Comparison of Three Moment-Based Estimation Methods For FGM Copula In the Presence of Outlier

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Taheri B.M.; JABBARI H.; AMINI M.

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Journal: JOURNAL OF STATISTICAL SCIENCES Year:2022 | Volume:16 | Issue:1 Page(s): 149-163

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Title

Comparison of Three Moment-Based Estimation Methods For FGM Copula In the Presence of Outlier

Author(s)

Taheri B.M. | JABBARI H. | AMINI M. | Issue Writer Certificate

Keywords

Copula

Outlier

Moment-based estimate

Dependence parameter

Bivariate analysis

Abstract

Introduction Using the Copulas function is a particular form of introducing variables and their dependency. Also, paying attention to Copulas for estimating the Dependence parameters has become popular in recent decades. As a semiparametric technique, Berahimi and Necir (2012), introduced Copula moment (CM) and compared it with PMLE and ,and ,inversion methods, also Kojadinovic and Yan (2010) used three semi-parametric methods based on Copula models. Taheri et al. (2018) studied the dependence of bivariate Copulas in the presence of Outliers. This article used three moment-based estimation methods in the presence of Outliers. The moment method and two other estimation methods are related to moment, Copula moment (CM) and their mixture. Material and Methods Let (X,Y ) be a random vector with Copula function C and Dependence parameter , . Also, let (X1,Y1), : : :, (Xn,Yn) are a random sample from (X,Y ). We assume that the random vector (X,Y ) is in the presence of Outliers. In other words, assume that n 􀀀,k (k is an unknown and a random integer) elements of the random sample have true Copula function C1 and Dependence parameter ,and the remind elements have another Copula function C2 and Dependence parameter , , , where ,is an unknown real value called as a noised parameter. The Copula functions C1 and C2 can have completely different structures. For estimating ,in the presence of Outliers, we may obtain the joint density function of a random sample. A simulation study is used to select the best estimation method, and the estimators are compared based on MSE. Also, to illustrate the results achieved in a simulation study, we applied a real example for the tourists who visit the ”, Tomb of Ayub Prophet”,(TAP) and the ”, Imamzadeh Asgari Tomb”,(IAT) in North Khorasan, IRAN. Here, we test the dependence between the number of visitors to TAP and IAT at weekends and holidays. Results and Discussion For estimating parameters and choosing which value of ,is good for FGM Copula in the presence of Outliers, we test various ,and use them in the likelihood function for Copulas in the presence of Outliers. Using ١, ٠, ٠, ٠,independent repetitions of the likelihood function for FGM Copula in a simulation study suggests that ,= 0: 1 is the best value. The Dependence parameter is estimated by substituting ,= 0: 1 in the likelihood function of Copulas in the presence of Outliers. The simulation results show that when we use CM and mixed methods in the presence of Outliers, the empirical MSEs are reasonable. Also, CM is the best estimator based on MSE. Conclusion Taheri et al. (٢, ٠, ١, ٨, ) showed that the best methods for estimating the Dependence parameter in the presence of Outliers are the MLE, PMLE, the inverse of , , and the inverse of ,and CM, respectively. In this paper, based on three moment methods, MM, Mixture and CM, the CM method is the best one according to their MSEs. From a practical point of view, it was also concluded that the estimations of the Dependence parameter in the presence of Outliers do not show a big difference in MLE and CM methods.

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APA: Copy

Taheri, B.M., JABBARI, H., & AMINI, M.. (2022). Comparison of Three Moment-Based Estimation Methods For FGM Copula In the Presence of Outlier. JOURNAL OF STATISTICAL SCIENCES, 16(1 ), 149-163. SID. https://sid.ir/paper/1021437/en

Vancouver: Copy

Taheri B.M., JABBARI H., AMINI M.. Comparison of Three Moment-Based Estimation Methods For FGM Copula In the Presence of Outlier. JOURNAL OF STATISTICAL SCIENCES[Internet]. 2022;16(1 ):149-163. Available from: https://sid.ir/paper/1021437/en

IEEE: Copy

B.M. Taheri, H. JABBARI, and M. AMINI, “Comparison of Three Moment-Based Estimation Methods For FGM Copula In the Presence of Outlier,” JOURNAL OF STATISTICAL SCIENCES, vol. 16, no. 1 , pp. 149–163, 2022, [Online]. Available: https://sid.ir/paper/1021437/en

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