# scipy.stats.ppcc_max

scipy.stats.ppcc_max(x, brack=(0.0, 1.0), dist='tukeylambda')

  Calculate the shape parameter that maximizes the PPCC.

  The probability plot correlation coefficient (PPCC) plot can be used to determine the optimal shape parameter for a one-parameter family of distributions. `ppcc_max` returns the shape parameter that would maximize the probability plot correlation coefficient for the given data to a one-parameter family of distributions.

  Parameters:
    x array_like

      Input array.

    brack tuple, optional

      Triple (a,b,c) where (a<b<c). If bracket consists of two numbers (a, c) then they are assumed to be a starting interval for a downhill bracket search (see `scipy.optimize.brent`).

    dist str or stats.distributions instance, optional

      Distribution or distribution function name. Objects that look enough like a stats.distributions instance (i.e. they have a `ppf` method) are also accepted. The default is `'tukeylambda'`.

  Returns:
    shape_value float

      The shape parameter at which the probability plot correlation coefficient reaches its max value.

  See also

  `ppcc_plot`, `probplot`, `boxcox`

  Notes

  The brack keyword serves as a starting point which is useful in corner cases. One can use a plot to obtain a rough visual estimate of the location for the maximum to start the search near it.

  References

  [1]

  J.J. Filliben, “The Probability Plot Correlation Coefficient Test for Normality”, Technometrics, Vol. 17, pp. 111-117, 1975.

  [2]

  Engineering Statistics Handbook, NIST/SEMATEC, https://www.itl.nist.gov/div898/handbook/eda/section3/ppccplot.htm

  Examples

  First we generate some random data from a Weibull distribution with shape parameter 2.5:

      >>> import numpy as np
      >>> from scipy import stats
      >>> import matplotlib.pyplot as plt
      >>> rng = np.random.default_rng()
      >>> c = 2.5
      >>> x = stats.weibull_min.rvs(c, scale=4, size=2000, random_state=rng)

  Generate the PPCC plot for this data with the Weibull distribution.

      >>> fig, ax = plt.subplots(figsize=(8, 6))
      >>> res = stats.ppcc_plot(x, c/2, 2*c, dist='weibull_min', plot=ax)

  We calculate the value where the shape should reach its maximum and a red line is drawn there. The line should coincide with the highest point in the PPCC graph.

      >>> cmax = stats.ppcc_max(x, brack=(c/2, 2*c), dist='weibull_min')
      >>> ax.axvline(cmax, color='r')
      >>> plt.show()