CDF                     Estimate stable distribution parameters using
                        classical ECF regression
DONNEE_with_serial_interval
                        Example serial interval data
Im                      Imaginary part of the ECF integral
Int_Im                  Integrate imaginary component over \mathbb{R}
Int_Re                  Integrate real component over \mathbb{R}
L_stable                Negative log-likelihood for stable distribution
                        using dstable
Max_vrai                Maximum likelihood estimation using Nelder-Mead
N_epanechnikov          Epanechnikov kernel
N_gaussian              Gaussian kernel
N_uniform               Uniform kernel
RT                      Compute effective reproduction number Rt
Re                      Real part of the ECF integral
TableS2_serial_interval_mean_
                        Example transmission pair data with mean serial
                        interval
aic                     Akaike Information Criterion (AIC)
analyse_stable_distribution
                        Perform full stability analysis and export
                        results
bayesian_mixture_model
                        Bayesian mixture model using normal components
                        (simplified)
bic                     Bayesian Information Criterion (BIC)
build_mcculloch_interpolators
                        Build interpolation functions from McCulloch
                        table
calculate_log_likelihood
                        Calculate simplified log-likelihood
clip                    Clip values between lower and upper bounds
compare_em_vs_em_gibbs
                        Compare standard EM and EM with Gibbs sampling
                        using kernel ECF
compare_estimators_on_simulations
                        Compare MLE, ECF, and McCulloch estimators on
                        simulated data
compare_methods_across_configs
                        Compare McCulloch, ECF, and MLE methods across
                        parameter configurations
compare_methods_with_gibbs
                        Compare estimation methods with and without
                        Gibbs sampling
compute_model_metrics   Compute log-likelihood, AIC, and BIC for
                        alpha-stable model
compute_quantile_ratios
                        Compute McCulloch quantile ratios from sample
                        data
compute_serial_interval
                        Compute serial interval from CSV file
cosine_exp_ralpha       Cosine exponential function
cosine_log_weighted_exp_ralpha
                        Cosine-log-weighted exponential with r^(-alpha)
                        term
ecf_components          Extract magnitude and phase components from ECF
ecf_empirical           Compute empirical characteristic function
ecf_estimate_all        Estimate all stable parameters from empirical
                        characteristic function
ecf_fn                  Empirical Characteristic Function
ecf_regression          Estimate stable parameters using weighted ECF
                        regression
em_alpha_stable         EM algorithm for alpha-stable mixture
em_estimate_stable_from_cdf
                        EM algorithm for mixture of alpha-stable
                        distributions using CDF-based ECF
em_estimate_stable_from_cdf_with_gibbs
                        EM algorithm for alpha-stable mixture using
                        CDF-based ECF and Gibbs M-step
em_estimate_stable_kernel_ecf
                        EM algorithm for mixture of alpha-stable
                        distributions using kernel ECF
em_estimate_stable_kernel_ecf_with_gibbs
                        EM algorithm for alpha-stable mixture using
                        kernel ECF and Gibbs M-step
em_estimate_stable_recursive_ecf
                        EM algorithm for mixture of alpha-stable
                        distributions using recursive ECF
em_estimate_stable_recursive_ecf_with_gibbs
                        EM algorithm for alpha-stable mixture using
                        recursive ECF and Gibbs M-step
em_estimate_stable_weighted_ols
                        EM algorithm for mixture of alpha-stable
                        distributions using weighted OLS
em_estimate_stable_weighted_ols_with_gibbs
                        EM algorithm for alpha-stable mixture using
                        weighted OLS and Gibbs M-step
em_estimation_mixture   EM algorithm for two-component Gaussian mixture
em_fit_alpha_stable_mixture
                        EM algorithm for two-component alpha-stable
                        mixture using MLE
em_stable_mixture       EM algorithm for alpha-stable mixture using a
                        custom estimator
empirical_r0            Empirical R0 estimation using growth model
ensure_positive_scale   Ensure positive scale parameter
est_r0_ml               Estimate R0 using maximum likelihood
est_r0_mle              MLE estimation of R0 using generation time
estimate_alpha_gamma    Estimate alpha and gamma from ECF modulus
estimate_beta_delta     Estimate beta and delta from ECF phase
estimate_mixture_params
                        Estimate mixture of two stable distributions
estimate_stable_from_cdf
                        Estimate stable parameters using CDF-based ECF
                        regression
estimate_stable_kernel_ecf
                        Estimate stable parameters using kernel-based
                        ECF method
estimate_stable_params
                        Estimate single stable distribution parameters
estimate_stable_r       Estimate stable parameters using method of
                        moments
estimate_stable_recursive_ecf
                        Estimate stable parameters using recursive ECF
                        method
estimate_stable_weighted_ols
                        Estimate stable parameters using weighted OLS
                        on recursive ECF
eta0                    Helper function for eta0 computation
eta_func                General eta function
evaluate_estimation_method
                        Evaluate estimation method using MSE over
                        multiple trials
evaluate_fit            Evaluate fit quality using RMSE and
                        log-likelihood
export_analysis_report
                        Export analysis report to JSON and Excel
false_position_update   False position method update step
fast_integrate          Fast numerical integration using trapezoidal
                        rule
fit_alpha_stable_mle    Fit Alpha-Stable Distribution using MLE
                        (L-BFGS-B)
fit_mle_mixture         Fit MLE Mixture of Two Stable Distributions
fit_stable_ecf          Estimate stable parameters using filtered and
                        weighted ECF regression
generate_alpha_stable_mixture
                        Generate samples from a predefined alpha-stable
                        mixture
generate_mcculloch_table
                        Generate McCulloch lookup table from simulated
                        stable samples
generate_mixture_data   Simulates a mixture of alpha-stable
                        distributions with randomly sampled parameters.
generate_synthetic_data
                        Generate synthetic data from two alpha-stable
                        components
gibbs_sampler           Gibbs sampler for Gaussian mixture model
grad_loglik_alpha       Log-likelihood gradient with respect to alpha
grad_loglik_beta        Log-likelihood gradient with respect to beta
grad_loglik_delta       Log-likelihood gradient with respect to delta
                        (scale)
grad_loglik_omega       Log-likelihood gradient with respect to omega
                        (location)
integrate_cosine        Integrate cosine exponential
integrate_cosine_log_weighted
                        Integrate cosine-log-weighted exponential
integrate_function      Robust integration helper function
integrate_sine          Integrate sin exponential
integrate_sine_log_weighted
                        Integrate sine-log-weighted exponential
integrate_sine_r_weighted
                        Integrate sine-r-weighted exponential
integrate_sine_weighted
                        Integration wrappers for specific integrands
kde_bandwidth_plugin    KDE bandwidth selection using plugin method
log_likelihood_mixture
                        Log-likelihood for mixture of stable
                        distributions
mcculloch_lookup_estimate
                        Estimate stable parameters using McCulloch
                        lookup
mcculloch_quantile_init
                        Initialization using McCulloch quantile method
metropolis_hastings     Metropolis-Hastings MCMC for stable mixture
                        clustering
mixture_stable_pdf      Mixture of two stable PDFs
mle_estimate            Simple MLE estimation with default starting
                        values
mock_gibbs_sampling     Mock Gibbs sampling for alpha-stable mixture
                        estimation
mock_lookup_alpha_beta
                        Mock lookup for alpha and beta (fallback)
negative_log_likelihood
                        Negative log-likelihood for single stable
                        distribution
normalized_grad_alpha   Normalized gradient for alpha parameter
                        Computes the normalized gradient of the
                        log-likelihood with respect to the alpha
                        parameter over a set of observations. This is
                        useful for optimization routines where
                        scale-invariant updates are preferred.
normalized_objective_beta
                        Normalized objective for beta parameter
normalized_objective_delta
                        Normalized objective for delta parameter
normalized_objective_omega
                        Normalized objective for omega parameter
plot_comparison         Compare EM-estimated mixture with a
                        non-optimized reference model
plot_distributions      Plot histogram with normal and stable PDF
                        overlays
plot_effective_reproduction_number
                        Plot effective reproduction number (Re) over
                        time
plot_final_mixture_fit
                        Plot final fitted mixture of alpha-stable
                        distributions
plot_fit_vs_true        Plot true vs estimated mixture density
plot_fit_vs_true_methods
                        Compare estimated mixture densities from two
                        methods against the true density
plot_method_comparison
                        Plot RMSE and Log-Likelihood comparison across
                        methods
plot_mixture            Plot mixture of two alpha-stable distributions
plot_mixture_fit        Plot mixture fit with individual components
plot_real_mixture_fit   Plot fitted mixture on real dataset
plot_results            Plot posterior mixture density from MCMC
                        samples
plot_trace              Plot trace of a parameter across MCMC
                        iterations
plot_vs_normal_stable   Plot comparison between normal and stable
                        distributions
qcv_stat                QCV statistic for tail heaviness
r_stable_pdf            Robust stable PDF computation
recursive_weight        Recursive weight function
robust_ecf_regression   Estimate stable parameters using robust ECF
                        regression
robust_mle_estimate     Robust MLE estimation with multiple starting
                        points
rstable                 Generate random samples from stable
                        distribution
run_all_estimations     Run all EM-based estimations without Gibbs
                        sampling (CRAN-safe)
run_estimations_with_gibbs
                        Run all EM-based estimations with Gibbs
                        sampling (CRAN-safe)
safe_integrate          Safe integration wrapper with multiple fallback
                        strategies
simple_em_real          Simple 2-component EM using ECF initialization
simulate_mixture        Simulate mixture data from alpha-stable
                        components
sine_exp_ralpha         Sine exponential function
sine_log_weighted_exp_ralpha
                        Sine-log-weighted exponential with r^(-alpha)
                        term
sine_r_weighted_exp_ralpha
                        Sine-r-weighted exponential function
sine_weighted_exp_ralpha
                        Sine-weighted exponential with r^alpha term
skew_kurtosis           Calculate skewness and kurtosis
stable_fit_init         Initialize stable distribution parameters
test_normality          Test normality using multiple statistical tests
unpack_params           Helper function to unpack parameters
validate_params         Validate and clip parameters for stable
                        distribution
wasserstein_distance_mixture
                        Wasserstein distance between two mixture
                        distributions
