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1

A Class of Weighted Rank Correlation Measures

Sanatgar Majid, Dolati Ali, Amini Mohammad

Probability and Mathematical Statistics

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2021

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Vol. 41, Fasc. 1

39--54

EN

We propose a class of weighted rank correlation measures extending Spearman’s rho. This class consists of two types of measures. The first type, which extends Blest’s rank correlation, places more emphasis on the agreement in top ranks. The second one places more emphasis on the agreement in the bottom ranks. The asymptotic distribution of the proposed measures and some of their properties are studied. A simulation study is performed to compare the performance of the proposed statistics for testing independence by using asymptotic relative efficiency calculations.

2

Asymptotic behaviour of linear rank statistics for the two-sample problem

Inglot T.

Probability and Mathematical Statistics

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2012

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Vol. 32, Fasc. 1

93--116

EN

Applying the strong approximation technique we present a unified approach to asymptotic results for multivariate linear rank statistics for the two-sample problem. We reprove asymptotic normality of these statistics under the null hypothesis and under local alternatives convergent at a moderate rate to the null hypothesis. We also provide a moderate deviation theorem for these statistics under the null hypothesis. Proofs are short and use natural argumentation.

3

Weighted quantile correlation tests for Gumbel, Weibull and Pareto families

Csörgõ S., Szabó T.

Probability and Mathematical Statistics

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2009

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Vol. 29, Fasc. 2

227--250

EN

Weighted quantile correlation tests are worked out for the Gumbel location and location-scale families. Our theoretical emphasis is on the determination of computable forms of the asymptotic distributions under the null hypotheses, which forms are based on the solution of an associated eigenvalue-eigenfunction problem. Suitable transformations then yield corresponding composite goodness-of-fit tests for the Weibull family with unknown shape and scale parameters and for the Pareto family with an unknown shape parameter. Simulations demonstrate slow convergence under the null hypotheses, and hence the inadequacy of the asymptotic critical points. Other rounds of extensive simulations illustrate the power of all three tests: Gumbel against the other extreme-value distributions, Weibull against gamma distributions, and Pareto against generalized Pareto distributions with logarithmic slow variation.

4

Asymptotic distribution of unbiased linear estimators in the presence of heavy-tailed stochastic regressors and residuals

Samorodnitsky G., Rachey S. T., Kurz-Kim J.-R., Stoyanov S.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 2

275--302

EN

Under the symmetric а-stable distributional assumption for the disturbances, Blattberg and Sargent [3] consider unbiased line- ar estimators for a regression model with non-stochastic regressors. We study both the rate of convergence to the true value and the asymptotic distribution of the normalized error of the linear unbiased estimators. By doing this, we allow the regressors to be stochastic and disturbances to be heavy-tailed with either finite or infinite variances, where the tail-thickness parameters of the regressors and disturbances may be different.

5

Asymptotic Distribution of the Mutual Variogram Estimate

Troush N., Tsekhovaya T.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2003

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Vol. 9

109--114

EN

The article deals with the problem of a statistical analysis of time series connected with the estimation of mutual variogram, a measure of spatial correlation. G. Matheron [1] has coined the term variogram, although earlier appearances of this function can be found in the scientific literature [2, 3]. The problem of estimating the mutual variogram and examination the statistical properties of this statistics has been considered in [1, 4, 5]. We present the limiting expressions of the first two moments and the higher order cumulants of the mutual variogram estimate of the second-order-stationary stochastic process with discrete time. These expressions are then used to prove the theorem concerning the asymptotic distribution of the mutual variogram estimate. The approach is similar to the approach taken in the time series literature, and the reader is referred to D. Brillinger [6] for theorems regarding the asymptotic distribution of the spectral density estimate of a time series.

6

The asymptotic consistency and efficiency of fixed-size sequential confidence sets

Różański Roman

Probability and Mathematical Statistics

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1998

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Vol. 18, Fasc. 1

19--31

EN

In the paper, a sequential confidence set based on an estimation process of a multivariate parameter is constructed. Under the assumption that the estimation process scaled by an increasing positive process has an asymptotic distribution it is proved that the sequential confidence set is asymptotically consistent and asymptotic- ally efficient. The results are applied to the sequential confidence sets based on maximum likelihood estimators of a multivariate parameter in the iid case and in the exponential class of processes.