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# scipy.stats.dpareto_lognorm

scipy.stats.dpareto_lognorm = <scipy.stats._continuous_distns.dpareto_lognorm_gen object>

  A double Pareto lognormal continuous random variable.

  As an instance of the `rv_continuous` class, `dpareto_lognorm` object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

  Methods

  rvs(u, s, a, b, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, u, s, a, b, loc=0, scale=1) | Probability density function.
  logpdf(x, u, s, a, b, loc=0, scale=1) | Log of the probability density function.
  cdf(x, u, s, a, b, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, u, s, a, b, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, u, s, a, b, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, u, s, a, b, loc=0, scale=1) | Log of the survival function.
  ppf(q, u, s, a, b, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, u, s, a, b, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, u, s, a, b, loc=0, scale=1) | Non-central moment of the specified order.
  stats(u, s, a, b, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(u, s, a, b, loc=0, scale=1) | (Differential) entropy of the RV.
  fit(data) | Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments.
  expect(func, args=(u, s, a, b), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) | Expected value of a function (of one argument) with respect to the distribution.
  median(u, s, a, b, loc=0, scale=1) | Median of the distribution.
  mean(u, s, a, b, loc=0, scale=1) | Mean of the distribution.
  var(u, s, a, b, loc=0, scale=1) | Variance of the distribution.
  std(u, s, a, b, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, u, s, a, b, loc=0, scale=1) | Confidence interval with equal areas around the median.

  Notes

  The probability density function for `dpareto_lognorm` is:

  \[f(x, \mu, \sigma, \alpha, \beta) = \frac{\alpha \beta}{(\alpha + \beta) x} \phi\left( \frac{\log x - \mu}{\sigma} \right) \left( R(y_1) + R(y_2) \right)\]

  where \(R(t) = \frac{1 - \Phi(t)}{\phi(t)}\), \(\phi\) and \(\Phi\) are the normal PDF and CDF, respectively, \(y_1 = \alpha \sigma - \frac{\log x - \mu}{\sigma}\), and \(y_2 = \beta \sigma + \frac{\log x - \mu}{\sigma}\) for real numbers \(x\) and \(\mu\), \(\sigma > 0\), \(\alpha > 0\), and \(\beta > 0\) [1].

  `dpareto_lognorm` takes `u` as a shape parameter for \(\mu\), `s` as a shape parameter for \(\sigma\), `a` as a shape parameter for \(\alpha\), and `b` as a shape parameter for \(\beta\).

  A random variable \(X\) distributed according to the PDF above can be represented as \(X = U \frac{V_1}{V_2}\) where \(U\), \(V_1\), and \(V_2\) are independent, \(U\) is lognormally distributed such that \(\log U \sim N(\mu, \sigma^2)\), and \(V_1\) and \(V_2\) follow Pareto distributions with parameters \(\alpha\) and \(\beta\), respectively [2].

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `dpareto_lognorm.pdf(x, u, s, a, b, loc, scale)` is identically equivalent to `dpareto_lognorm.pdf(y, u, s, a, b) / scale` with `y = (x - loc) / scale`. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

  References

  [1]

  Hajargasht, Gholamreza, and William E. Griffiths. “Pareto-lognormal distributions: Inequality, poverty, and estimation from grouped income data.” Economic Modelling 33 (2013): 593-604.

  [2]

  Reed, William J., and Murray Jorgensen. “The double Pareto-lognormal distribution - a new parametric model for size distributions.” Communications in Statistics - Theory and Methods 33.8 (2004): 1733-1753.

  Examples

      >>> import numpy as np
      >>> from scipy.stats import dpareto_lognorm
      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots(1, 1)

  Get the support:

      >>> u, s, a, b = 3, 1.2, 1.5, 2
      >>> lb, ub = dpareto_lognorm.support(u, s, a, b)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = dpareto_lognorm.stats(u, s, a, b, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(dpareto_lognorm.ppf(0.01, u, s, a, b),
      ...                 dpareto_lognorm.ppf(0.99, u, s, a, b), 100)
      >>> ax.plot(x, dpareto_lognorm.pdf(x, u, s, a, b),
      ...        'r-', lw=5, alpha=0.6, label='dpareto_lognorm pdf')

  Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

  Freeze the distribution and display the frozen `pdf`:

      >>> rv = dpareto_lognorm(u, s, a, b)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = dpareto_lognorm.ppf([0.001, 0.5, 0.999], u, s, a, b)
      >>> np.allclose([0.001, 0.5, 0.999], dpareto_lognorm.cdf(vals, u, s, a, b))
      True

  Generate random numbers:

      >>> r = dpareto_lognorm.rvs(u, s, a, b, size=1000)

  And compare the histogram:

      >>> ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2)
      >>> ax.set_xlim([x[0], x[-1]])
      >>> ax.legend(loc='best', frameon=False)
      >>> plt.show()