# Akaike's Information Criterion for GKw Regression Models Calculates the Akaike Information Criterion (AIC) for one or more fitted Generalized Kumaraswamy (GKw) regression model objects (class `"gkwreg"`). AIC is commonly used for model selection, penalizing model complexity. ```r ## S3 method for class 'gkwreg' AIC(object, ..., k = 2) ``` ## 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 AIC should also be calculated. - `k`: Numeric, the penalty per parameter. The default `k = 2` corresponds to the traditional AIC. Using `k = log(nobs)` would yield BIC (though using `BIC.gkwreg` is preferred for that). ## Returns If just one `object` is provided, returns a single numeric AIC 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 AIC values, sorted by AIC. ## Details The AIC is calculated based on the maximized log-likelihood ($L$) and the number of estimated parameters ($p$) in the model: $$ AIC = -2 \log(L) + k \times p $$ This function retrieves the log-likelihood and the number of parameters (`df`) using the `logLik.gkwreg` method for the fitted `gkwreg` object(s). Models with lower AIC values are generally preferred, as they indicate a better balance between goodness of fit and model parsimony. When comparing multiple models passed via `...`, the function relies on `AIC`'s default method for creating a comparison table, which in turn calls `logLik` for each provided object. For small sample sizes relative to the number of parameters, the second-order AIC (AICc) might be more appropriate: $$ AICc = AIC + \frac{2p(p+1)}{n-p-1} $$ where $n$ is the number of observations. AICc is not directly computed by this function but can be calculated manually using the returned AIC, $p$ (from `attr(logLik(object), "df")`), and $n$ (from `attr(logLik(object), "nobs")`). ## 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 AIC for a single model aic1 <- AIC(kw_reg1) print(aic1) # Compare models using AIC (lower is better) model_comparison_aic <- c(AIC(kw_reg1), AIC(kw_reg2), AIC(kw_reg3)) print(model_comparison_aic) # Calculate AIC with a different penalty (e.g., k=4) aic1_k4 <- AIC(kw_reg1, k = 4) print(aic1_k4) ``` ## References Akaike, H. (1974). A new look at the statistical model identification. **IEEE Transactions on Automatic Control**, 19 (6), 716-723. Burnham, K. P., & Anderson, D. R. (2002). **Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach** (2nd ed.). Springer-Verlag. ## See Also `gkwreg`, `logLik.gkwreg`, `BIC.gkwreg`, `AIC` ## Author(s) Lopes, J. E.

AIC.gkwreg function