# Calculating MLEces for three different distributions The closed-form estimators (MLEces) are calculated for three distributions: bivariate gamma, bivariate Weibull and multivariate Dirichlet. ```r MLEce(data, distname) ``` ## Arguments - `data`: a numeric matrix. - `distname`: a character indicating which distribution to be fitted. `BiGam` stands for the bivariate gamma distribution, `BiWei` stands for the bivariate Weibull distribution, and `Dirichlet` stands for the multivariate Dirichlet distribution. ## Returns `MLEce` returns an object of class `"MLEce"`. The object class `"MLEce"` is a list containing the following components. - **distribution**: a character string of a distribution assuming that data set comes from and the data was fitted to. - **estimation**: the estimated values of parameters in assigned distribution. ## Details Based on root n-consistent estimators, the closed-form estimators (MLEces) are calculated for the parameters in bivariate gamma, bivariate Weibull and multivariate Dirichlet distributions whose maximum likelihood estimators (MLEs) are not in closed forms. The MLEces are strong consistent and asymptotic normally like the corresponding MLEs, but their calculation are much faster than MLEs. For the bivariate gamma and multivariate Dirichlet distribution, their root n-consistent estimators are the corresponding method of moments estimators (MMEs). The correlation-based estimators (CMEs) are applied as root n-consistent estimators in the bivariate Weibull distribution. ## Examples ```r #bivariate gamma distribution data_BiGam <- rBiGam(100, c(1,4,5)) res_BiGam <- MLEce(data_BiGam, "BiGam") print(res_BiGam) data(flood) est_BiGam <- MLEce(flood, "BiGam") print(est_BiGam) #bivariate Weibull distribution data_BiWei <- rBiWei(n=30, c(4,3,3,4,0.6)) res_BiWei <- MLEce(data_BiWei, "BiWei") print(res_BiWei) #real data example data(airquality) air_data <- airquality[ ,3:4] air_data[ ,2] <- air_data[ ,2]*0.1 est_BiWei <- MLEce(air_data, "BiWei") print(est_BiWei) #Dirichlet distribution data_Diri <- LaplacesDemon::rdirichlet(n=60, c(1,2,3)) res_Diri <- MLEce(data_Diri, "Dirichlet") print(res_Diri) data(fossil_pollen) #real data example fossil_data <- fossil_pollen/rowSums(fossil_pollen) eps <- 1e-10 fossil_data <- (fossil_data +eps)/(1+2*eps) est_fossil <- MLEce(fossil_data, "Dirichlet") print(est_fossil) ``` ## References Kim, H.-M., Jang, Y.-H., Arnold, B. C. and Zhao, J. (2023) New efficient estimators for the Weibull distribution. **Communications in Statistics - Theory and Methods**, 1-26. Jang, Y.-H., Zhao, J., Kim, H.-M., Yu, K., Kwon, S.and Kim, S. (2023) New closed-form efficient estimator for the multivariate gamma distribution. **Statistica Neerlandica**, 1–18. Chang, J. H., Lee, S. K. and Kim, H.-M. (2023) An asymptotically efficient closed–form estimator for the multivariate Dirichlet distribution. submitted.

MLEce function