Stable Distributions

Tail and body behavior of stable distributions.

Stable distributions are defined by tail index \(\alpha\in (0, 2]\), skewness \(\beta\in[-1,1]\), scale \(\gamma>0\) and location \(\delta\). The tail index and skewness are shape parameters. The properties of \(\mathrm{St}(\alpha, \beta, 1, 0)=\mathrm{St}(\alpha, \beta)\) depend critically on \(\alpha\). There are two aspects: body behavior which is determined by small jumps, and tail behavior determined by large. Figure 1 shows the options.

import matplotlib.pyplot as plt
import numpy as np

def anno(ax, txt, xy, xytext, opts, arrow_color='black'):
    return ax.annotate(
        txt,  # Text to display
        xy=xy,  # Point the arrow points to
        xytext=xytext,  # Offset label position (x, y)
        arrowprops=dict(arrowstyle="->", color=arrow_color, lw=.5,
                       connectionstyle="arc3,rad=0"),  # Arrow style
        **opts )

def make_dia():
    # Define x values
    x = np.hstack((np.linspace(1/1000, 1, 999, endpoint=False), np.linspace(1, 10, 1000)))

    # Define the functions
    f1 = 1 / x
    f2 = 1 / x**2
    f3 = 1 / x**3

    # Create the figure and axis
    fig, ax = plt.subplots(figsize=(5, 5))

    ax.axvline(1, ymin=0, ymax=.25, lw=.5, c='k', alpha=0.5, ls='--')

    # Plot the functions
    ax.plot(x, f1, label=r"$1/x$", color="blue")
    ax.plot(x, f2, label=r"$1/x^2$", color="red")
    ax.plot(x, f3, label=r"$1/x^3$", color="black")

    # Shade areas
    ax.fill_between(x, f2, f1, where=(f2 < f1), color="blue", alpha=0.2, label="A1")

    ax.fill_between(x, f3, f2, where=(x >= 1), color="red", alpha=0.2, label="A2")

    ax.fill_between(x, f3, 0, where=(x >= 1), color="yellow", alpha=0.2, label=r"A3")
    # hatch='//',
    ax.fill_between(x, f2, 0, where=(x < 1),  #hatch='\\\\\\',
                    lw=.5, ec='blue', fc='green', alpha=0.05, label="SA4")

    ax.fill_between(x, f2, f3, where=(x < 1), hatch='////', lw=.5, ec='red', color='red', alpha=0.2, label=r"A5")

    # Annotate regions
    opts = {'ha': 'center', 'va': 'center', 'fontsize': 8}
    ax.text(0.5, 0.65, 'Finite small\njump length\nsubordinators\nwith half-line\nsupport possible\n(green)', **opts)


    opts = {'ha': 'left', 'va': 'bottom', 'fontsize': 8}
    anno(ax, 'No mean\n(blue)', (2.5, 1/2.75 - 0.125), (2.75, 0.45), opts)
    sh = -0.1
    anno(ax, 'Finite mean\nno variance\n(orange)', (2+sh, (1/4+1/8)/2+0.0), (2.25+sh, .7), opts)
    opts['ha'] = 'right'
    anno(ax, 'Finite mean\nand variance\n(yellow)', (1.5, 1/(1.5**3) - 0.2), (2, 1), opts)

    opts['ha'] = 'left'
    eps = 0.015
    xpad = .6
    anno(ax, '$1/x$, $\\alpha=0$, e.g., Levy', (1/3.5-eps, 3.5), (1/3.5 + xpad, 3.5), opts, arrow_color='blue')
    anno(ax, '$1/x^2$, $\\alpha=1$, Cauchy', (1/3.25**0.5-eps, 3.25),
            (1/3.25**0.5 + xpad, 3.25), opts, arrow_color='red')
    anno(ax, '$1/x^3$, $\\alpha=2$', (1/3.0**(1/3)-eps, 3.0), (1/3.0**(1/3) + xpad, 3.0), opts, arrow_color='black')

    opts['ha'] = 'center'
    x = 3.4
    y = .7
    anno(ax, '$1/x$', (x, 1/x-eps), (x, y), opts, arrow_color='blue')
    x = 3.625
    y = .6
    anno(ax, '$1/x^2$', (x, 1/x**2-eps), (x, y), opts, arrow_color='red')
    x = 3.85
    y = .5
    anno(ax, '$1/x^3$', (x, 1/x**3-eps), (x, y), opts, arrow_color='black')

    y = 2.
    x = (y**(-1/3) + y**-.5) / 2
    opts['ha'] = 'left'
    opts['va'] = 'center'
    anno(ax, 'Infinite activity\nsupport $(-\\infty, \\infty)$\n(orange hatched)', (x-eps, y), (x + .4, y), opts)

    # Label axes
    ax.set_xlabel(r"Jump size, $x$", fontsize=9)
    ax.set_ylabel(r"Jump frequency, $\nu(dx)$", fontsize=10)

    # Add legend
    # ax.legend(loc='upper right', fontsize=9)

    ax.set(ylim=[-0.05, 4], xlim=[-0.05, 4])

    ax.set_xticks([0, .5, 1, 2.5, 4])
    ax.set_xticklabels(['0', 'small\njumps', '1', 'large jumps\ncontrol tail', '$\\cdots$'])
    ax.set_yticks([0, 1, 4])
    ax.set_yticklabels(['0',  '1',  '$\\infty$'])
    ax.set_xticks([1.5, 2, 3, 3.5], minor=True)
    ax.set_yticks([2, 3, 4], minor=True)


make_dia()

Figure 1: Behavior of stable distributions. Small jumps determine the body and support. Large jumps determine tails.