# Tests of homogeneity and independence in bivariate sine von Mises and wrapped Cauchy distributions Performs the following likelihood ratio tests for the concentrations in bivariate sine von Mises and wrapped Cauchy distributions: (1) **homogeneity**: $H_0:\kappa_1=\kappa_2$ vs. $H_1:\kappa_1\neq\kappa_2$, and $H_0:\xi_1=\xi_2$ vs. $H_1:\xi_1\neq\xi_2$, respectively; (2) **independence**: $H_0:\lambda=0$ vs. $H_1:\lambda\neq0$, and $H_0:\rho=0$ vs. $H_1:\rho\neq0$. The tests (1) and (2) can be performed simultaneously. ```r biv_lrt(x, hom = FALSE, indep = FALSE, fit_mle = NULL, type, ...) ``` ## Arguments - `x`: matrix of dimension `c(n, 2)` containing the `n` observations of the pair of angles. - `hom`: test the homogeneity hypothesis? Defaults to `FALSE`. - `indep`: test the independence hypothesis? Defaults to `FALSE`. - `fit_mle`: output of `fit_bvm_mle` or `fit_bwc_mle` with `hom = FALSE`. Computed internally if not provided. - `type`: either `"bvm"` (bivariate sine von Mises) or `"bwc"` (bivariate wrapped Cauchy). - `...`: optional parameters passed to `fit_bvm_mle` and `fit_bwc_mle`, such as `start`, `lower`, or `upper`. ## Returns A list with class `htest`: - **statistic**: the value of the likelihood ratio test statistic. - **p.value**: the $p$-value of the test (computed using the asymptotic distribution). - **alternative**: a character string describing the alternative hypothesis. - **method**: description of the type of test performed. - **df**: degrees of freedom. - **data.name**: a character string giving the name of `theta`. - **fit_mle**: maximum likelihood fit. - **fit_null**: maximum likelihood fit under the null hypothesis. ## Examples ```r ## Bivariate sine von Mises # Homogeneity n <- 200 mu <- c(0, 0) kappa_0 <- c(1, 1, 0.5) kappa_1 <- c(0.7, 0.1, 0.25) samp_0 <- r_bvm(n = n, mu = mu, kappa = kappa_0) samp_1 <- r_bvm(n = n, mu = mu, kappa = kappa_1) biv_lrt(x = samp_0, hom = TRUE, type = "bvm") biv_lrt(x = samp_1, hom = TRUE, type = "bvm") # Independence kappa_0 <- c(0, 1, 0) kappa_1 <- c(1, 0, 1) samp_0 <- r_bvm(n = n, mu = mu, kappa = kappa_0) samp_1 <- r_bvm(n = n, mu = mu, kappa = kappa_1) biv_lrt(x = samp_0, indep = TRUE, type = "bvm") biv_lrt(x = samp_1, indep = TRUE, type = "bvm") # Independence and homogeneity kappa_0 <- c(3, 3, 0) kappa_1 <- c(3, 1, 0) samp_0 <- r_bvm(n = n, mu = mu, kappa = kappa_0) samp_1 <- r_bvm(n = n, mu = mu, kappa = kappa_1) biv_lrt(x = samp_0, indep = TRUE, hom = TRUE, type = "bvm") biv_lrt(x = samp_1, indep = TRUE, hom = TRUE, type = "bvm") ## Bivariate wrapped Cauchy # Homogeneity xi_0 <- c(0.5, 0.5, 0.25) xi_1 <- c(0.7, 0.1, 0.5) samp_0 <- r_bwc(n = n, mu = mu, xi = xi_0) samp_1 <- r_bwc(n = n, mu = mu, xi = xi_1) biv_lrt(x = samp_0, hom = TRUE, type = "bwc") biv_lrt(x = samp_1, hom = TRUE, type = "bwc") # Independence xi_0 <- c(0.1, 0.5, 0) xi_1 <- c(0.3, 0.5, 0.2) samp_0 <- r_bwc(n = n, mu = mu, xi = xi_0) samp_1 <- r_bwc(n = n, mu = mu, xi = xi_1) biv_lrt(x = samp_0, indep = TRUE, type = "bwc") biv_lrt(x = samp_1, indep = TRUE, type = "bwc") # Independence and homogeneity xi_0 <- c(0.2, 0.2, 0) xi_1 <- c(0.1, 0.2, 0.1) samp_0 <- r_bwc(n = n, mu = mu, xi = xi_0) samp_1 <- r_bwc(n = n, mu = mu, xi = xi_1) biv_lrt(x = samp_0, indep = TRUE, hom = TRUE, type = "bwc") biv_lrt(x = samp_1, indep = TRUE, hom = TRUE, type = "bwc") ``` ## References Kato, S. and Pewsey, A. (2015). A Möbius transformation-induced distribution on the torus. **Biometrika**, 102(2):359--370. tools:::Rd_expr_doi("10.1093/biomet/asv003") Singh, H., Hnizdo, V., and Demchuk, E. (2002). Probabilistic model for two dependent circular variables. **Biometrika**, 89(3):719--723. tools:::Rd_expr_doi("10.1093/biomet/89.3.719")

biv_lrt function