Research Article
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Year 2018, Volume: 47 Issue: 2, 347 - 363, 01.04.2018

Abstract

References

  • Acitas, S. Factorial designs in the presence of covariates, Unpublished M.S. thesis, Anadolu University, Eskisehir, Turkey, 2010.
  • Acitas, S. and Senoglu, B. Robust factorial ANCOVA: The case of one covariate, New Developments in Theory and Applications of Statistics: An International Conference in Memory of Professor Moti Lal Tiku (NEDETAS), Ankara, Turkey, 2011.
  • Bhattacharrya, G.K. The asymptotics of maximum likelihood and related estimators based on type II censored data, Journal of the American Statistical Association 80, 398–404, 1985.
  • Bowman, K.O. and Shenton, L.R.Weibull distributions when the shape parameter is defined, Computational Statistics & Data Analysis 36, 299–310, 2001.
  • Box, G.E.P. Nonnormality and tests on variances, Biometrika 40, 318–335, 1953.
  • Box, G.E.P. and Tiao, G.C. A note on criterion robustness and inference robustness, Biometrika 51, 169–173, 1964.
  • Ekiz, U.O. and Ekiz, M. A small-sample correction factor for S-estimators,, Journal of Statistical Computation and Simulation 85, 794-801, 2015.
  • Fisher, R.A. The Design of Experiments (Edinburgh: Oliver & Boyd, 1935).
  • Geary, R.C. Testing for normality, Biometrika 34, 209-242, 1947.
  • Huber, P.J. Robust Statistics (Wiley, 1981).
  • Islam, M.Q., Tiku, M.L., and Yildirim, F. Nonnormal regression I. Skew distributions, Communications in Statistics–Theory and Methods 30, 993–1020, 2001.
  • Islam, M.Q. and Tiku, M.L. Multiple Linear Regression Model Under Nonnormality, Communications in Statistics–Theory and Methods 33, 2443–2467, 2004.
  • Kantar, Y.M. and Senoglu, B. A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter, Computers & Geosciences 34, 1900– 1909, 2008.
  • Kendall, M.G. and Stuart, A. The Advanced Theory of Statistics (Charles Griffin, 1979).
  • Montgomery, D.C. Design and Analysis of Experiments, Fifth edition (John Wiley & Sons, 2001).
  • Puthenpura, S. and Sinha, N.K. Modified maximum likelihood method for the robust estimation of system parameters from very noisy data, Automatica 22, 231–235, 1986.
  • Senoglu, B. Experimental design under nonnormality: skew distributions, PhD thesis, METU, Ankara, Turkey, 2000.
  • Senoglu, B. and Tiku, M.L. Analysis of variance in experimental design with nonnormal error distributions, Communications in Statistics–Theory and Methods 30, 1335–1352, 2001.
  • Senoglu, B. and Tiku, M.L. Linear contrasts in experimental design with non-identical error distributions, Biometrical Journal 44, 359–374, 2002.
  • Senoglu, B. Robust 2k factorial design with Weibull error distributions, Journal of Applied Statistics 32, 1051–1066, 2005.
  • Senoglu, B. Estimating parameters in one-way analysis of covariance model with shorttailed symmetric error distributions, Journal of Computational and Applied Mathematics 201, 275–283, 2007.
  • Senoglu, B. and Avcioglu, M.D. Analysis of Covariance with Non-normal Errors, International Statistical Review 77, 366–377, 2009.
  • Senoglu, B. Robust estimation and hypothesis testing in 2k factorial design, Commun. Fac. Sci. Univ. Ank. Series A1 56, 39–50, 2009.
  • Senoglu, B. and Acitas, S. Statistical Experimental Design: Fixed Effect Models (Nobel, 2010).
  • Senoglu, B., Acitas, S. and Kasap, P. Robust 2k Factorial Design: Non-Normal Symmetric Distributions Errors, Pakistan Journal of Statistics 28, 93–114, 2011.
  • Tiku, M.L. Estimating the mean and standard deviation from a censored normal sample, Biometrika 54, 155–165, 1967.
  • Tiku, M.L. Estimating the parameters of normal and logistic distributions from a censored normal sample, Austral. J. Stat. 10, 64–74, 1968.
  • Tiku, M.L. and Suresh, R.P. A new method of estimation for location and scale parameters, Journal of Statistical Planning and Inference, 30(2), 281-292, 1992.
  • Tiku, M.L. and Kumra, S. Expected values and variances and covariances of order statistics for a family of symmetric distributions (Student’s t), Selected Tables in Mathematical Statistics 8, 141–270 American Mathematical Society, Providence RI, USA, 1981.
  • Vaughan, D.C. On the Tiku-Suresh method of estimation, Communications in Statistics– Theory and Methods 21, 451–469, 1992.
  • Tiku, M.L. and Vaughan, D.C. Logistic and nonlogistic density functions in binary regression with nonstochastic covariates, Journal of the American Statistical Association 39, 883–898, 1997.
  • Tiku, M.L., Islam, M.Q. and Selcuk, S. Nonnormal Regression. II. Symmetric Distributions, Communications in Statistics–Theory and Methods 30, 1021–1045, 2001.
  • Wildt, A.R. and Ahtola, O.T. Analysis of covariance (Sage Publications, 1978).
  • Vaughan, D.C. and Tiku, M.L. Estimation and Hypothesis Testing for a Non-normal Bivariate Distribution with Applications, J. Mathematical and Computer Modelling 32, 53–67, 2000.
  • Vaughan, D.C. The generalized secant hyperbolic distribution and its properties, Communications in Statistics–Theory and Methods 31, 219–238, 2002.
  • Yates, F. Design and analysis of factorial experiments, Tech. Comm. No. 35, Imperial Bureau of Soil Sciences, London, 1937.

Robust factorial ANCOVA with LTS error distributions

Year 2018, Volume: 47 Issue: 2, 347 - 363, 01.04.2018

Abstract

In this study, parameter estimation and hypotheses testing in the balanced factorial analysis of covariance (ANCOVA) model, when the distribution of error terms is long-tailed symmetric (LTS) are considered. The unknown model parameters are estimated using the methodology known as modified maximum likelihood (MML). New test statistics based on these estimators are also proposed for testing the main effects, interaction effect and slope parameter. Assuming LTS distributions for the error term, a Monte-Carlo simulation study is conducted to compare the efficiencies of MML estimators with corresponding least squares (LS) estimators. Power and the robustness properties of the proposed test statistics are also compared with traditional normal theory test statistics. The results of the simulation study show that MML estimators are more efficient than corresponding LS estimators. Furthermore, proposed test statistics are shown to be more powerful and robust than normal theory test statistics. In the application part, a data set, taken from the literature, is analyzed to show the implementation of the methodology presented in the study.

References

  • Acitas, S. Factorial designs in the presence of covariates, Unpublished M.S. thesis, Anadolu University, Eskisehir, Turkey, 2010.
  • Acitas, S. and Senoglu, B. Robust factorial ANCOVA: The case of one covariate, New Developments in Theory and Applications of Statistics: An International Conference in Memory of Professor Moti Lal Tiku (NEDETAS), Ankara, Turkey, 2011.
  • Bhattacharrya, G.K. The asymptotics of maximum likelihood and related estimators based on type II censored data, Journal of the American Statistical Association 80, 398–404, 1985.
  • Bowman, K.O. and Shenton, L.R.Weibull distributions when the shape parameter is defined, Computational Statistics & Data Analysis 36, 299–310, 2001.
  • Box, G.E.P. Nonnormality and tests on variances, Biometrika 40, 318–335, 1953.
  • Box, G.E.P. and Tiao, G.C. A note on criterion robustness and inference robustness, Biometrika 51, 169–173, 1964.
  • Ekiz, U.O. and Ekiz, M. A small-sample correction factor for S-estimators,, Journal of Statistical Computation and Simulation 85, 794-801, 2015.
  • Fisher, R.A. The Design of Experiments (Edinburgh: Oliver & Boyd, 1935).
  • Geary, R.C. Testing for normality, Biometrika 34, 209-242, 1947.
  • Huber, P.J. Robust Statistics (Wiley, 1981).
  • Islam, M.Q., Tiku, M.L., and Yildirim, F. Nonnormal regression I. Skew distributions, Communications in Statistics–Theory and Methods 30, 993–1020, 2001.
  • Islam, M.Q. and Tiku, M.L. Multiple Linear Regression Model Under Nonnormality, Communications in Statistics–Theory and Methods 33, 2443–2467, 2004.
  • Kantar, Y.M. and Senoglu, B. A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter, Computers & Geosciences 34, 1900– 1909, 2008.
  • Kendall, M.G. and Stuart, A. The Advanced Theory of Statistics (Charles Griffin, 1979).
  • Montgomery, D.C. Design and Analysis of Experiments, Fifth edition (John Wiley & Sons, 2001).
  • Puthenpura, S. and Sinha, N.K. Modified maximum likelihood method for the robust estimation of system parameters from very noisy data, Automatica 22, 231–235, 1986.
  • Senoglu, B. Experimental design under nonnormality: skew distributions, PhD thesis, METU, Ankara, Turkey, 2000.
  • Senoglu, B. and Tiku, M.L. Analysis of variance in experimental design with nonnormal error distributions, Communications in Statistics–Theory and Methods 30, 1335–1352, 2001.
  • Senoglu, B. and Tiku, M.L. Linear contrasts in experimental design with non-identical error distributions, Biometrical Journal 44, 359–374, 2002.
  • Senoglu, B. Robust 2k factorial design with Weibull error distributions, Journal of Applied Statistics 32, 1051–1066, 2005.
  • Senoglu, B. Estimating parameters in one-way analysis of covariance model with shorttailed symmetric error distributions, Journal of Computational and Applied Mathematics 201, 275–283, 2007.
  • Senoglu, B. and Avcioglu, M.D. Analysis of Covariance with Non-normal Errors, International Statistical Review 77, 366–377, 2009.
  • Senoglu, B. Robust estimation and hypothesis testing in 2k factorial design, Commun. Fac. Sci. Univ. Ank. Series A1 56, 39–50, 2009.
  • Senoglu, B. and Acitas, S. Statistical Experimental Design: Fixed Effect Models (Nobel, 2010).
  • Senoglu, B., Acitas, S. and Kasap, P. Robust 2k Factorial Design: Non-Normal Symmetric Distributions Errors, Pakistan Journal of Statistics 28, 93–114, 2011.
  • Tiku, M.L. Estimating the mean and standard deviation from a censored normal sample, Biometrika 54, 155–165, 1967.
  • Tiku, M.L. Estimating the parameters of normal and logistic distributions from a censored normal sample, Austral. J. Stat. 10, 64–74, 1968.
  • Tiku, M.L. and Suresh, R.P. A new method of estimation for location and scale parameters, Journal of Statistical Planning and Inference, 30(2), 281-292, 1992.
  • Tiku, M.L. and Kumra, S. Expected values and variances and covariances of order statistics for a family of symmetric distributions (Student’s t), Selected Tables in Mathematical Statistics 8, 141–270 American Mathematical Society, Providence RI, USA, 1981.
  • Vaughan, D.C. On the Tiku-Suresh method of estimation, Communications in Statistics– Theory and Methods 21, 451–469, 1992.
  • Tiku, M.L. and Vaughan, D.C. Logistic and nonlogistic density functions in binary regression with nonstochastic covariates, Journal of the American Statistical Association 39, 883–898, 1997.
  • Tiku, M.L., Islam, M.Q. and Selcuk, S. Nonnormal Regression. II. Symmetric Distributions, Communications in Statistics–Theory and Methods 30, 1021–1045, 2001.
  • Wildt, A.R. and Ahtola, O.T. Analysis of covariance (Sage Publications, 1978).
  • Vaughan, D.C. and Tiku, M.L. Estimation and Hypothesis Testing for a Non-normal Bivariate Distribution with Applications, J. Mathematical and Computer Modelling 32, 53–67, 2000.
  • Vaughan, D.C. The generalized secant hyperbolic distribution and its properties, Communications in Statistics–Theory and Methods 31, 219–238, 2002.
  • Yates, F. Design and analysis of factorial experiments, Tech. Comm. No. 35, Imperial Bureau of Soil Sciences, London, 1937.
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Statistics
Authors

Sükrü Acıtas This is me

Birdal Senoglu

Publication Date April 1, 2018
Published in Issue Year 2018 Volume: 47 Issue: 2

Cite

APA Acıtas, S., & Senoglu, B. (2018). Robust factorial ANCOVA with LTS error distributions. Hacettepe Journal of Mathematics and Statistics, 47(2), 347-363.
AMA Acıtas S, Senoglu B. Robust factorial ANCOVA with LTS error distributions. Hacettepe Journal of Mathematics and Statistics. April 2018;47(2):347-363.
Chicago Acıtas, Sükrü, and Birdal Senoglu. “Robust Factorial ANCOVA With LTS Error Distributions”. Hacettepe Journal of Mathematics and Statistics 47, no. 2 (April 2018): 347-63.
EndNote Acıtas S, Senoglu B (April 1, 2018) Robust factorial ANCOVA with LTS error distributions. Hacettepe Journal of Mathematics and Statistics 47 2 347–363.
IEEE S. Acıtas and B. Senoglu, “Robust factorial ANCOVA with LTS error distributions”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 2, pp. 347–363, 2018.
ISNAD Acıtas, Sükrü - Senoglu, Birdal. “Robust Factorial ANCOVA With LTS Error Distributions”. Hacettepe Journal of Mathematics and Statistics 47/2 (April 2018), 347-363.
JAMA Acıtas S, Senoglu B. Robust factorial ANCOVA with LTS error distributions. Hacettepe Journal of Mathematics and Statistics. 2018;47:347–363.
MLA Acıtas, Sükrü and Birdal Senoglu. “Robust Factorial ANCOVA With LTS Error Distributions”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 2, 2018, pp. 347-63.
Vancouver Acıtas S, Senoglu B. Robust factorial ANCOVA with LTS error distributions. Hacettepe Journal of Mathematics and Statistics. 2018;47(2):347-63.