Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model

Sthitadhi Das

doi:doi:10.11648/j.ajtas.20251402.12

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Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model

Sthitadhi Das^*

Published in American Journal of Theoretical and Applied Statistics (Volume 14, Issue 2)

Received: 12 February 2025 Accepted: 3 March 2025 Published: 24 March 2025

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Abstract

Under the setup of a generalized partially linear model Y = β₁X₁+ ... + β_pX_p+ m(W₁,...,W_q) + ϵ with p parametric regressors X₁,..., X_p and q nonparametric regressors W₁,..., W_q, we are motivated to test on the independence between all the (p + q) regressors and the random error ϵ. Since one obtains unbiased prediction of the study variable of interest when the regressors under consideration are independent of the random error, such testing of independence is a vital objective indeed. Here, m(W₁,..., W_q) is a Lipschitz continuous function defined on ℝ^q→ℝ. To carry out the prescribed testing scheme, some test statistics are formed based on the nonparametric measure of association Kendall’s (1938) τ. Moreover, as an implication of the null hypothesis suggesting independence between joint (p + q) regressors and ϵ, we further modify the hypothesis as ϵ is not observable at all. Eventually, independence among the r-th order difference of estimated response and the r-th order difference of observed response is implied from the original null hypothesis. Later, the concept of V-statistic is applied to propose the test statistics based on the paired observations on the r-th differences of the estimated and observed Y. Their consistent power performances are achieved under a sequence of contiguous alternatives stating complete dependence between error and regressors while testing the independence of regressors with ϵ. Le Cam (1960)’s theory on contiguity is required to develop a sequence of contiguous alternative hypotheses in this regard. The asymptotic powers of the test statistics are evaluated with the help of their asymptotic distributions under null hypothesis of independence and contiguous alternatives. Subsequently, a data analysis is performed to substantiate the eligibility of the proposed test statistics in such testing setup.

Published in	American Journal of Theoretical and Applied Statistics (Volume 14, Issue 2)
DOI	10.11648/j.ajtas.20251402.12
Page(s)	61-76
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Generalized Partially Linear Regression Model, Nonparametric Regression Model, Kendall's τ, Measures of Association, Contiguous Alternatives, Asymptotic Power, V-statistic

References

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[3]	Bergsma, W. and Dassios, A. (2014). A consistent test of independence based on a sign covariance related to Kendall’s tau. Bernoulli, 1006-1028. https://doi.org/10.3150/13-BEJ514
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[7]	Das, S., Halder, S., and Maiti, S. I. (2023). An Extended Approach to Test of Independence between Error and Covariates under Nonparametric Regression Model. Thailand Statistician, 21(1): 19-36.
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Das, S. (2025). Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model. American Journal of Theoretical and Applied Statistics, 14(2), 61-76. https://doi.org/10.11648/j.ajtas.20251402.12

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ACS Style

Das, S. Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model. Am. J. Theor. Appl. Stat. 2025, 14(2), 61-76. doi: 10.11648/j.ajtas.20251402.12

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AMA Style

Das S. Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model. Am J Theor Appl Stat. 2025;14(2):61-76. doi: 10.11648/j.ajtas.20251402.12

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@article{10.11648/j.ajtas.20251402.12,
  author = {Sthitadhi Das},
  title = {Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {14},
  number = {2},
  pages = {61-76},
  doi = {10.11648/j.ajtas.20251402.12},
  url = {https://doi.org/10.11648/j.ajtas.20251402.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20251402.12},
  abstract = {Under the setup of a generalized partially linear model Y = β1X1+ ... + βpXp+ m(W1,...,Wq) + ϵ with p parametric regressors X1,..., Xp and q nonparametric regressors W1,..., Wq, we are motivated to test on the independence between all the (p + q) regressors and the random error ϵ. Since one obtains unbiased prediction of the study variable of interest when the regressors under consideration are independent of the random error, such testing of independence is a vital objective indeed. Here, m(W1,..., Wq) is a Lipschitz continuous function defined on ℝq→ℝ. To carry out the prescribed testing scheme, some test statistics are formed based on the nonparametric measure of association Kendall’s (1938) τ. Moreover, as an implication of the null hypothesis suggesting independence between joint (p + q) regressors and ϵ, we further modify the hypothesis as ϵ is not observable at all. Eventually, independence among the r-th order difference of estimated response and the r-th order difference of observed response is implied from the original null hypothesis. Later, the concept of V-statistic is applied to propose the test statistics based on the paired observations on the r-th differences of the estimated and observed Y. Their consistent power performances are achieved under a sequence of contiguous alternatives stating complete dependence between error and regressors while testing the independence of regressors with ϵ. Le Cam (1960)’s theory on contiguity is required to develop a sequence of contiguous alternative hypotheses in this regard. The asymptotic powers of the test statistics are evaluated with the help of their asymptotic distributions under null hypothesis of independence and contiguous alternatives. Subsequently, a data analysis is performed to substantiate the eligibility of the proposed test statistics in such testing setup.},
 year = {2025}
}

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TY  - JOUR
T1  - Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model
AU  - Sthitadhi Das
Y1  - 2025/03/24
PY  - 2025
N1  - https://doi.org/10.11648/j.ajtas.20251402.12
DO  - 10.11648/j.ajtas.20251402.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 61
EP  - 76
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20251402.12
AB  - Under the setup of a generalized partially linear model Y = β1X1+ ... + βpXp+ m(W1,...,Wq) + ϵ with p parametric regressors X1,..., Xp and q nonparametric regressors W1,..., Wq, we are motivated to test on the independence between all the (p + q) regressors and the random error ϵ. Since one obtains unbiased prediction of the study variable of interest when the regressors under consideration are independent of the random error, such testing of independence is a vital objective indeed. Here, m(W1,..., Wq) is a Lipschitz continuous function defined on ℝq→ℝ. To carry out the prescribed testing scheme, some test statistics are formed based on the nonparametric measure of association Kendall’s (1938) τ. Moreover, as an implication of the null hypothesis suggesting independence between joint (p + q) regressors and ϵ, we further modify the hypothesis as ϵ is not observable at all. Eventually, independence among the r-th order difference of estimated response and the r-th order difference of observed response is implied from the original null hypothesis. Later, the concept of V-statistic is applied to propose the test statistics based on the paired observations on the r-th differences of the estimated and observed Y. Their consistent power performances are achieved under a sequence of contiguous alternatives stating complete dependence between error and regressors while testing the independence of regressors with ϵ. Le Cam (1960)’s theory on contiguity is required to develop a sequence of contiguous alternative hypotheses in this regard. The asymptotic powers of the test statistics are evaluated with the help of their asymptotic distributions under null hypothesis of independence and contiguous alternatives. Subsequently, a data analysis is performed to substantiate the eligibility of the proposed test statistics in such testing setup.
VL  - 14
IS  - 2
ER  -

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Sthitadhi Das

Department of Mathematics, Brainware University, Kolkata, India

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Das, S. (2025). Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model. American Journal of Theoretical and Applied Statistics, 14(2), 61-76. https://doi.org/10.11648/j.ajtas.20251402.12

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ACS Style

Das, S. Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model. Am. J. Theor. Appl. Stat. 2025, 14(2), 61-76. doi: 10.11648/j.ajtas.20251402.12

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AMA Style

Das S. Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model. Am J Theor Appl Stat. 2025;14(2):61-76. doi: 10.11648/j.ajtas.20251402.12

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@article{10.11648/j.ajtas.20251402.12,
  author = {Sthitadhi Das},
  title = {Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {14},
  number = {2},
  pages = {61-76},
  doi = {10.11648/j.ajtas.20251402.12},
  url = {https://doi.org/10.11648/j.ajtas.20251402.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20251402.12},
  abstract = {Under the setup of a generalized partially linear model Y = β1X1+ ... + βpXp+ m(W1,...,Wq) + ϵ with p parametric regressors X1,..., Xp and q nonparametric regressors W1,..., Wq, we are motivated to test on the independence between all the (p + q) regressors and the random error ϵ. Since one obtains unbiased prediction of the study variable of interest when the regressors under consideration are independent of the random error, such testing of independence is a vital objective indeed. Here, m(W1,..., Wq) is a Lipschitz continuous function defined on ℝq→ℝ. To carry out the prescribed testing scheme, some test statistics are formed based on the nonparametric measure of association Kendall’s (1938) τ. Moreover, as an implication of the null hypothesis suggesting independence between joint (p + q) regressors and ϵ, we further modify the hypothesis as ϵ is not observable at all. Eventually, independence among the r-th order difference of estimated response and the r-th order difference of observed response is implied from the original null hypothesis. Later, the concept of V-statistic is applied to propose the test statistics based on the paired observations on the r-th differences of the estimated and observed Y. Their consistent power performances are achieved under a sequence of contiguous alternatives stating complete dependence between error and regressors while testing the independence of regressors with ϵ. Le Cam (1960)’s theory on contiguity is required to develop a sequence of contiguous alternative hypotheses in this regard. The asymptotic powers of the test statistics are evaluated with the help of their asymptotic distributions under null hypothesis of independence and contiguous alternatives. Subsequently, a data analysis is performed to substantiate the eligibility of the proposed test statistics in such testing setup.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Study on Efficacy of Kendall’s τ Based Test Statistic in Generalized Partially Linear Regression Model
AU  - Sthitadhi Das
Y1  - 2025/03/24
PY  - 2025
N1  - https://doi.org/10.11648/j.ajtas.20251402.12
DO  - 10.11648/j.ajtas.20251402.12
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 61
EP  - 76
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20251402.12
AB  - Under the setup of a generalized partially linear model Y = β1X1+ ... + βpXp+ m(W1,...,Wq) + ϵ with p parametric regressors X1,..., Xp and q nonparametric regressors W1,..., Wq, we are motivated to test on the independence between all the (p + q) regressors and the random error ϵ. Since one obtains unbiased prediction of the study variable of interest when the regressors under consideration are independent of the random error, such testing of independence is a vital objective indeed. Here, m(W1,..., Wq) is a Lipschitz continuous function defined on ℝq→ℝ. To carry out the prescribed testing scheme, some test statistics are formed based on the nonparametric measure of association Kendall’s (1938) τ. Moreover, as an implication of the null hypothesis suggesting independence between joint (p + q) regressors and ϵ, we further modify the hypothesis as ϵ is not observable at all. Eventually, independence among the r-th order difference of estimated response and the r-th order difference of observed response is implied from the original null hypothesis. Later, the concept of V-statistic is applied to propose the test statistics based on the paired observations on the r-th differences of the estimated and observed Y. Their consistent power performances are achieved under a sequence of contiguous alternatives stating complete dependence between error and regressors while testing the independence of regressors with ϵ. Le Cam (1960)’s theory on contiguity is required to develop a sequence of contiguous alternative hypotheses in this regard. The asymptotic powers of the test statistics are evaluated with the help of their asymptotic distributions under null hypothesis of independence and contiguous alternatives. Subsequently, a data analysis is performed to substantiate the eligibility of the proposed test statistics in such testing setup.
VL  - 14
IS  - 2
ER  -

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