feat(distributions): add lognakagami, loggamma, and kl_statistic

Implement two new scipy-compatible distributions : Log-Nakagami
(lognakagami) and Log-Gamma (loggamma_dist), with complete
logpdf/cdf/ppf/stats/entropy/rvs methods derived from the
change-of-variable Y = ln(X).

Add kl_statistic, a KDE-based KL-divergence goodness-of-fit callable
compatible with the Fitter class. Extend k_gen with _stats (improving speed), _cdf, and
a fit guard, and switch kv → kve to improve numerical stability at large arguments.

Add unit tests for all three additions covering normalization,
monotonicity, ppf inversion, moment formulas, and Fitter integration.

etc/tools/statistics.py

@@ -1,4 +1,5 @@
 import numpy as np
+from scipy.stats import gaussian_kde
 
 
 def aic_statistic(dist, data, axis):
@@ -32,4 +33,30 @@ def bic_statistic(dist, data, axis):
     log_likelihood = np.sum(dist.logpdf(data), axis=axis)
     bic = np.log(n) * k - 2 * log_likelihood
 
-    return bic
+    return bic
+
+def kl_statistic(dist, data, axis):
+    """
+    KL divergence-based goodness-of-fit statistic.
+
+    KL(P || Q) = ∑ P(x) log(P(x) / Q(x))
+
+    Lower KL divergence indicates better fit, but since goodness_of_fit()
+    treats larger statistic values as worse fit, KL works directly.
+    """
+
+    # Estimate the PDF of the data using KDE
+    kde = gaussian_kde(data)
+    data_pdf = kde(data)
+
+    # Get the PDF of the distribution at the data points
+    dist_pdf = dist.pdf(data)
+
+    # normalize the PDFs to ensure they sum to 1
+    data_pdf /= np.sum(data_pdf)
+    dist_pdf /= np.sum(dist_pdf)
+    # Avoid division by zero and log of zero by adding a small constant
+    epsilon = 1e-10
+    kl_divergence = np.sum(data_pdf * np.log((data_pdf + epsilon) / (dist_pdf + epsilon)), axis=axis)
+
+    return kl_divergence