Properties of Correlation Determination Based on Qn Estimator for Standard Logistic Distribution with Contamination
DOI:
https://doi.org/10.33102/mjosht.574Keywords:
Qn estimator, goodness of fit, correlation of determination, contamination, robust estimatorAbstract
Robust statistical methods have become increasingly important in the presence of contaminated data, particularly in regression-based goodness of fit testing. The classical coefficient of determination is highly sensitive to deviations from assumptions such as normality and the absence of contaminated data. To address this limitation, this study investigates the properties of a robust coefficient of determination derived from the Qn estimator, which is a high-breakdown-point scale estimator. This research focuses on the behaviour of the sampling distribution under the standard logistic distribution with varying contamination structures. A comprehensive simulation study is conducted under different sample sizes, contamination levels and distributions of the contaminants. Bootstrap bias-corrected and accelerated (BCa) confidence intervals are constructed to evaluate the estimator’s statistical properties in the presence of contaminated data. Results demonstrate that contamination significantly influences skewness and variability, although the estimator remains bounded and robust.
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[1] G. L. Shevlyakov and P. O. Smirnov, “Robust estimation of the correlation coefficient: An attempt of survey,” Austrian Journal of Statistics, vol. 40, no. 1–2, pp. 147–156, 2011.
[2] G. L. Shevlyakov and P. O. Smirnov, “Robust estimation of the correlation coefficient: An attempt of survey,” Austrian Journal of Statistics, vol. 40, no. 1–2, pp. 147–156, 2011.
[3] P. J. Rousseeuw and C. Croux, “Alternatives to the median absolute deviation,” Journal of the American Statistical Association, vol. 88, no. 424, pp. 1273–1283, Dec. 1993.
[4] L. Giraitis, Y. Li, and P. C. B. Phillips, “Robust inference on correlation under general heterogeneity,” Journal of Econometrics, vol. 240, no. 1, Art. no. 105691, Mar. 2024.
[5] J. Qiu, Z. Li, and J. Yao, “Robust estimation for number of factors in high-dimensional factor modeling via Spearman correlation matrix,” Journal of the American Statistical Association, vol. 120, no. 550, pp. 1139–1151, 2025.
[6] S. Jang and C. Choi, “FORCE: Fast outlier-robust correlation estimation via streaming quantile approximation for high-dimensional data streams,” Mathematics, vol. 14, no. 1, Art. no. 191, Jan. 2026.
[7] J. Koerts and A. P. J. Abrahamse, “The correlation coefficient in the general linear model,” European Economic Review, vol. 1, pp. 401–427, 1970.
[8] A. C. Rencher and F. C. Pun, “Inflation of R² in best subset regression,” Technometrics, vol. 22, no. 1, pp. 49–53, 1980.
[9] J. S. Cramer, “Mean and variance of R² in small and moderate samples,” Journal of Econometrics, vol. 35, no. 2–3, pp. 253–266, 1987.
[10] M. L. Carrodus and D. E. A. Giles, “The exact distribution of R² when regression disturbances are autocorrelated,” Economics Letters, vol. 38, no. 4, pp. 375–380, 1992.
[11] G. Meepagala, “The small sample properties of R² in a misspecified regression model with stochastic regressors,” Economics Letters, vol. 40, no. 1, pp. 1–6, 1992.
[12] K. Ohtani and H. Hasegawa, “On small sample properties of R² in a linear regression model with multivariate t errors and proxy variables,” Econometric Theory, vol. 9, no. 4, pp. 504–515, 1993.
[13] K. Ohtani, “The density function of R² and adjusted R², and their risk performance under asymmetric loss in mis specified linear regression models,” Economic Modelling, vol. 11, no. 4, pp. 463–471, 1994.
[14] A. K. Srivastava and A. Ullah, “The coefficient of determination and its adjusted version in linear regression models,” Econometric Reviews, vol. 14, no. 2, pp. 229–240, 1995.
[15] K. Ohtani, “Bootstrapping R² and adjusted R² in regression analysis,” Economic Modelling, vol. 17, no. 4, pp. 473–483, 2000.
[16] I. S. Helland, “On the interpretation and use of R² in regression analysis,” Biometrics, vol. 43, no. 1, pp. 61–69, 1987.
[17] Q. Heng and K. Lange, “Bootstrap estimation of the proportion of outliers in robust regression,” Statistics and Computing, vol. 35, no. 1, Art. no. 3, 2025.
[18] M. Usman, M. Kashif, M. Z. Iqbal, and G. Farid,
“Evaluating robust process capability indices for Weibull process in presence of outliers and their bootstrap confidence intervals,” Journal of Statistical Computation and Simulation, vol. 95, no. 16, pp. 3497–3525, 2025.
[19] H. Choi and K. Jung, “Impact of data distribution and bootstrap setting on anomaly detection using isolation forest in process quality control,” Entropy, vol. 27, no. 7, Art. no. 761, 2025.
[20] C. Croux, C. Flandre, and G. Haesbroeck, “The breakdown behavior of the maximum likelihood estimator in the logistic regression model,” Statistics & Probability Letters, vol. 60, no. 4, pp. 377–386, 2002.
[21] H. D. Bondell and B. J. Reich, “Simultaneous regression shrinkage, variable selection, and supervised clustering of predictors with OSCAR,” Biometrics, vol. 64, no. 1, pp. 115–123, 2008.
[22] P. J. Bickel and E. L. Lehmann, “Descriptive statistics for non-parametric models. IV. Spread,” in Contributions to Statistics: Jaroslav Hájek Memorial Volume, J. Jurečková, Ed. Prague, Czech Republic: Academia, 1979, pp. 33–40.
[23] N. A. Zakaria, S. Abdullah, and N. A. Ahad, “Robust correlation coefficient based on estimator,” AIP Conference Proceedings, pp. 1–6, 2017.
[24] F. Maarof, L. F. Peng, and N. A. Ibrahim, “Bootstrapping the confidence intervals for samples from contaminated standard logistic distribution,” Pertanika Journal of Science & Technology, vol. 18, no. 1, pp. 209–221, 2010.
[25] F. P. Lim, I. Mohamed, N. Daud, and S. L. Goh, “Comparison of outlier detection methods in standard 2×2 crossover design,” Sains Malaysiana, vol. 45, no. 3, pp. 499–506, 2016.
[26] F. P. Lim, I. Mohamed, A. I. N. Ibrahim, S. L. Goh, N. A. Mohamed, and A. Rahman, “Outlier detection in 2 × 2 crossover design using Bayesian framework,” Sains Malaysiana, vol. 48, no. 4, pp. 893–899, 2019.
[27] R. B. D’Agostino and M. A. Stephens, Goodness-of-Fit Techniques. New York, NY, USA: Marcel Dekker, 1986.
[28] B. Efron, “Better bootstrap confidence intervals,” Journal of the American Statistical Association, vol. 82, no. 397, pp. 171–200, 1987.
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Copyright (c) 2026 Nor Adilla Che Ahmad, Lim Fong Peng, Yow Kai Siong, Yap Hong Keat
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