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the median of all the pairwise slopes between n points has a numerical complexity of O ( n 2 ) $\text{O}\left({n}^{2}\right)$ , better algorithms have only a complexity of O ( n ln n ) $\text{O}\left(n\text{ln}n\right)$ . The problem can be shown to be equivalent to that of computing Kendall’s tau, for which an efficient solution was given by Knight in 1966 [ 21 ]. It is based on the observation that the evaluation of Kendall’s τ ( X ″ ,   Y ″ ) $\tau \left({X}^{{\prime\prime}},\,{Y}^{{\prime\prime}}\right)$ is equivalent to that of counting the inversions

] Derumigny, A. and J.-D. Fermanian (2019). A classification point-of-view about conditional Kendall’s tau. Comput. Statist. Data Anal. 135 , 70–94. [10] Dony, J. and D. M. Mason (2008). Uniform in bandwidth consistency of conditional U-statistics. Bernoulli 14 (4), 1108–1133. [11] Einmahl, U. and D. M. Mason (2005). Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33 (3), 1380–1403. [12] Fermanian, J.-D. and O. Lopez (2018). Single-index copulas. J. Multivariate Anal. 165 , 27–55. [13] Fermanian, J.-D. and M. H. Wegkamp (2012). Time

Abstract

Several successful approaches to structure determination of hierarchical Archimedean copulas (HACs) proposed in the literature rely on agglomerative clustering and Kendall’s correlation coefficient. However, there has not been presented any theoretical proof justifying such approaches. This work fills this gap and introduces a theorem showing that, given the matrix of the pairwise Kendall correlation coefficients corresponding to a HAC, its structure can be recovered by an agglomerative clustering technique.

References [1] Berg, D. (2009). Copula goodness-of-fit testing: an overview and power comparison. Eur. J. Finance 15(7-8), 675-701. [2] Bilodeau, M. and D. Brenner (1999). Theory of Multivariate Statistics. Springer, New York. [3] Blomqvist, N. (1950). On a measure of dependence between two random variables. Ann. Math. Stat. 21(4), 593-600. [4] Bücher, A. (2011). Statistical Inference for Copulas and Extremes. PhD thesis, Ruhr-Universität Bochum, Germany. [5] Dengler, B. (2010). On the Asymptotic Behaviour of the Estimator of Kendall’s Tau. PhD thesis, Vienna

Abstract

A statistical method of defining the impact of real estate attributes based on individual capacities of Hellwig’s information carriers was proposed in the paper. The method may be used for defining the impact of attributes in the Szczecin algorithm of real estate mass appraisal. The proposed procedure refers to the so-called Hellwig’s method, used for the selection of explanatory variables in an econometric model. In the case of real estate attributes, we typically deal with variables measured on an ordinal scale. Therefore, Kendall’s tau coefficients (tau a, tau b, tau c) will be applied in order to determine the strength and direction of a relation between variables. These coefficients enable the measuring of the strength and direction of a relation between variables measured on an ordinal scale. After valuating proper matrices of Kendall’s tau coefficients, individual capacities of Hellwig’s information carriers were determined, on the basis of which the impact of attributes in the Szczecin algorithm of real estate mass appraisal was defined. The proposed procedure was supported with an empirical example using a real data base that comprises 99 undeveloped land properties of industrial designation, including depots, storehouses, warehouses, and yards. After determining the impact of each attribute, the Szczecin mass appraisal algorithm was used to evaluate values. The obtained real estate values were very close to the values obtained by real estate experts.

, Academic Press, London, 1979. [11] NUSSBAUM,M.: An asymptotic minimax risk bound for estimation of a linear functional relationship, J. Multivariate Anal. 14 (1984), 300-314. [12] SEN, P. K.: Estimates of the regression coefficient based on Kendall’s tau, J. Amer. Statist. Assoc. 63 (1968), 1379-1389. [13] SEN, P. K.-SALEH, A. K. MD. E.: The Theil-Sen estimator in a measurement error perspective. In: Nonparametrics and robustness in modern statistical inference and time series analysis: Festschrift for Jana Jureˇckov´a, Inst. Math. Stat. Collect., Vol. 7, Inst. Math

References [1] Alsina, C., M. J. Frank, and B. Schweizer (2006). Associative Functions: Triangular Norms and Copulas. World Scientific Publishing, Hackensack NJ. [2] Barbe, P., C. Genest, K. Ghoudi, and B. Rémillard (1996). On Kendall’s process. J. Multivariate Anal. 58(2), 197-229. [3] Capéraà, P. and C. Genest (1993). Spearman’s rho is larger than Kendall’s tau for positively dependent random variables. J. Nonparametr. Stat. 2(2), 183-194. [4] Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies

conclude with a discussion in Section 5. 2 Estimation of τ Let (X1, Y1) and (X2, Y2) be two independent replications of (X, Y ), a bivariate lifetime random variable with continuous marginals SX(x) = P (X > x) and SY (y) = P (Y > y). This pair is said to be concordant if (X1−X2)(Y1−Y2) > 0 and discordant if (X1 −X2)(Y1 − Y2) < 0. Kendall’s tau (Kendall & Gibbons, 1990) is defined by τ = P{(X1 − X2)(Y1 − Y2) > 0} − P{(X1 − X2)(Y1 − Y2) < 0} = 2P{(X1 − X2)(Y1 − Y2) > 0} − 1 = E(a12b12) = 4 ∫ ∞ 0 ∫ ∞ 0 π(x, y) ∂2π(x, y) ∂x∂y dxdy − 1 (1) where aij = 2×I{Xi−Xj>0}−1, bij = 2

–Kendall trend test. Test statistics for both tests are presented in Table 4 . Table 4 Values of non-parametric tests and trend status Period Streamflow Kendall’s tau p Trend Spearman’s rho p Trend Annual Annual −0.055 0.784 ▼ −0.055 0.852 ▼ Seasonal Spring −0.209 0.298 ▼ −0.297 0.303 ▼ Summer 0.143 0.477 ▲ 0.240 0.409 ▲ Autumn −0.297 0.169 ▼ −0.437 0.118 ▼ Winter −0.099 0.622 ▼ −0.108 0.714 ▼ Monthly January −0.209 0.298 ▼ −0.262 0.366 ▼ February −0.165 0.412 ▼ −0.204 0.483 ▼ March −0.055 0.784 ▼ −0.099 0.737 ▼ April −0.209 0.298 ▼ −0.288 0.318 ▼ May −0.143 0.477 ▼ −0.143 0

Kendall’s tau (τ) correlation coefficient was used to calculate the correlations between the time changes in relative number of tourists, immigrants and patients with scabies by Croatian counties. Kendall’s tau was used due to small sample size (n=21) and non-normality of the parameter distributions. Levels of significance were set to p<0.05 or p<0.01. 3 Results 3.1 Temporal Pattern of Scabies Incidence in Croatia, 2007-2017 Our results show an increasing trend in scabies incidence across Croatia between 2007 and 2017. The incidence rate increased 6-fold during that 11