# Bayesian Information Criterion for GKw Regression Models Calculates the Bayesian Information Criterion (BIC), also known as Schwarz's Bayesian Criterion (SBC), for one or more fitted Generalized Kumaraswamy (GKw) regression model objects (class `"gkwreg"`). BIC is used for model selection and tends to penalize model complexity more heavily than AIC, especially for larger datasets. ```r ## S3 method for class 'gkwreg' BIC(object, ...) ``` ## Arguments - `object`: An object of class `"gkwreg"`, typically the result of a call to `gkwreg`. - `...`: Optionally, one or more additional fitted model objects of class `"gkwreg"`, for which BIC should also be calculated. ## Returns If just one `object` is provided, returns a single numeric BIC value. If multiple objects are provided via `...`, returns a `data.frame` with rows corresponding to the models and columns for the degrees of freedom (`df`) and the BIC values, sorted by BIC. ## Details The BIC is calculated based on the maximized log-likelihood ($L$), the number of estimated parameters ($p$) in the model, and the number of observations ($n$): $$ BIC = -2 \log(L) + p \times \log(n) $$ This function retrieves the log-likelihood, the number of parameters (`df`), and the number of observations (`nobs`) using the `logLik.gkwreg` method for the fitted `gkwreg` object(s). Models with lower BIC values are generally preferred. The penalty term $p \log(n)$ increases more rapidly with sample size $n$ compared to AIC's penalty $2p$, meaning BIC favors simpler models more strongly in larger samples. BIC can be motivated from a Bayesian perspective as an approximation related to Bayes factors. When comparing multiple models passed via `...`, the function relies on `BIC`'s default method for creating a comparison table, which in turn calls `logLik` for each provided object. ## Examples ```r # Assume 'df' exists with response 'y' and predictors 'x1', 'x2', 'x3' # and that rkw() is available and data is appropriate (0 < y < 1). set.seed(123) n <- 100 x1 <- runif(n) x2 <- rnorm(n) x3 <- factor(rbinom(n, 1, 0.4)) alpha <- exp(0.5 + 0.2 * x1) beta <- exp(1.0 - 0.1 * x2 + 0.3 * (x3 == "1")) y <- rkw(n, alpha = alpha, beta = beta) # Placeholder if rkw not available y <- pmax(pmin(y, 1 - 1e-7), 1e-7) df <- data.frame(y = y, x1 = x1, x2 = x2, x3 = x3) # Fit two competing models kw_reg1 <- gkwreg(y ~ x1 | x2, data = df, family = "kw") kw_reg2 <- gkwreg(y ~ x1 | x2 + x3, data = df, family = "kw") # More complex beta model kw_reg3 <- gkwreg(y ~ 1 | x2 + x3, data = df, family = "kw") # Simpler alpha model # Calculate BIC for a single model bic1 <- BIC(kw_reg1) print(bic1) # Compare models using BIC (lower is better) model_comparison_bic <- c(BIC(kw_reg1), BIC(kw_reg2), BIC(kw_reg3)) print(model_comparison_bic) ``` ## References Schwarz, G. (1978). Estimating the dimension of a model. **The Annals of Statistics**, 6 (2), 461-464. ## See Also `gkwreg`, `logLik.gkwreg`, `AIC.gkwreg`, `BIC` ## Author(s) Lopes, J. E.

BIC.gkwreg function