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Testing distributional assumption of unit-Lindley regression model

Year 2022, Volume: 51 Issue: 3, 882 - 899, 01.06.2022

Deniz Özonur

https://doi.org/10.15672/hujms.932811

Abstract

This paper proposes smooth goodness of fit test statistic and its components to test the distributional assumption of the unit-Lindley regression model, which is useful for describing data measured between zero and one. Orthonormal polynomials on the unit-Lindley distribution, score functions and Fisher's information matrix are provided for the smooth test. Deviance and Pearson's chi-square tests are also adapted to the unit-Lindley regression model. A parametric bootstrap simulation study is conducted to compare type I errors and powers of the tests under different scenarios. Empirical findings demonstrate that the first smooth component, deviance, and chi-square tests have undesirable behavior for the unit-Lindley regression model. A real data set is analyzed by using the developed tests to show the adequacy of the unit-Lindley regression model. Model selection criteria and residual analysis prove that the unit-Lindley regression model provides a better fit than the Beta and simplex regression models for the real data set.

Keywords

Chi-square test deviance test power of test smooth test unit-Lindley distribution parametric bootstrap

References

  • [1] H. Akaike, A new look at the statistical model identification, IEEE Trans. Automat. Contr. 19 (6), 716-723, 1974.
  • [2] E. Altun, The log-weighted exponential regression model: alternative to the beta regression model, Comm. Statist. Theory Methods 50 (10), 2306-2321, 2021.
  • [3] E. Altun and G.M. Cordeiro, The unit-improved second-degree Lindley distribution: inference and regression modeling, Comput. Statist. 35 (1), 259-279, 2020.
  • [4] E. Altun, M. El-Morshedy and M.S. Eliwa, A new regression model for bounded response variable: an alternative to the beta and unit-Lindley regression models, PloS one 16 (1), e0245627, 2021.
  • [5] H.S. Bakouch, B.M. Al-Zahrani, A.A. Al-Shomrani, V.A. Marchi and F. Louzada, An extended Lindley distribution, J. Korean Statist. Soc. 41 (1), 75-85, 2012.
  • [6] W. Barreto-Souza and H.S. Bakouch, A new lifetime model with decreasing failure rate, Stats. 47 (2), 465-476, 2013.
  • [7] D.J. Best and J.C.W. Rayner, Smooth tests of fit for the Lindley distribution, Stats 1 (1), 92-97, 2018.
  • [8] J.M. Carrasco and N. Reid, Simplex regression models with measurement error, Comm. Statist. Simulation Comput. 50 (11), 3420-3435, 2021.
  • [9] D.R. Cox and E.J. Snell, A general definition of residuals, J. R. Stat. Soc. Ser. B. Stat. Methodol. 30 (2), 248-275, 1968.
  • [10] B. De Boeck, O. Thas, J.C.W. Rayner and D.J. Best, Smooth tests for the gamma distribution, J. Stat. Comput. Simul. 81 (7), 843-855, 2011.
  • [11] R.S. Defries, M.C. Hansen, J.R. Townshend, A.C. Janetos and T.R. Loveland, A new global 1km dataset of percentage tree cover derived from remote sensing, Glob. Change Biol. 6 (2), 247-254, 2000.
  • [12] A.J. Dobson and A.G. Barnett, An Introduction to Generalized Linear Models, Chapman and Hall/CRC, 2008.
  • [13] P.K. Dunn and G.K. Smyth, Randomized quantile residuals, J. Comput. Graph. Statist. 5 (3), 236-244, 1996.
  • [14] S. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (7), 799-815, 2004.
  • [15] M.E. Ghitany, D.K. Al-Mutairi, N. Balakrishnan and L.J. Al-Enezi, Power Lindley distribution and associated inference, Comput. Statist. Data Anal. 64, 20-33, 2013.
  • [16] M.E. Ghitany, J. Mazucheli, A.F.B. Menezes and F. Alqallaf, The unit-inverse Gaussian distribution: a new alternative to two-parameter distributions on the unit interval, Comm. Statist. Theory Methods 48 (14), 3423-3438, 2019.
  • [17] E.J. Hannan and B.G. Quinn, The determination of the order of an autoregression, J. R. Stat. Soc. Ser. B. Stat. Methodol. 41 (2), 190-195, 1979.
  • [18] C.M. Jarque and A.K. Bera, A test for normality of observations and regression residuals, Int. Stat. Rev. 55 (2), 163-177, 1987.
  • [19] R. Kieschnick and B.D. McCullough, Regression analysis of variates observed on (0, 1): percentages, proportions and fractions, Stat. Model. 3 (3), 193-213, 2003.
  • [20] M.Ç. Korkmaz, E. Altun, M. Alizadeh and M. El-Morshedy, The log exponential- power distribution: properties, estimations and quantile regression model, Mathematics 9 (21), 1-19, 2021.
  • [21] J.A. Koziol, Assessing multivariate normality: a compendium, Comm. Statist. Theory Methods 15 (9), 2763-2783, 1986.
  • [22] J.A. Koziol, An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives, Metrika 34 (1), 17-24, 1987.
  • [23] D.V. Lindley, Fiducial distributions and Bayes’ theorem, J. R. Stat. Soc. Ser. B. Stat. Methodol. 20 (1), 102-107, 1958.
  • [24] J. Mazucheli, A.F.B. Menezes and S. Chakraborty, On the one parameter unit-Lindley distribution and its associated regression model for proportion data, J. Appl. Stat. 46 (4), 700-714, 2019.
  • [25] P. McCullagh and J.A. Nelder, Generalized Linear Models, Chapman and Hall, 1989.
  • [26] A.M. Mousa, A.A. El-Sheikh and M.A. Abdel-Fattah, A gamma regression for bounded continuous variables, Adv. Appl. Stat. 49 (4), 305-326, 2016.
  • [27] L.R. Nakamura, P.H. Cerqueira, T.G. Ramires, R.R. Pescim, R.A. Rigby and D.M. Stasinopoulos, A new continuous distribution on the unit interval applied to modelling the points ratio of football teams, J. Appl. Stat. 46 (3), 416-431, 2019.
  • [28] J. Neyman, Smooth test for goodness of fit, Scand. Actuar. J. 1937 (3-4), 149-199, 1937.
  • [29] D. Ozonur, H.T.K. Akdur and H. Bayrak, Comparisons of tests of distributional assumption in Poisson regression model, Comm. Statist. Simulation Comput. 46 (8), 6197-6207, 2017.
  • [30] D. Özonur, F. Gökpnar, E. Gökpnar and H. Bayrak, Goodness of fit tests for Nakagami distribution based on smooth tests, Comm. Statist. Theory Methods 45 (7), 1876-1886, 2016.
  • [31] H. Poorter and L. Sack, Pitfalls and possibilities in the analysis of biomass allocation patterns in plants, Front. Plant Sci. 3 (259), 1-10, 2012.
  • [32] G. Pumi, C. Rauber and F.M. Bayer, Kumaraswamy regression model with Aranda- Ordaz link function, Test 29, 1051-1071, 2020.
  • [33] P.L. Ramos and F. Louzada, The generalized weighted Lindley distribution: properties, estimation, and applications, Cogent Math. 3 (1), 1-18, 2016.
  • [34] P.L. Ramos, F. Louzada, T.K. Shimizu and A.O. Luiz, The inverse weighted Lindley distribution: properties, estimation and an application on a failure time data, Comm. Statist. Theory Methods 48 (10), 2372-2389, 2019.
  • [35] J.C.W. Rayner and D.J. Best, Neyman-type smooth tests for location-scale families, Biometrika 73 (2), 437-446, 1986.
  • [36] J.C.W. Rayner, O. Thas and D.J. Best, Smooth Tests of Goodness of Fit: Using R, John Wiley and Sons, 2009.
  • [37] P. Rippon, Application of smooth tests of goodness of fit to generalized linear models, PhD thesis, University of Newcastle, 2013.
  • [38] G. Schwarz, Estimating the dimension of a model, Ann. Statist. 6 (2), 461-464, 1978.
  • [39] M. Smithson and J. Verkuilen, A better lemon-squeezer? Maximum likelihood regression with beta-distribuited dependent variables, Psycholog. Meth. 11 (1), 54-71, 2006.
  • [40] M.G. Swainson, A.M. Batterham, C. Tsakirides, Z.H. Rutherford and K. Hind, Prediction of whole-body fat percentage and visceral adipose tissue mass from five anthro- pometric variables, PloS one 12 (5), 1-12, 2017.
  • [41] O. Thas and J.C.W. Rayner, Smooth tests for the zero-inflated Poisson distribution, Biometrics 61 (3), 808-815, 2005.
  • [42] C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Amer. Statist. Assoc. 50 (269), 209-219, 1955.
  • [43] T.W. Yee, The VGAM package for categorical data analysis, J. Stat. Softw. 32 (10), 1-34, 2010.
  • [44] H. Zakerzadeh and A. Dolati, Generalized Lindley distribution, J. Math. Ext. 3 (2), 1-17, 2009.
  • [45] A. Zeileis, F. Cribari-Neto, B. Grün and I. Kos-midis, Beta regression in R, J. Statist. Softw. 34 (2), 1-24, 2010.

Year 2022, Volume: 51 Issue: 3, 882 - 899, 01.06.2022

Deniz Özonur

https://doi.org/10.15672/hujms.932811

Abstract

References

  • [1] H. Akaike, A new look at the statistical model identification, IEEE Trans. Automat. Contr. 19 (6), 716-723, 1974.
  • [2] E. Altun, The log-weighted exponential regression model: alternative to the beta regression model, Comm. Statist. Theory Methods 50 (10), 2306-2321, 2021.
  • [3] E. Altun and G.M. Cordeiro, The unit-improved second-degree Lindley distribution: inference and regression modeling, Comput. Statist. 35 (1), 259-279, 2020.
  • [4] E. Altun, M. El-Morshedy and M.S. Eliwa, A new regression model for bounded response variable: an alternative to the beta and unit-Lindley regression models, PloS one 16 (1), e0245627, 2021.
  • [5] H.S. Bakouch, B.M. Al-Zahrani, A.A. Al-Shomrani, V.A. Marchi and F. Louzada, An extended Lindley distribution, J. Korean Statist. Soc. 41 (1), 75-85, 2012.
  • [6] W. Barreto-Souza and H.S. Bakouch, A new lifetime model with decreasing failure rate, Stats. 47 (2), 465-476, 2013.
  • [7] D.J. Best and J.C.W. Rayner, Smooth tests of fit for the Lindley distribution, Stats 1 (1), 92-97, 2018.
  • [8] J.M. Carrasco and N. Reid, Simplex regression models with measurement error, Comm. Statist. Simulation Comput. 50 (11), 3420-3435, 2021.
  • [9] D.R. Cox and E.J. Snell, A general definition of residuals, J. R. Stat. Soc. Ser. B. Stat. Methodol. 30 (2), 248-275, 1968.
  • [10] B. De Boeck, O. Thas, J.C.W. Rayner and D.J. Best, Smooth tests for the gamma distribution, J. Stat. Comput. Simul. 81 (7), 843-855, 2011.
  • [11] R.S. Defries, M.C. Hansen, J.R. Townshend, A.C. Janetos and T.R. Loveland, A new global 1km dataset of percentage tree cover derived from remote sensing, Glob. Change Biol. 6 (2), 247-254, 2000.
  • [12] A.J. Dobson and A.G. Barnett, An Introduction to Generalized Linear Models, Chapman and Hall/CRC, 2008.
  • [13] P.K. Dunn and G.K. Smyth, Randomized quantile residuals, J. Comput. Graph. Statist. 5 (3), 236-244, 1996.
  • [14] S. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (7), 799-815, 2004.
  • [15] M.E. Ghitany, D.K. Al-Mutairi, N. Balakrishnan and L.J. Al-Enezi, Power Lindley distribution and associated inference, Comput. Statist. Data Anal. 64, 20-33, 2013.
  • [16] M.E. Ghitany, J. Mazucheli, A.F.B. Menezes and F. Alqallaf, The unit-inverse Gaussian distribution: a new alternative to two-parameter distributions on the unit interval, Comm. Statist. Theory Methods 48 (14), 3423-3438, 2019.
  • [17] E.J. Hannan and B.G. Quinn, The determination of the order of an autoregression, J. R. Stat. Soc. Ser. B. Stat. Methodol. 41 (2), 190-195, 1979.
  • [18] C.M. Jarque and A.K. Bera, A test for normality of observations and regression residuals, Int. Stat. Rev. 55 (2), 163-177, 1987.
  • [19] R. Kieschnick and B.D. McCullough, Regression analysis of variates observed on (0, 1): percentages, proportions and fractions, Stat. Model. 3 (3), 193-213, 2003.
  • [20] M.Ç. Korkmaz, E. Altun, M. Alizadeh and M. El-Morshedy, The log exponential- power distribution: properties, estimations and quantile regression model, Mathematics 9 (21), 1-19, 2021.
  • [21] J.A. Koziol, Assessing multivariate normality: a compendium, Comm. Statist. Theory Methods 15 (9), 2763-2783, 1986.
  • [22] J.A. Koziol, An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives, Metrika 34 (1), 17-24, 1987.
  • [23] D.V. Lindley, Fiducial distributions and Bayes’ theorem, J. R. Stat. Soc. Ser. B. Stat. Methodol. 20 (1), 102-107, 1958.
  • [24] J. Mazucheli, A.F.B. Menezes and S. Chakraborty, On the one parameter unit-Lindley distribution and its associated regression model for proportion data, J. Appl. Stat. 46 (4), 700-714, 2019.
  • [25] P. McCullagh and J.A. Nelder, Generalized Linear Models, Chapman and Hall, 1989.
  • [26] A.M. Mousa, A.A. El-Sheikh and M.A. Abdel-Fattah, A gamma regression for bounded continuous variables, Adv. Appl. Stat. 49 (4), 305-326, 2016.
  • [27] L.R. Nakamura, P.H. Cerqueira, T.G. Ramires, R.R. Pescim, R.A. Rigby and D.M. Stasinopoulos, A new continuous distribution on the unit interval applied to modelling the points ratio of football teams, J. Appl. Stat. 46 (3), 416-431, 2019.
  • [28] J. Neyman, Smooth test for goodness of fit, Scand. Actuar. J. 1937 (3-4), 149-199, 1937.
  • [29] D. Ozonur, H.T.K. Akdur and H. Bayrak, Comparisons of tests of distributional assumption in Poisson regression model, Comm. Statist. Simulation Comput. 46 (8), 6197-6207, 2017.
  • [30] D. Özonur, F. Gökpnar, E. Gökpnar and H. Bayrak, Goodness of fit tests for Nakagami distribution based on smooth tests, Comm. Statist. Theory Methods 45 (7), 1876-1886, 2016.
  • [31] H. Poorter and L. Sack, Pitfalls and possibilities in the analysis of biomass allocation patterns in plants, Front. Plant Sci. 3 (259), 1-10, 2012.
  • [32] G. Pumi, C. Rauber and F.M. Bayer, Kumaraswamy regression model with Aranda- Ordaz link function, Test 29, 1051-1071, 2020.
  • [33] P.L. Ramos and F. Louzada, The generalized weighted Lindley distribution: properties, estimation, and applications, Cogent Math. 3 (1), 1-18, 2016.
  • [34] P.L. Ramos, F. Louzada, T.K. Shimizu and A.O. Luiz, The inverse weighted Lindley distribution: properties, estimation and an application on a failure time data, Comm. Statist. Theory Methods 48 (10), 2372-2389, 2019.
  • [35] J.C.W. Rayner and D.J. Best, Neyman-type smooth tests for location-scale families, Biometrika 73 (2), 437-446, 1986.
  • [36] J.C.W. Rayner, O. Thas and D.J. Best, Smooth Tests of Goodness of Fit: Using R, John Wiley and Sons, 2009.
  • [37] P. Rippon, Application of smooth tests of goodness of fit to generalized linear models, PhD thesis, University of Newcastle, 2013.
  • [38] G. Schwarz, Estimating the dimension of a model, Ann. Statist. 6 (2), 461-464, 1978.
  • [39] M. Smithson and J. Verkuilen, A better lemon-squeezer? Maximum likelihood regression with beta-distribuited dependent variables, Psycholog. Meth. 11 (1), 54-71, 2006.
  • [40] M.G. Swainson, A.M. Batterham, C. Tsakirides, Z.H. Rutherford and K. Hind, Prediction of whole-body fat percentage and visceral adipose tissue mass from five anthro- pometric variables, PloS one 12 (5), 1-12, 2017.
  • [41] O. Thas and J.C.W. Rayner, Smooth tests for the zero-inflated Poisson distribution, Biometrics 61 (3), 808-815, 2005.
  • [42] C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Amer. Statist. Assoc. 50 (269), 209-219, 1955.
  • [43] T.W. Yee, The VGAM package for categorical data analysis, J. Stat. Softw. 32 (10), 1-34, 2010.
  • [44] H. Zakerzadeh and A. Dolati, Generalized Lindley distribution, J. Math. Ext. 3 (2), 1-17, 2009.
  • [45] A. Zeileis, F. Cribari-Neto, B. Grün and I. Kos-midis, Beta regression in R, J. Statist. Softw. 34 (2), 1-24, 2010.
There are 45 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Deniz Özonur 0000-0002-7622-1008

Publication Date June 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 3

Cite

APA Özonur, D. (2022). Testing distributional assumption of unit-Lindley regression model. Hacettepe Journal of Mathematics and Statistics, 51(3), 882-899. https://doi.org/10.15672/hujms.932811
AMA Özonur D. Testing distributional assumption of unit-Lindley regression model. Hacettepe Journal of Mathematics and Statistics. June 2022;51(3):882-899. doi:10.15672/hujms.932811
Chicago Özonur, Deniz. “Testing Distributional Assumption of Unit-Lindley Regression Model”. Hacettepe Journal of Mathematics and Statistics 51, no. 3 (June 2022): 882-99. https://doi.org/10.15672/hujms.932811.
EndNote Özonur D (June 1, 2022) Testing distributional assumption of unit-Lindley regression model. Hacettepe Journal of Mathematics and Statistics 51 3 882–899.
IEEE D. Özonur, “Testing distributional assumption of unit-Lindley regression model”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, pp. 882–899, 2022, doi: 10.15672/hujms.932811.
ISNAD Özonur, Deniz. “Testing Distributional Assumption of Unit-Lindley Regression Model”. Hacettepe Journal of Mathematics and Statistics 51/3 (June 2022), 882-899. https://doi.org/10.15672/hujms.932811.
JAMA Özonur D. Testing distributional assumption of unit-Lindley regression model. Hacettepe Journal of Mathematics and Statistics. 2022;51:882–899.
MLA Özonur, Deniz. “Testing Distributional Assumption of Unit-Lindley Regression Model”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, 2022, pp. 882-99, doi:10.15672/hujms.932811.
Vancouver Özonur D. Testing distributional assumption of unit-Lindley regression model. Hacettepe Journal of Mathematics and Statistics. 2022;51(3):882-99.

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