scipy.stats.dpareto_lognorm#
- scipy.stats.dpareto_lognorm = <scipy.stats._continuous_distns.dpareto_lognorm_gen object>[source]#
- A double Pareto lognormal continuous random variable. - As an instance of the - rv_continuousclass,- dpareto_lognormobject 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_lognormis:\[f(x, \mu, \sigma, \alpha, \beta) = \frac{\alpha \beta}{(\alpha + \beta) x} \phi\left( \frac{\log x - \mu}{\sigma} \right) \left( R(y_1) + R(y_2) \right)\]- 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_lognormtakes- uas a shape parameter for \(\mu\),- sas a shape parameter for \(\sigma\),- aas a shape parameter for \(\alpha\), and- bas 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 - locand- scaleparameters. Specifically,- dpareto_lognorm.pdf(x, u, s, a, b, loc, scale)is identically equivalent to- dpareto_lognorm.pdf(y, u, s, a, b) / scalewith- 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 - cdfand- 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() 