International Journal of Nonlinear Analysis and Applications
New estimation method to reduce the high leverage points effect in quantile regression
Document Type : Research Paper
Authors
Department of Statistics, Faculty of Administration and Economics, University of Al-Qadisyiah, Iraq
Abstract
Quantile regression is a powerful statistical method for modeling and analyzing the impact of explanatory and response variables at different points in the conditional distribution of the response variable. Many research papers have indicated that Quantile Regression (QR) estimator is only resistant to vertical outliers. Quantile regression like other regression M-estimators and Least Absolute Deviation LAD can be very sensitive to outliers in explanatory variables (Leverage Points). To~overcome this drawback, at first, we have to use a robust, effective and efficient method to identify high leverage points if there is masking and swamping problems. In literature, the usage of Generalized M-estimator (GM-estimator) is proposed to estimate the unknown parameters against high leverage points. In this paper, we proposed weighted~method's the generalized- M for quantile regression namely (GMQu), and improve the algorithm of this method by adapting the Improved Diagnostic Robust Generalized Potential (IDRGP) method. So that the calculation of the initial weights in this algorithm depends on (IDRGP), we're going to symbolize that method by (GMQuID). Simulation study and real data are considered to verify the performance of our proposed methods compared to other methods.
Keywords
[1] M. Abosaooda, W.J. Majid, E.A.Hussein, A.T. Jalil, M.M. Kadhim, M.M. Abdullah and H.A. Almashhadani,
Role of vitamin C in the protection of the gum and implants in the human body: theoretical and experimental
studies, Int. J. Corros. Scale Inhib. 10 (2021), no. 3, 1213–1229.
[2] J. Adrover, R. A. Maronna and V. J. Yohai, Robust regression quantiles, J. Statist. Plann. Infer. 122 (2004), no.
1-2, 187–202.
[3] R. Andersen, Modern methods for robust regression, Sage, 2008.
[4] S.H. Dilfy, M.J. Hanawi, A.W. Al-bideri and A.T. Jalil, Determination of chemical composition of cultivated
mushrooms in iraq with spectrophotometrically and high performance liquid chromatographic, J. Green Engin. 10
(2020), 6200–6216.
[5] D.L. Donoho and P.J. Huber, The notion of breakdown point, A Festschrift for Erich L. Lehmann, 1983.
[6] S.P. Ellis and S. Morgenthaler, Leverage and breakdown in L 1 regression, J. Amer. Statist. Assoc. 87 (1992), no.
417, 143–148.
[7] A. Giloni, J.S. Simonoff and B. Sengupta, Robust weighted LAD regression, Comput. Statist. Data Anal. 50
(2006), no. 11, 3124–3140.
[8] M.M.R. Habshah Norazan and A.H.M. Rahmatullah Imon, The performance of diagnostic-robust generalized
potentials for the identification of multiple high leverage points in linear regression, J. Appl. Statist. 36 (2009),
no. 5, 507–520.
[9] A.S. Hadi, A new measure of overall potential influence in linear regression, Comput. Statist. Data Anal. 14
(1992), 1–27.
[10] F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, The approach based on influence functions,
Robust Statistics, Wiley, 1986.
[11] Z.K. Hanan, M.B. Saleh, E.H. Mezal and A.T. Jalil, Detection of human genetic variation in VAC14 gene by
ARMA-PCR technique and relation with typhoid fever infection in patients with gallbladder diseases in Thi-Qar
province/Iraq, Materials Today, Proceedings, 2021.
[12] X. He, J. Jureˇckov´a, R. Koenker and S. Portnoy, Tail behavior of regression estimators and their breakdown points,
Econometrica 1990 (1990) 1195–1214.
[13] R.W. Hill, Robust regression when there are outliers in the carriers, Doctoral Dissertation, Harvard University,
1977.
[14] D.C. Hoaglin and R.E. Welsch, The hat matrix in regression and ANOVA, Amer. Statist. 32 (1978), 17–22.
[15] M. Hubert and P.J. Rousseeuw, The catline for deep regression, J. Multivariate Anal. 66 (1998), no. 2, 270–296.
[16] A.T. Jalil, A.H. D. Al-Khafaji, A. Karevskiy, S.H. Dilfy and Z.K. Hanan, Polymerase chain reaction technique
for molecular detection of HPV16 infections among women with cervical cancer in Dhi-Qar Province, Materials
Today: Proceedings, 2021.
[17] A. T. Jalil, W.R. Kadhum, M.U. Faryad Khan, Cancer stages and demographical study of HPV16 in gene L2 isolated from cervical cancer in Dhi-Qar province, Iraq, Appl. Nanosci, 2021.
[18] A. T. Jalil, S. H. Dilfy, A. Karevskiy and N. Najah, Viral Hepatitis in Dhi-Qar Province: Demographics and
Hematological Characteristics of Patients, Int. J. Pharmac. Res. 12 (2020), no. 1.
[19] A.T. Jalil, M.T. Shanshool, S.H. Dilfy, M.M. Saleh and A.A. Suleiman, Hematological and serological parameters
for detection of COVID-19, J. Microbio. Biotechnol. Food Sci. 11 (2022), no. 4, 4229–4229
[20] J. Jumintono, S. Alkubaisy, D. Y´anez Silva, K. Singh, A. Turki Jalil, S. Mutia Syarifah, Y. Fakri Mustafa, I.Mikolaychik, L. Morozova and M. Derkho, The effect of cystamine on sperm and antioxidant parameters of ram
semen stored at 4 °c for 50 hours, Arch. Razi Instit. 76 (2021), no. 4, 115.
[21] R. Koenker, Quantile regression, Cambridge University Press, Cambridge, UK., 2005.
[22] R. Koenker, and G. Bassett Jr, Regression quantiles, Econometrica J. Econ. Soc. 1978 (1978), 33–50.
[23] W.S. Krasker and R.E. Welsch, Efficient bounded-influence regression estimation, J. Amer. Statist. Assoc. 77
(1982), no. 379, 595–604.
[24] A.M. Leroy and P.J. Rousseeuw, Robust regression and outlier detection, Wiley series in probability and mathematical statistics, 1987.
[25] F. Marofi, O. Abdul-Rasheed, H. Sulaiman Rahman, H. Setia Budi, A.T. Jalil, Y. Valerievich and M. Jarahian,
CAR-NK cell in cancer immunotherapy: A promising frontier, Cancer Sci. 112 (2021), no. 9, 3427.
[26] F. Marofi, H.S. Rahman, Z.M.J. Al-Obaidi, A.T. Jalil, W.K. Abdelbasset, W. Suksatan and M. Jarahian, Novel
CART therapy is a ray of hope in the treatment of seriously ill AML patients, Stem Cell Res. Therapy, 12 (2021),
no. 1, 1–23.
[27] R.A. Maronna and V.J. Yohai, Asymptotic behavior of general M-estimates for regression and scale with random
carriers, Z. Wahrscheinl. verwandte Gebiete 58 (1981), no. 1, 7-20.
[28] H. Midi, T. Alshaybawee and M. Alguraibawi, Modified least trimmed quantile regression to overcome effects of
leverage points, Math. Probl. Engin. 2020 (2020).
[29] S. Moghadasi, M. Elveny and H.S. Rahman, A paradigm shift in cell-free approach: the emerging role of MSCsderived exosomes in regenerative medicine, J. Transl. Med. 19 (2021), no. 302.
[30] A.M. Mohammad, Rbust estimation methods and robust multicollinearity diagnostics for multiple regression model
in the presence of high leverage collinearity -influential observation, Thesis submitted to the School of Graduate
Studies, UPM., 2015.
[31] N.M. Neykov, P. C´ıˇzek, P. Filzmoser and P.M. Neytchev, ˇ The least trimmed quantile regression, Comput. Statist.
Data Anal. 56 (2012), no. 6, 1757–1770.
[32] N. Ngafwan, H. Rasyid, E.S. Abood, W.K. Abdelbasset, S.G. Al-Shawi, D. Bokov and A.T. Jalil, Study on novel
fluorescent carbon nanomaterials in food analysis, Food Sci. Technol. 42 (2022), 37821.
[33] A.H.M. Rahmatullah Imon, Identifying multiple influential observations in linear regression, J. Appl. Statist. 32
(2005), no. 9, 929–946.
[34] A.B. Roomi, G. Widjaja, D. Savitri, A. Turki Jalil, Y. Fakri Mustafa, L. Thangavelu and S. Aravindhan, SnO2:
Au/Carbon quantum dots nanocomposites: synthesis, characterization, and antibacterial activity, J. Nanostruct.
11 (2021), no. 3, 514–523.
[35] P. Rousseeuw and B. Van Zomeren, Unmasking multivariate outliers and leverage points, J. Amer. Statist. Assoc.
85 (1990), 633–639.
[36] M.M. Saleh, A.T. Jalil, R.A. Abdulkereem and A.A. Suleiman, Evaluation of immunoglobulins, CD4/CD8 T
lymphocyte ratio and interleukin-6 in COVID-19 patients, Turk. J. Immunol. 8 (2020), no. 3, 129–134.
[37] I. Sarjito, M. Elveny, A.T. Jalil, A. Davarpanah, M. Alfakeer, A.A.A. Bahajjaj and M. Ouladsmane, CFD-based
simulation to reduce greenhouse gas emissions from industrial plants, Int. J. Chem. Reactor Engin. 19 (2021),
no. 11, 1179–1186.
[38] A. Turki Jalil, S. Hussain Dilfy, S. Oudah Meza, S. M. Aravindhan, M. Kadhim and M. Aljeboree, CuO/ZrO2
nanocomposites: facile synthesis, characterization and photocatalytic degradation of tetracycline antibiotic, J.
Nanostruct. 11 (2021), no. 2, 333–346.
[39] S. Vakili-Samiani, A.T. Jalil, W.K. Abdelbasset, A.V. Yumashev, V. Karpisheh, P. Jalali and F. Jadidi-Niaragh,
Targeting Wee1 kinase as a therapeutic approach in Hematological Malignancies, DNA Repair 107 (2021), 103203.
[40] G. Widjaja, A.T. Jalil, H.S. Rahman, W.K. Abdelbasset, D.O. Bokov, W. Suksatan and M. Ahmadi, Humoral
Immune mechanisms involved in protective and pathological immunity during COVID-19, Human Immunol. 82
(2021), no. 10, 733–745.
Role of vitamin C in the protection of the gum and implants in the human body: theoretical and experimental
studies, Int. J. Corros. Scale Inhib. 10 (2021), no. 3, 1213–1229.
[2] J. Adrover, R. A. Maronna and V. J. Yohai, Robust regression quantiles, J. Statist. Plann. Infer. 122 (2004), no.
1-2, 187–202.
[3] R. Andersen, Modern methods for robust regression, Sage, 2008.
[4] S.H. Dilfy, M.J. Hanawi, A.W. Al-bideri and A.T. Jalil, Determination of chemical composition of cultivated
mushrooms in iraq with spectrophotometrically and high performance liquid chromatographic, J. Green Engin. 10
(2020), 6200–6216.
[5] D.L. Donoho and P.J. Huber, The notion of breakdown point, A Festschrift for Erich L. Lehmann, 1983.
[6] S.P. Ellis and S. Morgenthaler, Leverage and breakdown in L 1 regression, J. Amer. Statist. Assoc. 87 (1992), no.
417, 143–148.
[7] A. Giloni, J.S. Simonoff and B. Sengupta, Robust weighted LAD regression, Comput. Statist. Data Anal. 50
(2006), no. 11, 3124–3140.
[8] M.M.R. Habshah Norazan and A.H.M. Rahmatullah Imon, The performance of diagnostic-robust generalized
potentials for the identification of multiple high leverage points in linear regression, J. Appl. Statist. 36 (2009),
no. 5, 507–520.
[9] A.S. Hadi, A new measure of overall potential influence in linear regression, Comput. Statist. Data Anal. 14
(1992), 1–27.
[10] F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, The approach based on influence functions,
Robust Statistics, Wiley, 1986.
[11] Z.K. Hanan, M.B. Saleh, E.H. Mezal and A.T. Jalil, Detection of human genetic variation in VAC14 gene by
ARMA-PCR technique and relation with typhoid fever infection in patients with gallbladder diseases in Thi-Qar
province/Iraq, Materials Today, Proceedings, 2021.
[12] X. He, J. Jureˇckov´a, R. Koenker and S. Portnoy, Tail behavior of regression estimators and their breakdown points,
Econometrica 1990 (1990) 1195–1214.
[13] R.W. Hill, Robust regression when there are outliers in the carriers, Doctoral Dissertation, Harvard University,
1977.
[14] D.C. Hoaglin and R.E. Welsch, The hat matrix in regression and ANOVA, Amer. Statist. 32 (1978), 17–22.
[15] M. Hubert and P.J. Rousseeuw, The catline for deep regression, J. Multivariate Anal. 66 (1998), no. 2, 270–296.
[16] A.T. Jalil, A.H. D. Al-Khafaji, A. Karevskiy, S.H. Dilfy and Z.K. Hanan, Polymerase chain reaction technique
for molecular detection of HPV16 infections among women with cervical cancer in Dhi-Qar Province, Materials
Today: Proceedings, 2021.
[17] A. T. Jalil, W.R. Kadhum, M.U. Faryad Khan, Cancer stages and demographical study of HPV16 in gene L2 isolated from cervical cancer in Dhi-Qar province, Iraq, Appl. Nanosci, 2021.
[18] A. T. Jalil, S. H. Dilfy, A. Karevskiy and N. Najah, Viral Hepatitis in Dhi-Qar Province: Demographics and
Hematological Characteristics of Patients, Int. J. Pharmac. Res. 12 (2020), no. 1.
[19] A.T. Jalil, M.T. Shanshool, S.H. Dilfy, M.M. Saleh and A.A. Suleiman, Hematological and serological parameters
for detection of COVID-19, J. Microbio. Biotechnol. Food Sci. 11 (2022), no. 4, 4229–4229
[20] J. Jumintono, S. Alkubaisy, D. Y´anez Silva, K. Singh, A. Turki Jalil, S. Mutia Syarifah, Y. Fakri Mustafa, I.Mikolaychik, L. Morozova and M. Derkho, The effect of cystamine on sperm and antioxidant parameters of ram
semen stored at 4 °c for 50 hours, Arch. Razi Instit. 76 (2021), no. 4, 115.
[21] R. Koenker, Quantile regression, Cambridge University Press, Cambridge, UK., 2005.
[22] R. Koenker, and G. Bassett Jr, Regression quantiles, Econometrica J. Econ. Soc. 1978 (1978), 33–50.
[23] W.S. Krasker and R.E. Welsch, Efficient bounded-influence regression estimation, J. Amer. Statist. Assoc. 77
(1982), no. 379, 595–604.
[24] A.M. Leroy and P.J. Rousseeuw, Robust regression and outlier detection, Wiley series in probability and mathematical statistics, 1987.
[25] F. Marofi, O. Abdul-Rasheed, H. Sulaiman Rahman, H. Setia Budi, A.T. Jalil, Y. Valerievich and M. Jarahian,
CAR-NK cell in cancer immunotherapy: A promising frontier, Cancer Sci. 112 (2021), no. 9, 3427.
[26] F. Marofi, H.S. Rahman, Z.M.J. Al-Obaidi, A.T. Jalil, W.K. Abdelbasset, W. Suksatan and M. Jarahian, Novel
CART therapy is a ray of hope in the treatment of seriously ill AML patients, Stem Cell Res. Therapy, 12 (2021),
no. 1, 1–23.
[27] R.A. Maronna and V.J. Yohai, Asymptotic behavior of general M-estimates for regression and scale with random
carriers, Z. Wahrscheinl. verwandte Gebiete 58 (1981), no. 1, 7-20.
[28] H. Midi, T. Alshaybawee and M. Alguraibawi, Modified least trimmed quantile regression to overcome effects of
leverage points, Math. Probl. Engin. 2020 (2020).
[29] S. Moghadasi, M. Elveny and H.S. Rahman, A paradigm shift in cell-free approach: the emerging role of MSCsderived exosomes in regenerative medicine, J. Transl. Med. 19 (2021), no. 302.
[30] A.M. Mohammad, Rbust estimation methods and robust multicollinearity diagnostics for multiple regression model
in the presence of high leverage collinearity -influential observation, Thesis submitted to the School of Graduate
Studies, UPM., 2015.
[31] N.M. Neykov, P. C´ıˇzek, P. Filzmoser and P.M. Neytchev, ˇ The least trimmed quantile regression, Comput. Statist.
Data Anal. 56 (2012), no. 6, 1757–1770.
[32] N. Ngafwan, H. Rasyid, E.S. Abood, W.K. Abdelbasset, S.G. Al-Shawi, D. Bokov and A.T. Jalil, Study on novel
fluorescent carbon nanomaterials in food analysis, Food Sci. Technol. 42 (2022), 37821.
[33] A.H.M. Rahmatullah Imon, Identifying multiple influential observations in linear regression, J. Appl. Statist. 32
(2005), no. 9, 929–946.
[34] A.B. Roomi, G. Widjaja, D. Savitri, A. Turki Jalil, Y. Fakri Mustafa, L. Thangavelu and S. Aravindhan, SnO2:
Au/Carbon quantum dots nanocomposites: synthesis, characterization, and antibacterial activity, J. Nanostruct.
11 (2021), no. 3, 514–523.
[35] P. Rousseeuw and B. Van Zomeren, Unmasking multivariate outliers and leverage points, J. Amer. Statist. Assoc.
85 (1990), 633–639.
[36] M.M. Saleh, A.T. Jalil, R.A. Abdulkereem and A.A. Suleiman, Evaluation of immunoglobulins, CD4/CD8 T
lymphocyte ratio and interleukin-6 in COVID-19 patients, Turk. J. Immunol. 8 (2020), no. 3, 129–134.
[37] I. Sarjito, M. Elveny, A.T. Jalil, A. Davarpanah, M. Alfakeer, A.A.A. Bahajjaj and M. Ouladsmane, CFD-based
simulation to reduce greenhouse gas emissions from industrial plants, Int. J. Chem. Reactor Engin. 19 (2021),
no. 11, 1179–1186.
[38] A. Turki Jalil, S. Hussain Dilfy, S. Oudah Meza, S. M. Aravindhan, M. Kadhim and M. Aljeboree, CuO/ZrO2
nanocomposites: facile synthesis, characterization and photocatalytic degradation of tetracycline antibiotic, J.
Nanostruct. 11 (2021), no. 2, 333–346.
[39] S. Vakili-Samiani, A.T. Jalil, W.K. Abdelbasset, A.V. Yumashev, V. Karpisheh, P. Jalali and F. Jadidi-Niaragh,
Targeting Wee1 kinase as a therapeutic approach in Hematological Malignancies, DNA Repair 107 (2021), 103203.
[40] G. Widjaja, A.T. Jalil, H.S. Rahman, W.K. Abdelbasset, D.O. Bokov, W. Suksatan and M. Ahmadi, Humoral
Immune mechanisms involved in protective and pathological immunity during COVID-19, Human Immunol. 82
(2021), no. 10, 733–745.
)
Volume 13, Issue 2
July 2022Pages 1325-1333
- Receive Date: 08 November 2021
- Revise Date: 20 December 2021
- Accept Date: 02 January 2022