gkwgof: Comprehensive Goodness-of-Fit Analysis for GKw Family...

In gkwreg: Generalized Kumaraswamy Regression Models for Bounded Data

Comprehensive Goodness-of-Fit Analysis for GKw Family Distributions

Description

Computes and displays a comprehensive set of goodness-of-fit statistics for distributions from the Generalized Kumaraswamy (GKw) family fitted using gkwfit. This function provides various measures including distance-based tests, information criteria, moment comparisons, probability plot metrics, and additional visualization tools for model adequacy assessment.

Usage

gkwgof(
  object,
  simulate_p_values = FALSE,
  n_bootstrap = 1000,
  plot = TRUE,
  print_summary = TRUE,
  verbose = FALSE,
  theme = ggplot2::theme_bw(),
  ncols = 4,
  title = NULL,
  ...
)

Arguments

object	An object of class "gkwfit", typically the result of a call to gkwfit.
simulate_p_values	Logical; if TRUE, uses parametric bootstrap to compute approximate p-values for distance-based tests. Default is FALSE.
n_bootstrap	Number of bootstrap replicates for p-value simulation. Only used if simulate_p_values = TRUE. Default is 1000.
plot	Logical; if TRUE, generates additional diagnostic plots beyond those already in the gkwfit object. Default is TRUE.
print_summary	Logical; if TRUE, prints a formatted summary of the goodness-of-fit statistics. Default is TRUE.
verbose	Logical; if TRUE, provides additional details and explanations about the test statistics. Default is FALSE.
theme	A ggplot theme for all plots. default ggplot2::theme_bw()
ncols	Number of columns to draw plots in graphics window.
title	Plot title.
...	Additional arguments to be passed to plotting functions.

Details

This function calculates the following goodness-of-fit statistics:

Distance-based tests:

• Kolmogorov-Smirnov (KS) statistic: Measures the maximum absolute difference between the empirical and theoretical CDFs.

• Cramer-von Mises (CvM) statistic: Measures the integrated squared difference between the empirical and theoretical CDFs.

• Anderson-Darling (AD) statistic: Similar to CvM but places more weight on the tails of the distribution.

• Watson (W^2) statistic: A modification of CvM that is location-invariant on the circle.

Information criteria:

• Akaike Information Criterion (AIC): -2\log(L) + 2k

• Bayesian Information Criterion (BIC): -2\log(L) + k\log(n)

• Corrected AIC (AICc): AIC + \frac{2k(k+1)}{n-k-1}

• Consistent AIC (CAIC): -2\log(L) + k(\log(n) + 1)

• Hannan-Quinn IC (HQIC): -2\log(L) + 2k\log(\log(n))

Moment-based comparisons:

• Theoretical vs. sample mean, variance, skewness, and kurtosis

• Standardized moment differences (relative to sample standard deviation)

• Root mean squared error of moments (RMSE)

Probability plot metrics:

• Correlation coefficient from P-P plot (closer to 1 indicates better fit)

• Area between P-P curve and diagonal line

• Mean absolute error in Q-Q plot

Likelihood statistics:

• Log-likelihood

• Log-likelihood per observation

• Pseudo-R^2 measure for bounded distributions

Prediction accuracy metrics:

• Mean Absolute Error (MAE) between empirical and theoretical CDF

• Root Mean Squared Error (RMSE) between empirical and theoretical CDF

• Continuous Ranked Probability Score (CRPS)

For model selection, lower values of information criteria (AIC, BIC, etc.) indicate better fit, while higher values of correlation coefficients and pseudo-R^2 indicate better fit. The distance-based tests are primarily used for hypothesis testing rather than model selection.

When simulate_p_values = TRUE, the function performs parametric bootstrap to compute approximate p-values for the distance-based tests, which accounts for parameter estimation uncertainty.

Value

An object of class "gkwgof" (inheriting from "list") containing the following components:

• family: The fitted distribution family

• coefficients: The estimated model parameters

• sample_size: The number of observations used in fitting

• distance_tests: Results from KS, CvM, AD, and Watson tests

• information_criteria: AIC, BIC, AICc, CAIC, and HQIC values

• moments: Theoretical moments, sample moments, and their differences

• probability_plots: Metrics from P-P and Q-Q plots

• likelihood: Log-likelihood statistics and pseudo-R^2 measure

• prediction: Prediction accuracy metrics

• plots: Additional diagnostic plots (if plot = TRUE)

• p_values: Simulated p-values (if simulate_p_values = TRUE)

• bootstrap_stats: Bootstrap distribution of test statistics (if simulate_p_values = TRUE)

• call: The matched call

• gkwfit_object: The original gkwfit object used for analysis

References

Anderson, T. W., & Darling, D. A. (1952). Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes. Annals of Mathematical Statistics, 23(2), 193-212.

Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (2nd ed.). Springer.

D'Agostino, R. B., & Stephens, M. A. (1986). Goodness-of-fit techniques. Marcel Dekker, Inc.

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.

See Also

gkwfit, summary.gkwfit, print.gkwgof, plot.gkwgof, plotcompare

Examples

# Example 1: Simulate and analyze data from a Kumaraswamy (Kw) distribution
set.seed(2203) # Set seed for reproducibility
# Simulate 1000 observations from Kumaraswamy distribution with parameters alpha=2.5, beta=1.8
data_kw <- rkw(n = 1000, alpha = 2.5, beta = 1.8)
# Fit the Kumaraswamy distribution to the data
fit_kw <- gkwfit(data_kw, family = "kw")
# Basic goodness-of-fit analysis
gof_kw <- gkwgof(fit_kw)
# Analysis with bootstrap simulated p-values
gof_kw_bootstrap <- gkwgof(fit_kw, simulate_p_values = TRUE, n_bootstrap = 500)
# Detailed summary with additional explanations
gof_kw_verbose <- gkwgof(fit_kw, verbose = TRUE)

# Example 2: Comparing different distributions from the GKw family
# Simulate data from the Generalized Kumaraswamy (GKw) distribution
data_gkw <- rgkw(n = 1000, alpha = 2.0, beta = 1.5, gamma = 3.0, delta = 2.0, lambda = 1.2)
# Fit different models to the same data
fit_kw <- gkwfit(data_gkw, family = "kw") # Kumaraswamy model (simplified)
fit_bkw <- gkwfit(data_gkw, family = "bkw") # Beta-Kumaraswamy model
fit_ekw <- gkwfit(data_gkw, family = "ekw") # Exponentiated Kumaraswamy model
fit_gkw <- gkwfit(data_gkw, family = "gkw") # Full Generalized Kumaraswamy model
fit_beta <- gkwfit(data_gkw, family = "beta") # Standard Beta model
# Goodness-of-fit analysis for each model (without printing summaries)
gof_kw <- gkwgof(fit_kw, print_summary = FALSE)
gof_bkw <- gkwgof(fit_bkw, print_summary = FALSE)
gof_ekw <- gkwgof(fit_ekw, print_summary = FALSE)
gof_gkw <- gkwgof(fit_gkw, print_summary = FALSE)
gof_beta <- gkwgof(fit_beta, print_summary = FALSE)
# Information criteria comparison for model selection
ic_comparison <- data.frame(
  family = c("kw", "bkw", "ekw", "gkw", "beta"),
  n_params = c(
    length(gof_kw$coefficients),
    length(gof_bkw$coefficients),
    length(gof_ekw$coefficients),
    length(gof_gkw$coefficients),
    length(gof_beta$coefficients)
  ),
  logLik = c(
    gof_kw$likelihood$loglik,
    gof_bkw$likelihood$loglik,
    gof_ekw$likelihood$loglik,
    gof_gkw$likelihood$loglik,
    gof_beta$likelihood$loglik
  ),
  AIC = c(
    gof_kw$information_criteria$AIC,
    gof_bkw$information_criteria$AIC,
    gof_ekw$information_criteria$AIC,
    gof_gkw$information_criteria$AIC,
    gof_beta$information_criteria$AIC
  ),
  BIC = c(
    gof_kw$information_criteria$BIC,
    gof_bkw$information_criteria$BIC,
    gof_ekw$information_criteria$BIC,
    gof_gkw$information_criteria$BIC,
    gof_beta$information_criteria$BIC
  ),
  AICc = c(
    gof_kw$information_criteria$AICc,
    gof_bkw$information_criteria$AICc,
    gof_ekw$information_criteria$AICc,
    gof_gkw$information_criteria$AICc,
    gof_beta$information_criteria$AICc
  )
)
# Sort by AIC (lower is better)
ic_comparison <- ic_comparison[order(ic_comparison$AIC), ]
print(ic_comparison)

# Example 3: Comparative visualization
# Generate data from Beta distribution to demonstrate another case
set.seed(2203)
data_beta <- rbeta_(n = 1000, gamma = 2.5, delta = 1.5)
# Fit different distributions
fit_beta_true <- gkwfit(data_beta, family = "beta")
fit_kw_misspec <- gkwfit(data_beta, family = "kw")
fit_gkw_complex <- gkwfit(data_beta, family = "gkw")
# Goodness-of-fit analysis
gof_beta_true <- gkwgof(fit_beta_true, print_summary = FALSE, plot = FALSE)
gof_kw_misspec <- gkwgof(fit_kw_misspec, print_summary = FALSE, plot = FALSE)
gof_gkw_complex <- gkwgof(fit_gkw_complex, print_summary = FALSE, plot = FALSE)
# Comparative goodness-of-fit plot

plotcompare(
  list(
    "Beta (correct)" = gof_beta_true,
    "Kw (underspecified)" = gof_kw_misspec,
    "GKw (overspecified)" = gof_gkw_complex
  ),
  title = "Comparison of fits for Beta data"
)

# Example 5: Likelihood ratio tests for nested models
# Testing if the GKw distribution is significantly better than BKw (lambda = 1)
# Null hypothesis: lambda = 1 (BKw is adequate)
# Alternative hypothesis: lambda != 1 (GKw is necessary)
# Fitting nested models to data
nested_data <- rgkw(n = 1000, alpha = 2.0, beta = 1.5, gamma = 3.0, delta = 2.0, lambda = 1.0)
nested_fit_bkw <- gkwfit(nested_data, family = "bkw") # Restricted model (lambda = 1)
nested_fit_gkw <- gkwfit(nested_data, family = "gkw") # Unrestricted model

# Extracting log-likelihoods
ll_bkw <- nested_fit_bkw$loglik
ll_gkw <- nested_fit_gkw$loglik
# Calculating likelihood ratio test statistic
lr_stat <- 2 * (ll_gkw - ll_bkw)
# Calculating p-value (chi-square with 1 degree of freedom)
lr_pvalue <- 1 - pchisq(lr_stat, df = 1)
# Displaying test results
cat("Likelihood ratio test:\n")
cat("Test statistic:", round(lr_stat, 4), "\n")
cat("P-value:", format.pval(lr_pvalue), "\n")
cat("Conclusion:", ifelse(lr_pvalue < 0.05,
  "Reject H0 - GKw is necessary",
  "Fail to reject H0 - BKw is adequate"
), "\n")

# Example 6: Power simulation
# Checking the power of goodness-of-fit tests in detecting misspecifications
# Function to simulate power
simulate_power <- function(n_sim = 1000, n_obs = 1000, alpha_level = 0.05) {
  # Counters for rejections
  ks_rejections <- 0
  ad_rejections <- 0

  for (i in 1:n_sim) {
    # Simulating data from GKw, but fitting a Kw model (incorrect)
    sim_data <- rgkw(
      n = n_obs, alpha = 2.0, beta = 1.5,
      gamma = 3.0, delta = 2.0, lambda = 1.5
    )

    # Fitting incorrect model
    sim_fit_kw <- gkwfit(sim_data, family = "kw")

    # Calculating test statistics
    sim_gof <- gkwgof(sim_fit_kw,
      simulate_p_values = TRUE,
      n_bootstrap = 200, print_summary = FALSE, plot = FALSE
    )

    # Checking rejections in tests
    if (sim_gof$p_values$ks < alpha_level) ks_rejections <- ks_rejections + 1
    if (sim_gof$p_values$ad < alpha_level) ad_rejections <- ad_rejections + 1
  }

  # Calculating power
  power_ks <- ks_rejections / n_sim
  power_ad <- ad_rejections / n_sim

  return(list(
    power_ks = power_ks,
    power_ad = power_ad
  ))
}

# Run power simulation with reduced number of repetitions for example
power_results <- simulate_power(n_sim = 100)
cat("Estimated power (KS):", round(power_results$power_ks, 2), "\n")
cat("Estimated power (AD):", round(power_results$power_ad, 2), "\n")

gkwreg documentation built on April 16, 2025, 1:10 a.m.