# Negative Log-Likelihood for the Beta Distribution (gamma, delta+1 Parameterization) Computes the negative log-likelihood function for the standard Beta distribution, using a parameterization common in generalized distribution families. The distribution is parameterized by `gamma` ($\gamma$) and `delta` ($\delta$), corresponding to the standard Beta distribution with shape parameters `shape1 = gamma` and `shape2 = delta + 1`. This function is suitable for maximum likelihood estimation. ```r llbeta(par, data) ``` ## Arguments - `par`: A numeric vector of length 2 containing the distribution parameters in the order: `gamma` ($\gamma > 0$), `delta` ($\delta \ge 0$). - `data`: A numeric vector of observations. All values must be strictly between 0 and 1 (exclusive). ## Returns Returns a single `double` value representing the negative log-likelihood ($-\ell(\theta|\mathbf{x})$). Returns `Inf` if any parameter values in `par` are invalid according to their constraints, or if any value in `data` is not in the interval (0, 1). ## Details This function calculates the negative log-likelihood for a Beta distribution with parameters `shape1 = gamma` ($\gamma$) and `shape2 = delta + 1` ($\delta+1$). The probability density function (PDF) is: $$ f(x | \gamma, \delta) = \frac{x^{\gamma-1} (1-x)^{\delta}}{B(\gamma, \delta+1)} $$ for $0 < x < 1$, where $B(a,b)$ is the Beta function (`beta`). The log-likelihood function $\ell(\theta | \mathbf{x})$ for a sample $\mathbf{x} = (x_1, \dots, x_n)$ is $\sum_{i=1}^n \ln f(x_i | \theta)$: $$ \ell(\theta | \mathbf{x}) = \sum_{i=1}^{n} [(\gamma-1)\ln(x_i) + \delta\ln(1-x_i)] - n \ln B(\gamma, \delta+1) $$ where $\theta = (\gamma, \delta)$. This function computes and returns the **negative** log-likelihood, $-\ell(\theta|\mathbf{x})$, suitable for minimization using optimization routines like `optim`. It is equivalent to the negative log-likelihood of the GKw distribution (`llgkw`) evaluated at $\alpha=1, \beta=1, \lambda=1$, and also to the negative log-likelihood of the McDonald distribution (`llmc`) evaluated at $\lambda=1$. The term $\ln B(\gamma, \delta+1)$ is typically computed using log-gamma functions (`lgamma`) for numerical stability. ## Examples ```r # Assuming existence of rbeta_, llbeta, grbeta, hsbeta functions # Generate sample data from a Beta(2, 4) distribution # (gamma=2, delta=3 in this parameterization) set.seed(123) true_par_beta <- c(gamma = 2, delta = 3) sample_data_beta <- rbeta_(100, gamma = true_par_beta[1], delta = true_par_beta[2]) hist(sample_data_beta, breaks = 20, main = "Beta(2, 4) Sample") # --- Maximum Likelihood Estimation using optim --- # Initial parameter guess start_par_beta <- c(1.5, 2.5) # Perform optimization (minimizing negative log-likelihood) # Use method="L-BFGS-B" for box constraints (params > 0 / >= 0) mle_result_beta <- stats::optim(par = start_par_beta, fn = llbeta, # Use the custom Beta neg-log-likelihood method = "L-BFGS-B", lower = c(1e-6, 1e-6), # Bounds: gamma>0, delta>=0 hessian = TRUE, data = sample_data_beta) # Check convergence and results if (mle_result_beta$convergence == 0) { print("Optimization converged successfully.") mle_par_beta <- mle_result_beta$par print("Estimated Beta parameters (gamma, delta):") print(mle_par_beta) print("True Beta parameters (gamma, delta):") print(true_par_beta) cat(sprintf("MLE corresponds approx to Beta(%.2f, %.2f)\n", mle_par_beta[1], mle_par_beta[2] + 1)) } else { warning("Optimization did not converge!") print(mle_result_beta$message) } # --- Compare numerical and analytical derivatives (if available) --- # Requires 'numDeriv' package and analytical functions 'grbeta', 'hsbeta' if (mle_result_beta$convergence == 0 && requireNamespace("numDeriv", quietly = TRUE) && exists("grbeta") && exists("hsbeta")) { cat("\nComparing Derivatives at Beta MLE estimates:\n") # Numerical derivatives of llbeta num_grad_beta <- numDeriv::grad(func = llbeta, x = mle_par_beta, data = sample_data_beta) num_hess_beta <- numDeriv::hessian(func = llbeta, x = mle_par_beta, data = sample_data_beta) # Analytical derivatives (assuming they return derivatives of negative LL) ana_grad_beta <- grbeta(par = mle_par_beta, data = sample_data_beta) ana_hess_beta <- hsbeta(par = mle_par_beta, data = sample_data_beta) # Check differences cat("Max absolute difference between gradients:\n") print(max(abs(num_grad_beta - ana_grad_beta))) cat("Max absolute difference between Hessians:\n") print(max(abs(num_hess_beta - ana_hess_beta))) } else { cat("\nSkipping derivative comparison for Beta.\n") cat("Requires convergence, 'numDeriv' pkg & functions 'grbeta', 'hsbeta'.\n") } ``` ## References Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). **Continuous Univariate Distributions, Volume 2** (2nd ed.). Wiley. Cordeiro, G. M., & de Castro, M. (2011). A new family of generalized distributions. **Journal of Statistical Computation and Simulation**, ## See Also `llgkw`, `llmc` (related negative log-likelihoods), `dbeta_`, `pbeta_`, `qbeta_`, `rbeta_`, `grbeta` (gradient, if available), `hsbeta` (Hessian, if available), `optim`, `lbeta`. ## Author(s) Lopes, J. E.

llbeta function