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# scipy.stats.mielke

scipy.stats.mielke = <scipy.stats._continuous_distns.mielke_gen object>

  A Mielke Beta-Kappa / Dagum continuous random variable.

  As an instance of the `rv_continuous` class, `mielke` 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(k, s, loc=0, scale=1, size=1, random_state=None) | Random variates.
  pdf(x, k, s, loc=0, scale=1) | Probability density function.
  logpdf(x, k, s, loc=0, scale=1) | Log of the probability density function.
  cdf(x, k, s, loc=0, scale=1) | Cumulative distribution function.
  logcdf(x, k, s, loc=0, scale=1) | Log of the cumulative distribution function.
  sf(x, k, s, loc=0, scale=1) | Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
  logsf(x, k, s, loc=0, scale=1) | Log of the survival function.
  ppf(q, k, s, loc=0, scale=1) | Percent point function (inverse of `cdf` — percentiles).
  isf(q, k, s, loc=0, scale=1) | Inverse survival function (inverse of `sf`).
  moment(order, k, s, loc=0, scale=1) | Non-central moment of the specified order.
  stats(k, s, loc=0, scale=1, moments=’mv’) | Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
  entropy(k, s, 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=(k, s), 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(k, s, loc=0, scale=1) | Median of the distribution.
  mean(k, s, loc=0, scale=1) | Mean of the distribution.
  var(k, s, loc=0, scale=1) | Variance of the distribution.
  std(k, s, loc=0, scale=1) | Standard deviation of the distribution.
  interval(confidence, k, s, loc=0, scale=1) | Confidence interval with equal areas around the median.

  Notes

  The probability density function for `mielke` is:

  \[f(x, k, s) = \frac{k x^{k-1}}{(1+x^s)^{1+k/s}}\]

  for \(x > 0\) and \(k, s > 0\). The distribution is sometimes called Dagum distribution ([2]). It was already defined in [3], called a Burr Type III distribution (`burr` with parameters `c=s` and `d=k/s`).

  `mielke` takes `k` and `s` as shape parameters.

  The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the `loc` and `scale` parameters. Specifically, `mielke.pdf(x, k, s, loc, scale)` is identically equivalent to `mielke.pdf(y, k, s) / 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]

  Mielke, P.W., 1973 “Another Family of Distributions for Describing and Analyzing Precipitation Data.” J. Appl. Meteor., 12, 275-280

  [2]

  Dagum, C., 1977 “A new model for personal income distribution.” Economie Appliquee, 33, 327-367.

  [3]

  Burr, I. W. “Cumulative frequency functions”, Annals of Mathematical Statistics, 13(2), pp 215-232 (1942).

  Examples

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

  Get the support:

      >>> k, s = 10.4, 4.6
      >>> lb, ub = mielke.support(k, s)

  Calculate the first four moments:

      >>> mean, var, skew, kurt = mielke.stats(k, s, moments='mvsk')

  Display the probability density function (`pdf`):

      >>> x = np.linspace(mielke.ppf(0.01, k, s),
      ...                 mielke.ppf(0.99, k, s), 100)
      >>> ax.plot(x, mielke.pdf(x, k, s),
      ...        'r-', lw=5, alpha=0.6, label='mielke 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 = mielke(k, s)
      >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

  Check accuracy of `cdf` and `ppf`:

      >>> vals = mielke.ppf([0.001, 0.5, 0.999], k, s)
      >>> np.allclose([0.001, 0.5, 0.999], mielke.cdf(vals, k, s))
      True

  Generate random numbers:

      >>> r = mielke.rvs(k, s, 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()