feat(distributions): add logk distribution and k_dist compound sampler

Introduce logk_gen (Y = ln X where X ~ K) with analytically derived
mean/variance via the CGF, numerically stable logpdf using an asymptotic
Bessel expansion for large arguments, CDF delegation to k_dist, and a
compound-gamma rvs sampler.

Add _rvs to k_dist via the same compound-gamma algorithm and extend
TestKDistPdf with stats and rvs coverage. Add a full TestLogK suite
covering pdf normalization, change-of-variable identity, CDF consistency,
analytical moment checks, and rvs moment checks.

Module-level docstring added to distributions.py

etc/tests/test_distributions.py

@@ -7,7 +7,7 @@ import os
 
 sys.path.insert(0, os.path.join(os.path.dirname(__file__), ".."))
 
-from tools.distributions import k_dist, lognakagami, loggamma_dist, lograyleigh, logrice
+from tools.distributions import k_dist, logk, lognakagami, loggamma_dist, lograyleigh, logrice
 
 
 X = np.linspace(0.01, 10.0, 500)
@@ -73,6 +73,48 @@ class TestKDistPdf:
         pdf_ba = k_dist.pdf(x_test, mu=1.0, alpha=3.0, beta=2.0)
         np.testing.assert_allclose(pdf_ab, pdf_ba, rtol=1e-6)
 
+    def test_stats_mean_equals_mu(self):
+        """Analytical mean must equal mu."""
+        for mu in [0.5, 1.0, 3.0]:
+            dist_mean = float(k_dist.stats(mu=mu, alpha=2.0, beta=3.0, moments="m"))
+            assert pytest.approx(dist_mean, rel=1e-10) == mu
+
+    def test_stats_variance_formula_eq4(self):
+        """Variance must equal mu^2*(alpha+beta+1)/(alpha*beta) (equation 4)."""
+        mu, alpha, beta = 2.0, 3.0, 2.0
+        expected_var = mu**2 * (alpha + beta + 1) / (alpha * beta)
+        _, dist_var, *_ = k_dist.stats(mu=mu, alpha=alpha, beta=beta, moments="mv")
+        assert pytest.approx(float(dist_var), rel=1e-10) == expected_var
+
+    def test_stats_variance_symmetric_in_alpha_beta(self):
+        """Variance is symmetric in alpha and beta."""
+        mu = 2.0
+        _, var_ab, *_ = k_dist.stats(mu=mu, alpha=2.0, beta=3.0, moments="mv")
+        _, var_ba, *_ = k_dist.stats(mu=mu, alpha=3.0, beta=2.0, moments="mv")
+        assert pytest.approx(float(var_ab), rel=1e-10) == float(var_ba)
+
+    def test_stats_variance_numerical(self):
+        """Analytical variance should match sample variance from rvs."""
+        mu, alpha, beta = 2.0, 2.0, 3.0
+        rng = np.random.default_rng(42)
+        samples = k_dist.rvs(mu=mu, alpha=alpha, beta=beta, size=100_000, random_state=rng)
+        _, dist_var, *_ = k_dist.stats(mu=mu, alpha=alpha, beta=beta, moments="mv")
+        assert pytest.approx(float(np.var(samples)), rel=5e-2) == float(dist_var)
+
+    def test_rvs_samples_are_positive_and_finite(self):
+        """K distribution samples must be positive and finite."""
+        rng = np.random.default_rng(7)
+        samples = k_dist.rvs(mu=1.0, alpha=2.0, beta=2.0, size=500, random_state=rng)
+        assert np.all(samples > 0)
+        assert np.all(np.isfinite(samples))
+
+    def test_rvs_sample_mean_near_mu(self):
+        """Sample mean of k_dist rvs should be close to mu."""
+        mu, alpha, beta = 2.0, 3.0, 2.0
+        rng = np.random.default_rng(0)
+        samples = k_dist.rvs(mu=mu, alpha=alpha, beta=beta, size=100_000, random_state=rng)
+        assert pytest.approx(float(np.mean(samples)), rel=5e-2) == mu
+
 
 # ── Parametric curve plots ───────────────────────────────────────────────────
 
@@ -514,6 +556,127 @@ class TestLogRice:
         assert pytest.approx(samples.mean(), rel=5e-2) == numerical_mean
 
 
+# ── logk unit tests ───────────────────────────────────────────────────────────
+
+Y_LOGK = np.linspace(-10.0, 10.0, 500)
+
+
+class TestLogK:
+    def test_logpdf_is_finite_on_real_line(self):
+        """logpdf must be finite for all real y."""
+        log_vals = logk.logpdf(Y_LOGK, mu=2.0, alpha=3.0, beta=2.0)
+        assert np.all(np.isfinite(log_vals))
+
+    def test_pdf_integrates_to_one(self):
+        """Numerical integral of PDF over the real line should be ≈ 1."""
+        y_fine = np.linspace(-30, 20, 200_000)
+        integral = np.trapezoid(logk.pdf(y_fine, mu=2.0, alpha=3.0, beta=2.0), y_fine)
+        assert pytest.approx(integral, abs=1e-3) == 1.0
+
+    def test_logpdf_equals_log_pdf(self):
+        """logpdf must equal log(pdf) at points where pdf does not underflow."""
+        y_bulk = np.linspace(-5.0, 5.0, 30)
+        log_via_pdf = np.log(logk.pdf(y_bulk, mu=2.0, alpha=3.0, beta=2.0))
+        log_direct = logk.logpdf(y_bulk, mu=2.0, alpha=3.0, beta=2.0)
+        np.testing.assert_allclose(log_direct, log_via_pdf, rtol=1e-6)
+
+    def test_log_transform_relation_to_k_dist(self):
+        """logk.pdf(y) must equal k_dist.pdf(exp(y)) * exp(y) (change-of-variable)."""
+        y_test = np.linspace(-3.0, 5.0, 20)
+        mu, alpha, beta = 2.0, 3.0, 2.0
+        direct = logk.pdf(y_test, mu=mu, alpha=alpha, beta=beta)
+        via_k = k_dist.pdf(np.exp(y_test), mu=mu, alpha=alpha, beta=beta) * np.exp(y_test)
+        np.testing.assert_allclose(direct, via_k, rtol=1e-6)
+
+    def test_cdf_consistent_with_k_dist(self):
+        """logk.cdf(y) must equal k_dist.cdf(exp(y))."""
+        y_test = np.array([-2.0, 0.0, 1.0, 3.0])
+        mu, alpha, beta = 2.0, 2.0, 3.0
+        cdf_logk = logk.cdf(y_test, mu=mu, alpha=alpha, beta=beta)
+        cdf_k = k_dist.cdf(np.exp(y_test), mu=mu, alpha=alpha, beta=beta)
+        np.testing.assert_allclose(cdf_logk, cdf_k, rtol=1e-6)
+
+    def test_cdf_is_monotone_increasing(self):
+        """CDF must be strictly non-decreasing (up to floating-point noise in the saturated tail)."""
+        y_grid = np.linspace(-5.0, 8.0, 30)
+        cdf_vals = logk.cdf(y_grid, mu=2.0, alpha=3.0, beta=2.0)
+        assert np.all(np.diff(cdf_vals) >= -1e-10)
+
+    def test_stats_mean_analytical(self):
+        """Mean must equal ln(mu) - ln(alpha) - ln(beta) + psi(alpha) + psi(beta)."""
+        mu, alpha, beta = 2.0, 3.0, 2.0
+        expected = np.log(mu) - np.log(alpha) - np.log(beta) + sc.digamma(alpha) + sc.digamma(beta)
+        dist_mean = float(logk.stats(mu=mu, alpha=alpha, beta=beta, moments="m"))
+        assert pytest.approx(dist_mean, rel=1e-10) == expected
+
+    def test_stats_variance_analytical(self):
+        """Variance must equal psi_1(alpha) + psi_1(beta)."""
+        mu, alpha, beta = 2.0, 3.0, 2.0
+        expected_var = sc.polygamma(1, alpha) + sc.polygamma(1, beta)
+        _, dist_var, *_ = logk.stats(mu=mu, alpha=alpha, beta=beta, moments="mv")
+        assert pytest.approx(float(dist_var), rel=1e-10) == expected_var
+
+    def test_stats_variance_mu_independent(self):
+        """Variance must not depend on mu (mu is a pure shift in log-space)."""
+        alpha, beta = 3.0, 2.0
+        _, var1, *_ = logk.stats(mu=1.0, alpha=alpha, beta=beta, moments="mv")
+        _, var4, *_ = logk.stats(mu=4.0, alpha=alpha, beta=beta, moments="mv")
+        assert pytest.approx(float(var1), rel=1e-10) == float(var4)
+
+    def test_stats_mean_shifts_by_log_mu(self):
+        """Doubling mu shifts the mean by ln(2) and leaves variance unchanged."""
+        alpha, beta = 3.0, 2.0
+        mean1 = float(logk.stats(mu=1.0, alpha=alpha, beta=beta, moments="m"))
+        mean2 = float(logk.stats(mu=2.0, alpha=alpha, beta=beta, moments="m"))
+        assert pytest.approx(mean2 - mean1, rel=1e-10) == np.log(2.0)
+
+    def test_argcheck_rejects_non_positive_mu(self):
+        """mu <= 0 must not produce a valid PDF value."""
+        val = logk.pdf(0.0, mu=-1.0, alpha=2.0, beta=2.0)
+        assert not (np.isfinite(val) and val > 0)
+
+    def test_argcheck_rejects_non_positive_alpha(self):
+        """alpha <= 0 must not produce a valid PDF value."""
+        val = logk.pdf(0.0, mu=1.0, alpha=-1.0, beta=2.0)
+        assert not (np.isfinite(val) and val > 0)
+
+    def test_argcheck_rejects_non_positive_beta(self):
+        """beta <= 0 must not produce a valid PDF value."""
+        val = logk.pdf(0.0, mu=1.0, alpha=2.0, beta=-1.0)
+        assert not (np.isfinite(val) and val > 0)
+
+    def test_rvs_samples_are_finite(self):
+        """Random samples must be finite real numbers."""
+        rng = np.random.default_rng(42)
+        samples = logk.rvs(mu=2.0, alpha=3.0, beta=2.0, size=200, random_state=rng)
+        assert samples.shape == (200,)
+        assert np.all(np.isfinite(samples))
+
+    def test_rvs_sample_mean_near_analytical(self):
+        """Sample mean of many RVS should be close to the analytical mean."""
+        mu, alpha, beta = 2.0, 3.0, 2.0
+        rng = np.random.default_rng(0)
+        samples = logk.rvs(mu=mu, alpha=alpha, beta=beta, size=100_000, random_state=rng)
+        expected_mean = float(logk.stats(mu=mu, alpha=alpha, beta=beta, moments="m"))
+        assert pytest.approx(float(samples.mean()), rel=5e-2) == expected_mean
+
+    def test_rvs_sample_variance_near_analytical(self):
+        """Sample variance of many RVS should be close to the analytical variance."""
+        mu, alpha, beta = 2.0, 3.0, 2.0
+        rng = np.random.default_rng(1)
+        samples = logk.rvs(mu=mu, alpha=alpha, beta=beta, size=100_000, random_state=rng)
+        _, expected_var, *_ = logk.stats(mu=mu, alpha=alpha, beta=beta, moments="mv")
+        assert pytest.approx(float(np.var(samples)), rel=5e-2) == float(expected_var)
+
+    def test_symmetry_in_alpha_beta(self):
+        """logk PDF is symmetric in alpha and beta."""
+        y_test = np.linspace(-3.0, 5.0, 20)
+        mu = 2.0
+        pdf_ab = logk.pdf(y_test, mu=mu, alpha=2.0, beta=3.0)
+        pdf_ba = logk.pdf(y_test, mu=mu, alpha=3.0, beta=2.0)
+        np.testing.assert_allclose(pdf_ab, pdf_ba, rtol=1e-6)
+
+
 if __name__ == "__main__":
     plot_k_dist_varying_alpha()
     plot_k_dist_varying_mu()

etc/tools/distributions.py

@@ -1,3 +1,164 @@
+"""
+Custom continuous probability distributions for radar clutter modelling.
+
+All classes inherit from ``scipy.stats.rv_continuous``.  After instantiation
+each distribution object exposes the full scipy public API summarised below.
+Parameters ``loc`` and ``scale`` shift and rescale the support; shape
+parameters (e.g. *mu*, *alpha*, *beta*) are passed as keyword arguments.
+
+Public methods (from rv_continuous)
+------------------------------------
+rvs(*args, loc=0, scale=1, size=1, random_state=None)
+    Draw random variates.
+
+pdf(x, *args, loc=0, scale=1)
+    Probability density function evaluated at *x*.
+
+logpdf(x, *args, loc=0, scale=1)
+    Natural logarithm of the PDF; numerically preferred when values are small.
+
+cdf(x, *args, loc=0, scale=1)
+    Cumulative distribution function: P(X <= x).
+
+logcdf(x, *args, loc=0, scale=1)
+    Log of the CDF; avoids underflow in the far left tail.
+
+sf(x, *args, loc=0, scale=1)
+    Survival function: 1 - CDF, i.e. P(X > x).
+
+logsf(x, *args, loc=0, scale=1)
+    Log of the survival function; avoids underflow in the far right tail.
+
+ppf(q, *args, loc=0, scale=1)
+    Percent-point function (quantile): inverse of the CDF.
+
+isf(q, *args, loc=0, scale=1)
+    Inverse survival function: inverse of sf.
+
+moment(order, *args, loc=0, scale=1)
+    Non-central raw moment of the specified integer order.
+
+stats(*args, loc=0, scale=1, moments='mv')
+    Summary statistics selected by *moments* string: 'm' mean, 'v' variance,
+    's' skewness, 'k' excess kurtosis.
+
+entropy(*args, loc=0, scale=1)
+    Differential (Shannon) entropy of the distribution.
+
+expect(func, args, loc=0, scale=1, lb=None, ub=None, conditional=False)
+    Expected value of *func(x)* computed by numerical integration.
+
+median(*args, loc=0, scale=1)
+    Median of the distribution.
+
+mean(*args, loc=0, scale=1)
+    Mean of the distribution.
+
+std(*args, loc=0, scale=1)
+    Standard deviation of the distribution.
+
+var(*args, loc=0, scale=1)
+    Variance of the distribution.
+
+interval(confidence, *args, loc=0, scale=1)
+    Confidence interval with equal probability mass on each side of the median.
+
+__call__(*args, loc=0, scale=1)
+    Freeze the distribution — returns a frozen instance with fixed parameters,
+    so methods can be called without repeating shape arguments.
+
+fit(data, *args, **kwds)
+    Maximum-likelihood estimates of shape, loc, and scale from *data*.
+
+fit_loc_scale(data, *args)
+    Quick loc and scale estimates via method of moments (mean and variance).
+
+nnlf(theta, x)
+    Negative log-likelihood for parameter vector *theta* and observations *x*.
+
+support(*args, loc=0, scale=1)
+    Lower and upper endpoints of the distribution's support.
+
+Overrideable private methods
+-----------------------------
+Subclasses customise behaviour by implementing the private counterparts
+listed below.  When a private method is *not* overridden scipy falls back to
+a default implementation — often a slower or less precise one.  Override to
+gain speed or numerical accuracy.
+
+_argcheck(*args)
+    Validate shape parameters; return a boolean array.
+    Default: always ``True``.  Should be overridden to reject invalid values.
+
+_get_support(*args)
+    Return ``(lower, upper)`` endpoints of the support as a function of the
+    shape parameters.  Default: returns the ``(a, b)`` constants passed at
+    construction time.
+
+_pdf(x, *args)
+    Core of the density.  No default — must be implemented (or ``_logpdf``).
+
+_logpdf(x, *args)
+    Log-density.  Default: ``log(_pdf(x))``, which loses precision when
+    ``_pdf`` underflows to zero.  Override whenever a stable closed form
+    exists.
+
+_cdf(x, *args)
+    Core of the cumulative distribution function.  Default: numerical
+    integration of ``_pdf`` from the lower support boundary — slow.
+
+_sf(x, *args)
+    Survival function P(X > x).  Default: ``1 - _cdf(x)``, which loses
+    significant digits when ``_cdf(x)`` is close to 1 (far right tail).
+    Override with a direct formula whenever one exists.
+
+_logcdf(x, *args)
+    Log of the CDF.  Default: ``log(_cdf(x))``.
+
+_logsf(x, *args)
+    Log of the survival function.  Default: ``log(_sf(x))`` = ``log(1 -
+    _cdf(x))``, which is catastrophically inaccurate in the far right tail.
+    Override to avoid cancellation, e.g. via ``log1p(-_cdf(x))`` or a
+    complementary special function.
+
+_ppf(q, *args)
+    Percent-point (quantile) function.  Default: numerical inversion of
+    ``_cdf`` — slow.  Override with a closed-form inverse when available.
+
+_isf(q, *args)
+    Inverse survival function.  Default: ``_ppf(1 - q)``, which loses
+    precision for small *q* (extreme quantiles).  Override when the
+    complementary special function provides a direct inverse.
+
+_stats(*args, moments='mv')
+    Return ``(mean, variance, skewness, excess_kurtosis)``; use ``None`` for
+    moments you do not compute.  Default: numerical integration — override
+    with analytical expressions for speed and accuracy.
+
+_munp(n, *args)
+    Non-central raw moment of integer order *n*.  Default: numerical
+    integration.  Implement when closed-form moments are available and
+    ``_stats`` is not sufficient.
+
+_entropy(*args)
+    Differential entropy.  Default: numerical integration of ``-f log f``.
+    Override with a closed-form expression when one exists.
+
+_rvs(*args, size=None, random_state=None)
+    Random variate sampler.  Default: CDF-inversion via ``_ppf`` — slow for
+    distributions without a closed-form quantile function.  Override with
+    a direct simulation algorithm (e.g. a compound-distribution sampler).
+
+Numerical accuracy summary
+~~~~~~~~~~~~~~~~~~~~~~~~~~
+Priority order for overriding to improve tail accuracy:
+
+1. ``_logsf`` / ``_logcdf`` — first line of defence against underflow.
+2. ``_sf``                   — avoids ``1 - cdf`` cancellation.
+3. ``_isf``                  — avoids ``ppf(1 - q)`` cancellation.
+
+"""
+
 from scipy.stats import rv_continuous, ncx2
 from scipy.special import kve, gammaln
 from scipy.integrate import quad
@@ -88,6 +249,11 @@ class k_gen(rv_continuous):
             return val
         return np.vectorize(_scalar)(x, mu, alpha, beta)
 
+    def _rvs(self, mu, alpha, beta, size=None, random_state=None):
+        # Compound gamma: tau ~ Gamma(beta, mu/beta), X|tau ~ Gamma(alpha, tau/alpha)
+        tau = random_state.gamma(beta, mu / beta, size=size)
+        return random_state.gamma(alpha, tau / alpha)
+
     def fit(self, data, *args, **kwds):
         if ("loc" in kwds and kwds["loc"] != 0.0) or ("floc" in kwds and kwds["floc"] != 0.0):
             raise ValueError("k_distribution uses a fixed loc=0; use mu to control the mean/scale.")
@@ -102,6 +268,117 @@ class k_gen(rv_continuous):
 k_dist = k_gen(a=0.0, name="k_distribution", shapes="mu, alpha, beta")
 
 
+class logk_gen(rv_continuous):
+    """Log-K continuous random variable.
+
+    Y = ln(X) where X ~ K(mu, alpha, beta). The PDF is:
+
+        f(y; mu, alpha, beta) = 2 / (Gamma(alpha) * Gamma(beta))
+                                * (alpha*beta/mu)^((alpha+beta)/2)
+                                * exp((alpha+beta)/2 * y)
+                                * K_{alpha-beta}(2*sqrt(alpha*beta*exp(y)/mu))
+
+    for y in (-inf, +inf), mu > 0, alpha > 0, beta > 0.
+
+    The cumulants of Y follow from the CGF
+        K(t) = t*ln(mu/(alpha*beta)) + lnGamma(alpha+t) - lnGamma(alpha)
+                                     + lnGamma(beta+t)  - lnGamma(beta),
+    giving:
+        E[Y]   = ln(mu) - ln(alpha) - ln(beta) + psi(alpha) + psi(beta)
+        Var[Y] = psi_1(alpha) + psi_1(beta)
+    """
+
+    def _shape_info(self):
+        return [
+            _ShapeInfo("mu", domain=(0, np.inf), inclusive=(False, True)),
+            _ShapeInfo("alpha", domain=(0, np.inf), inclusive=(True, True)),
+            _ShapeInfo("beta", domain=(0, np.inf), inclusive=(True, True)),
+        ]
+
+    def _argcheck(self, mu, alpha, beta):
+        return (mu > 0) & (alpha > 0) & (beta > 0)
+
+    @staticmethod
+    def _log_kve(v, z):
+        """log(kve(v, z)) = log(Kv(z)) + z, stable for all z > 0.
+
+        For large z, kve(v, z) ≈ sqrt(π/(2z)) underflows to 0 in float64,
+        making log(kve) return -inf/-nan.  The asymptotic expansion:
+
+            log(kve(v,z)) ≈ 0.5*log(π/(2z)) + log1p((4v²-1)/(8z) +
+                             (4v²-1)(4v²-9)/(128z²))
+
+        is used whenever the direct evaluation would underflow.
+        """
+        z = np.asarray(z, dtype=float)
+        v = np.asarray(v, dtype=float)
+        kve_val = kve(v, z)
+        # Asymptotic (2-term Hankel expansion) — accurate to O(z^{-3})
+        mu_v = 4.0 * v ** 2
+        log_asymp = (
+            0.5 * (np.log(np.pi) - np.log(2.0) - np.log(np.where(z > 0, z, 1.0)))
+            + np.log1p((mu_v - 1.0) / (8.0 * np.where(z > 0, z, 1.0))
+                       + (mu_v - 1.0) * (mu_v - 9.0) / (128.0 * np.where(z > 0, z**2, 1.0)))
+        )
+        return np.where(kve_val > 0, np.log(np.where(kve_val > 0, kve_val, 1.0)), log_asymp)
+
+    def _pdf(self, y, mu, alpha, beta):
+        return np.exp(self._logpdf(y, mu, alpha, beta))
+
+    def _logpdf(self, y, mu, alpha, beta):
+        half_sum = (alpha + beta) / 2.0
+        log_ab_over_mu = np.log(alpha) + np.log(beta) - np.log(mu)
+        z = 2.0 * np.sqrt(alpha * beta * np.exp(y) / mu)
+        return (
+            np.log(2.0)
+            - gammaln(alpha)
+            - gammaln(beta)
+            + half_sum * log_ab_over_mu
+            + half_sum * y
+            + self._log_kve(alpha - beta, z)
+            - z
+        )
+
+    def _stats(self, mu, alpha, beta):
+        mean = np.log(mu) - np.log(alpha) - np.log(beta) + sc.digamma(alpha) + sc.digamma(beta)
+        var = sc.polygamma(1, alpha) + sc.polygamma(1, beta)
+        g1 = (sc.polygamma(2, alpha) + sc.polygamma(2, beta)) / var**1.5
+        g2 = (sc.polygamma(3, alpha) + sc.polygamma(3, beta)) / var**2
+        return mean, var, g1, g2
+
+    def _cdf(self, y, mu, alpha, beta):
+        return k_dist.cdf(np.exp(y), mu, alpha, beta)
+
+    def _rvs(self, mu, alpha, beta, size=None, random_state=None):
+        tau = random_state.gamma(beta, mu / beta, size=size)
+        sample = random_state.gamma(alpha, tau / alpha)
+        return np.log(np.clip(sample, np.finfo(float).tiny, None))
+
+    def fit(self, data, *args, **kwds):
+        if ("loc" in kwds and kwds["loc"] != 0.0) or ("floc" in kwds and kwds["floc"] != 0.0):
+            raise ValueError("logk uses a fixed loc=0.")
+        if ("scale" in kwds and kwds["scale"] != 1.0) or ("fscale" in kwds and kwds["fscale"] != 1.0):
+            raise ValueError("logk uses a fixed scale=1.")
+        kwds.pop("loc", None)
+        kwds.pop("scale", None)
+        # Supply data-driven initial guesses when none are provided so the
+        # optimizer starts close to the data instead of the default (1,1,1).
+        # E[Y] = ln(mu) + ln(alpha) + ln(beta) - digamma(alpha) - digamma(beta)
+        # => mu0 = exp(mean(data) + ln(a0) + ln(b0) - psi(a0) - psi(b0))
+        if not args:
+            alpha0, beta0 = 1.0, 1.0
+            mu0 = float(np.exp(
+                np.mean(data)
+                + np.log(alpha0) + np.log(beta0)
+                - sc.digamma(alpha0) - sc.digamma(beta0)
+            ))
+            args = (mu0, alpha0, beta0)
+        return super().fit(data, *args, floc=0.0, fscale=1.0, **kwds)
+
+
+logk = logk_gen(name="logk", shapes="mu, alpha, beta")
+
+
 class lognakagami_gen(rv_continuous):
     """Log-Nakagami continuous random variable.