scipy.stats.studentized_range#
- scipy.stats.studentized_range = <scipy.stats._continuous_distns.studentized_range_gen object>[source]#
- A studentized range continuous random variable. - As an instance of the - rv_continuousclass,- studentized_rangeobject 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 - t
- Student’s t distribution 
 - Notes - The probability density function for - studentized_rangeis:\[f(x; k, \nu) = \frac{k(k-1)\nu^{\nu/2}}{\Gamma(\nu/2) 2^{\nu/2-1}} \int_{0}^{\infty} \int_{-\infty}^{\infty} s^{\nu} e^{-\nu s^2/2} \phi(z) \phi(sx + z) [\Phi(sx + z) - \Phi(z)]^{k-2} \,dz \,ds\]- for \(x ≥ 0\), \(k > 1\), and \(\nu > 0\). - studentized_rangetakes- kfor \(k\) and- dffor \(\nu\) as shape parameters.- When \(\nu\) exceeds 100,000, an asymptotic approximation (infinite degrees of freedom) is used to compute the cumulative distribution function [4] and probability distribution function. - The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- studentized_range.pdf(x, k, df, loc, scale)is identically equivalent to- studentized_range.pdf(y, k, df) / 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]- “Studentized range distribution”, https://en.wikipedia.org/wiki/Studentized_range_distribution [2]- Batista, Ben Dêivide, et al. “Externally Studentized Normal Midrange Distribution.” Ciência e Agrotecnologia, vol. 41, no. 4, 2017, pp. 378-389., doi:10.1590/1413-70542017414047716. [3]- Harter, H. Leon. “Tables of Range and Studentized Range.” The Annals of Mathematical Statistics, vol. 31, no. 4, 1960, pp. 1122-1147. JSTOR, www.jstor.org/stable/2237810. Accessed 18 Feb. 2021. [4]- Lund, R. E., and J. R. Lund. “Algorithm AS 190: Probabilities and Upper Quantiles for the Studentized Range.” Journal of the Royal Statistical Society. Series C (Applied Statistics), vol. 32, no. 2, 1983, pp. 204-210. JSTOR, www.jstor.org/stable/2347300. Accessed 18 Feb. 2021. - Examples - >>> import numpy as np >>> from scipy.stats import studentized_range >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots(1, 1) - Display the probability density function ( - pdf):- >>> k, df = 3, 10 >>> x = np.linspace(studentized_range.ppf(0.01, k, df), ... studentized_range.ppf(0.99, k, df), 100) >>> ax.plot(x, studentized_range.pdf(x, k, df), ... 'r-', lw=5, alpha=0.6, label='studentized_range 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 = studentized_range(k, df) >>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf') - Check accuracy of - cdfand- ppf:- >>> vals = studentized_range.ppf([0.001, 0.5, 0.999], k, df) >>> np.allclose([0.001, 0.5, 0.999], studentized_range.cdf(vals, k, df)) True - Rather than using ( - studentized_range.rvs) to generate random variates, which is very slow for this distribution, we can approximate the inverse CDF using an interpolator, and then perform inverse transform sampling with this approximate inverse CDF.- This distribution has an infinite but thin right tail, so we focus our attention on the leftmost 99.9 percent. - >>> a, b = studentized_range.ppf([0, .999], k, df) >>> a, b 0, 7.41058083802274 - >>> from scipy.interpolate import interp1d >>> rng = np.random.default_rng() >>> xs = np.linspace(a, b, 50) >>> cdf = studentized_range.cdf(xs, k, df) # Create an interpolant of the inverse CDF >>> ppf = interp1d(cdf, xs, fill_value='extrapolate') # Perform inverse transform sampling using the interpolant >>> r = ppf(rng.uniform(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(k, df, loc=0, scale=1, size=1, random_state=None) - Random variates. - pdf(x, k, df, loc=0, scale=1) - Probability density function. - logpdf(x, k, df, loc=0, scale=1) - Log of the probability density function. - cdf(x, k, df, loc=0, scale=1) - Cumulative distribution function. - logcdf(x, k, df, loc=0, scale=1) - Log of the cumulative distribution function. - sf(x, k, df, loc=0, scale=1) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(x, k, df, loc=0, scale=1) - Log of the survival function. - ppf(q, k, df, loc=0, scale=1) - Percent point function (inverse of - cdf— percentiles).- isf(q, k, df, loc=0, scale=1) - Inverse survival function (inverse of - sf).- moment(order, k, df, loc=0, scale=1) - Non-central moment of the specified order. - stats(k, df, loc=0, scale=1, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(k, df, 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, df), 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, df, loc=0, scale=1) - Median of the distribution. - mean(k, df, loc=0, scale=1) - Mean of the distribution. - var(k, df, loc=0, scale=1) - Variance of the distribution. - std(k, df, loc=0, scale=1) - Standard deviation of the distribution. - interval(confidence, k, df, loc=0, scale=1) - Confidence interval with equal areas around the median.