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Register a new userA General Class of Weighted Rank Correlation Measures
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Added by
Ali Dolati
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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In this paper we propose a class of weighted rank correlation coefficients extending the Spearmans rho. The proposed class constructed by giving suitable weights to the distance between two sets of ranks to place more emphasis on items having low rankings than those have high rankings or vice versa. The asymptotic distribution of the proposed measures and properties of the parameters estimated by them are studied through the associated copula. A simulation study is performed to compare the performance of the proposed statistics for testing independence using asymptotic relative efficiency calculations.
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In the multivariate one-sample location model, we propose a class of flexible robust, affine-equivariant L-estimators of location, for distributions invoking affine-invariance of Mahalanobis distances of individual observations. An involved iteration process for their computation is numerically illustrated.
Chatterjee (2021) introduced a simple new rank correlation coefficient that has attracted much recent attention. The coefficient has the unusual appeal that it not only estimates a population quantity first proposed by Dette et al. (2013) that is zero if and only if the underlying pair of random variables is independent, but also is asymptotically normal under independence. This paper compares Chatterjees new correlation coefficient to three established rank correlations that also facilitate consistent tests of independence, namely, Hoeffdings $D$, Blum-Kiefer-Rosenblatts $R$, and Bergsma-Dassios-Yanagimotos $tau^*$. We contrast their computational efficiency in light of recent advances, and investigate their power against local rotation and mixture alternatives. Our main results show that Chatterjees coefficient is unfortunately rate sub-optimal compared to $D$, $R$, and $tau^*$. The situation is more subtle for a related earlier estimator of Dette et al. (2013). These results favor $D$, $R$, and $tau^*$ over Chatterjees new correlation coefficient for the purpose of testing independence.
Gini-type correlation coefficients have become increasingly important in a variety of research areas, including economics, insurance and finance, where modelling with heavy-tailed distributions is of pivotal importance. In such situations, naturally, the classical Pearson correlation coefficient is of little use. On the other hand, it has been observed that when light-tailed situations are of interest, and hence when both the Gini-type and Pearson correlation coefficients are well-defined and finite, then these coefficients are related and sometimes even coincide. In general, understanding how the correlation coefficients above are related has been an illusive task. In this paper we put forward arguments that establish such a connection via certain regression-type equations. This, in turn, allows us to introduce a Gini-type Weighted Insurance Pricing Model that works in heavy-tailed situation and thus provides a natural alternative to the classical Capital Asset Pricing Model. We illustrate our theoretical considerations using several bivariate distributions, such as elliptical and those with heavy-tailed Pareto margins.
Consider a Gaussian vector $mathbf{z}=(mathbf{x},mathbf{y})$, consisting of two sub-vectors $mathbf{x}$ and $mathbf{y}$ with dimensions $p$ and $q$ respectively, where both $p$ and $q$ are proportional to the sample size $n$. Denote by $Sigma_{mathbf{u}mathbf{v}}$ the population cross-covariance matrix of random vectors $mathbf{u}$ and $mathbf{v}$, and denote by $S_{mathbf{u}mathbf{v}}$ the sample counterpart. The canonical correlation coefficients between $mathbf{x}$ and $mathbf{y}$ are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix $Sigma_{mathbf{x}mathbf{x}}^{-1}Sigma_{mathbf{x}mathbf{y}}Sigma_{mathbf{y}mathbf{y}}^{-1}Sigma_{mathbf{y}mathbf{x}}$. In this paper, we focus on the case that $Sigma_{mathbf{x}mathbf{y}}$ is of finite rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_1geqcdotsgeq r_k>0$. We study the sample counterparts of $r_i,i=1,ldots,k$, i.e. the largest $k$ eigenvalues of the sample canonical correlation matrix $S_{mathbf{x}mathbf{x}}^{-1}S_{mathbf{x}mathbf{y}}S_{mathbf{y}mathbf{y}}^{-1}S_{mathbf{y}mathbf{x}}$, denoted by $lambda_1geqcdotsgeq lambda_k$. We show that there exists a threshold $r_cin(0,1)$, such that for each $iin{1,ldots,k}$, when $r_ileq r_c$, $lambda_i$ converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by $d_{+}$. When $r_i>r_c$, $lambda_i$ possesses an almost sure limit in $(d_{+},1]$. We also obtain the limiting distribution of $lambda_i$s under appropriate normalization. Specifically, $lambda_i$ possesses Gaussian type fluctuation if $r_i>r_c$, and follows Tracy-Widom distribution if $r_i<r_c$. Some applications of our results are also discussed.
This paper discusses a nonparametric regression model that naturally generalizes neural network models. The model is based on a finite number of one-dimensional transformations and can be estimated with a one-dimensional rate of convergence. The model contains the generalized additive model with unknown link function as a special case. For this case, it is shown that the additive components and link function can be estimated with the optimal rate by a smoothing spline that is the solution of a penalized least squares criterion.
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