scipy.stats.burr#
- scipy.stats.burr = <scipy.stats._continuous_distns.burr_gen object>[source]#
A Burr (Type III) continuous random variable.
As an instance of the
rv_continuous
class,burr
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.See also
Notes
The probability density function for
burr
is:\[f(x; c, d) = c d \frac{x^{-c - 1}} {{(1 + x^{-c})}^{d + 1}}\]for \(x >= 0\) and \(c, d > 0\).
burr
takesc
andd
as shape parameters for \(c\) and \(d\).This is the PDF corresponding to the third CDF given in Burr’s list; specifically, it is equation (11) in Burr’s paper [1]. The distribution is also commonly referred to as the Dagum distribution [2]. If the parameter \(c < 1\) then the mean of the distribution does not exist and if \(c < 2\) the variance does not exist [2]. The PDF is finite at the left endpoint \(x = 0\) if \(c * d >= 1\).
The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the
loc
andscale
parameters. Specifically,burr.pdf(x, c, d, loc, scale)
is identically equivalent toburr.pdf(y, c, d) / scale
withy = (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
Burr, I. W. “Cumulative frequency functions”, Annals of Mathematical Statistics, 13(2), pp 215-232 (1942).
- 2(1,2)
- 3
Kleiber, Christian. “A guide to the Dagum distributions.” Modeling Income Distributions and Lorenz Curves pp 97-117 (2008).
Examples
>>> from scipy.stats import burr >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> c, d = 10.5, 4.3 >>> mean, var, skew, kurt = burr.stats(c, d, moments='mvsk')
Display the probability density function (
pdf
):>>> x = np.linspace(burr.ppf(0.01, c, d), ... burr.ppf(0.99, c, d), 100) >>> ax.plot(x, burr.pdf(x, c, d), ... 'r-', lw=5, alpha=0.6, label='burr 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 = burr(c, d) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')
Check accuracy of
cdf
andppf
:>>> vals = burr.ppf([0.001, 0.5, 0.999], c, d) >>> np.allclose([0.001, 0.5, 0.999], burr.cdf(vals, c, d)) True
Generate random numbers:
>>> r = burr.rvs(c, d, size=1000)
And compare the histogram:
>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2) >>> ax.legend(loc='best', frameon=False) >>> plt.show()
Methods
rvs(c, d, loc=0, scale=1, size=1, random_state=None)
Random variates.
pdf(x, c, d, loc=0, scale=1)
Probability density function.
logpdf(x, c, d, loc=0, scale=1)
Log of the probability density function.
cdf(x, c, d, loc=0, scale=1)
Cumulative distribution function.
logcdf(x, c, d, loc=0, scale=1)
Log of the cumulative distribution function.
sf(x, c, d, loc=0, scale=1)
Survival function (also defined as
1 - cdf
, but sf is sometimes more accurate).logsf(x, c, d, loc=0, scale=1)
Log of the survival function.
ppf(q, c, d, loc=0, scale=1)
Percent point function (inverse of
cdf
— percentiles).isf(q, c, d, loc=0, scale=1)
Inverse survival function (inverse of
sf
).moment(order, c, d, loc=0, scale=1)
Non-central moment of the specified order.
stats(c, d, loc=0, scale=1, moments=’mv’)
Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(c, d, 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=(c, d), 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(c, d, loc=0, scale=1)
Median of the distribution.
mean(c, d, loc=0, scale=1)
Mean of the distribution.
var(c, d, loc=0, scale=1)
Variance of the distribution.
std(c, d, loc=0, scale=1)
Standard deviation of the distribution.
interval(confidence, c, d, loc=0, scale=1)
Confidence interval with equal areas around the median.