scipy.stats.jf_skew_t#
- scipy.stats.jf_skew_t = <scipy.stats._continuous_distns.jf_skew_t_gen object>[source]#
- Jones and Faddy skew-t distribution. - As an instance of the - rv_continuousclass,- jf_skew_tobject 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(a, b, loc=0, scale=1, size=1, random_state=None) - Random variates. - pdf(x, a, b, loc=0, scale=1) - Probability density function. - logpdf(x, a, b, loc=0, scale=1) - Log of the probability density function. - cdf(x, a, b, loc=0, scale=1) - Cumulative distribution function. - logcdf(x, a, b, loc=0, scale=1) - Log of the cumulative distribution function. - sf(x, a, b, loc=0, scale=1) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(x, a, b, loc=0, scale=1) - Log of the survival function. - ppf(q, a, b, loc=0, scale=1) - Percent point function (inverse of - cdf— percentiles).- isf(q, a, b, loc=0, scale=1) - Inverse survival function (inverse of - sf).- moment(order, a, b, loc=0, scale=1) - Non-central moment of the specified order. - stats(a, b, loc=0, scale=1, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(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=(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(a, b, loc=0, scale=1) - Median of the distribution. - mean(a, b, loc=0, scale=1) - Mean of the distribution. - var(a, b, loc=0, scale=1) - Variance of the distribution. - std(a, b, loc=0, scale=1) - Standard deviation of the distribution. - interval(confidence, a, b, loc=0, scale=1) - Confidence interval with equal areas around the median. - Notes - The probability density function for - jf_skew_tis:\[f(x; a, b) = C_{a,b}^{-1} \left(1+\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{a+1/2} \left(1-\frac{x}{\left(a+b+x^2\right)^{1/2}}\right)^{b+1/2}\]- for real numbers \(a>0\) and \(b>0\), where \(C_{a,b} = 2^{a+b-1}B(a,b)(a+b)^{1/2}\), and \(B\) denotes the beta function ( - scipy.special.beta).- When \(a<b\), the distribution is negatively skewed, and when \(a>b\), the distribution is positively skewed. If \(a=b\), then we recover the - tdistribution with \(2a\) degrees of freedom.- jf_skew_ttakes \(a\) and \(b\) as shape parameters.- The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- jf_skew_t.pdf(x, a, b, loc, scale)is identically equivalent to- jf_skew_t.pdf(y, 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]- M.C. Jones and M.J. Faddy. “A skew extension of the t distribution, with applications” Journal of the Royal Statistical Society. Series B (Statistical Methodology) 65, no. 1 (2003): 159-174. DOI:10.1111/1467-9868.00378 - Examples - >>> import numpy as np >>> from scipy.stats import jf_skew_t >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Get the support: - >>> a, b = 8, 4 >>> lb, ub = jf_skew_t.support(a, b) - Calculate the first four moments: - >>> mean, var, skew, kurt = jf_skew_t.stats(a, b, moments='mvsk') - Display the probability density function ( - pdf):- >>> x = np.linspace(jf_skew_t.ppf(0.01, a, b), ... jf_skew_t.ppf(0.99, a, b), 100) >>> ax.plot(x, jf_skew_t.pdf(x, a, b), ... 'r-', lw=5, alpha=0.6, label='jf_skew_t 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 = jf_skew_t(a, b) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') - Check accuracy of - cdfand- ppf:- >>> vals = jf_skew_t.ppf([0.001, 0.5, 0.999], a, b) >>> np.allclose([0.001, 0.5, 0.999], jf_skew_t.cdf(vals, a, b)) True - Generate random numbers: - >>> r = jf_skew_t.rvs(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() 