ohtaki & Hirokazu Yanagihara, 2003. "Corrected versions of cross-validation criteria for selecting multivariate regression and growth curve models," Annals of the Institute of Statistical Mathematics, Springer, vol. 55(3), pages 537-553, September.
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