Optimal design for a Fréchet copula marginal regression with exponential marginals

Document Type : Original Scientific Paper

Authors

Department of Statistics‎, ‎Razi University‎, ‎Kermanshah‎, ‎Iran

Abstract

Modeling dependence using copula functions is a standard and widely used method in applied statistics‎. ‎In recent years‎, ‎experiments have been conducted in which several dependent responses are described by regression models‎, ‎and then experimental designs for these models have been carried out‎. ‎These regression models are often expressed as copula-based regression models‎, ‎so the copula parameters (if they exist) also affect the optimal design problem‎. ‎In this paper‎, ‎we consider the dependence structure of a random pair from the exponential distribution conditionally upon a covariate as a regression model‎, ‎then investigate the D-optimal design for this copula-based regression model‎. ‎The copula function that is used is the Fréchet‎‎ copula‎. ‎The optimal designs obtained all have a general form depending on the selected design space.

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References

Abdollahi, N.A., Jafari, H. and Khazaei, S. (2024). Locally, Bayesian and non parametric Bayesian optimal designs for unit exponential regression model. Communications in Statistics-Theory and Methods. https://doi.org/10.1080/ 03610926.2024.2328182
Aminnejad, M. and Jafari, H. (2017). Bayesian A- and D-optimal designs for gamma regression model with inverse link function. Communications in Statistics-Simulation and Computation, 46(10):8166–8189.
Atkinson, A.C., Donev, A.N. and Tobias, R.D. (2007). Optimum Experimental Design, with SAS. Oxford: Oxford University Press.
Balakrishnan, N. and Nevzorov, V.B. (2003). A Primer on Statistical Distributions. New Jersey: John Wiley & Sons.
Burghaus, I. and Dette, H. (2014). Optimal designs for nonlinear regression models with respect to non-informative priors. Journal of Statistical Planning and Inference, 154:12–25.
Burkner, P.C., Schwabe, R. and Holling, H. (2019). Optimal designs for the generalized partial credit model. British Journal of Mathematical and Statistical Psychology, 72(2):271–293.
Chaloner, K.M. and Duncan, G.T. (1983). Assessment of a beta prior distribution: PM elicitation. Journal of the Royal Statistical Society. Series D (The Statistician), 32(1-2):174–180.
Cherubini, U., Gobbi, F. and Mulinacci, S. (2016). Convolution Copula Econometrics. Cham: Springer.
Cherubini, U., Luciano, E. and Vecchiato, W. (2004). Copula Methods in Finance. John Wiley & Sons.
Chernoff, H. (1953). Locally optimal designs for estimating parameters. The Annals of Mathematical Statistics, 24(4):586–602.
Deldossi, L., Osmetti, S.A. and Tommasi, C. (2018). Optimal design to discriminate between rival copula models for a bivariate binary response. TEST, 28:147–165.
Denman, N.G., McGree, J.M., Eccleston, J.A. and Duffull, S.B. (2011). Design of experiments for bivariate binary responses modelled by Copula functions. Computational Statistics and Data Analysis, 55(4):1509–1520.
Dette, H., Melas, V.B. and Wong, W.K. (2006). Locally D-optimal designs for exponential regression. Statistica Sinica, 16(3):789–803.
Dette, H. and Neugebauer, H.-M. (1997). Bayesian D-optimal designs for exponential regression models. Journal of Statistical Planning and Inference, 60(2):331–349.
Durante, F. and Perrone, E. (2016). Asymmetric copulas and their application in design of experiments. In: Saminger-Platz S., Mesiar, R. (eds.) On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory. Vol. 336, 157–172, Cham: Springer.
Durante, F. and Sempi, C. (2015). Principles of Copula Theory. Vol. 474, Boca Raton, FL: CRC Press.
Fedorov, V. and Hackl, P. (2012). Model-oriented Design of Experiments. New York: Springer.
Firth, D. and Hinde, J.P. (1997). On Bayesian D-optimum design criteria and the equivalence theorem in non-linear models. Journal of the Royal Statistical Society Series B: Statistical Methodology, 59(4):793–797.
Gijbels, I., Veraverbeke, N. and Omelka, M. (2011). Conditional copulas, association measures and their applications. Computational Statistics & Data Analysis, 55(5):1919–1932.
Goudarzi, M., Jafari, H. and Khazaei, S. (2019). Nonparametric Bayesian optimal designs for exponential regression model. Communications in Statistics-Simulation and Computation, 50(6):1809–1819.
Graßhoff, U., Holling, H. and Schwabe, R. (2012). Optimal designs for the Rasch model. Psychometrika, 77(4):710–723.
Joag-Dev, K. and Proschan, P. (1983). Negative Association of Random Variables with Applications. The Annals of Statistics, 11(1):286–295.
Joe, H. (1990). Multivariate concordance. Journal of Multivariate Analysis, 35(1):12–30.
Kiefer, J. (1959). Optimum experimental design. Journal of the Royal Statistical Society Series B: Statistical Methodology, 21(2):272–319.
Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems. The Annals of Mathematical Statistics, 30(2):271–294.
Kiefer, J. and Wolfowitz, J. (1960). The equivalence of two extremum problems. Canadian Journal of Mathematics, 12:363–366.
Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). The Annals of Statistics, 2(5):849–879.
Lehmann, E.L. (1966). Some Concepts of Dependence. The Annals of Statistics, 37(5):1137–1153.
Li, J., Bárdossy, A., Guenni, L. and Liu, M. (2011). A copula based observation network design approach. Environmental Modelling & Software, 26(11):1349–1357.
Mukhopadhyay, S. and Haines, L.M. (1995). Bayesian D-optimal designs for the exponential growth model. Journal of Statistical Planning and Inference, 44(3):385–397.
Nelsen, R.B. (2006). An Introduction to Copulas. New York: Springer.
Parsa Maram, P. and Jafari, H. (2016). Bayesian D-optimal design for logistic regression model with exponential distribution for random intercept. Journal of Statistical Computation and Simulation, 86(10):1856–1868.
Perrone, E. and Müller, W.G. (2016). Optimal designs for Copula models. Statistics, 50(4):917–929.
Perrone, E., Rappold, A. and Müller, W.G. (2016). Ds-optimality in copula models. Statistical Methods & Applications, 26(3):403–418.
Pilz, J., Kazianka, H. and Spöck, G. (2012). Some advances in Bayesian spatial prediction and sampling design. Spatial Statistics, 1:65–81.
Pukelsheim, F. (1993). Optimal Design of Experiments. New York: John Wiley.
Rappold, A., Müller, W.G. and Woods, D.C. (2020). Copula-based robust optimal block designs. Applied Stochastic Models in Business and Industry, 36(1):210–219.
Salvadori, G., De Michele, C., Kottegoda, N.T. and Rosso, R. (2007). Extremes in Nature: An Approach Using Copulas. Dordrecht: Springer.
Schmidt, R., Faldum, A., Witt, O. and Gerss, J. (2014). Adaptive designs with arbitrary dependence structure. Biom J, 56(1):86–106.
Silvey, S.D. (1980). Optimal Design. London: Chapman and Hall.
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de l'institut de Statistique de l'Université de Paris, 8(3):229–231.
Smith, K. (1918). On the standard deviations of adjusted and interpolated values of an observed polynomial function and its constant and the guidance they give towards a proper choice of the distribution of observations. Biometrika, 12(1/2):1–85.
Xie, Z., Wang, F., Yang, Y. and Guo, N. (2022). Multivariate Bernstein Fréchet copulas. Statistics and Its Interface, 15(4):527–547.
Journal of Statistical Modelling: Theory and Applications
Volume 5, Issue 1
January 2024
Pages 117-130

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History

  • Receive Date: 19 July 2024
  • Revise Date: 12 October 2024
  • Accept Date: 28 November 2024

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