twdcdf

TWDCDF computes the cumulative distribution function of the Tweedie distribution.

Syntax

  • cdf=twdcdf(x,alpha,theta,delta)example

Description

This function returns the cdf of a Tweedie distribution evaluated on an array of values $x$. The description of the Tweedie distribution and its parameters is detailed in fuction twdrnd.

example

cdf =twdcdf(x, alpha, theta, delta) Estimating the empirical CDF passing from the integral of the pdf.

Examples

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  • Estimating the empirical CDF passing from the integral of the pdf.
  • Data taken from example by Barabesi et. al (2016).

    clear all
    close all
    n = 500;
    pMushrooms = [-0.0936 , 1.27 *10^(-6)  , 0.6145];
    param1 = pMushrooms;
    tit1 = 'Mushrooms';
    al = param1(1) ; th = param1(2) ; de = param1(3) ;
    x = twdrnd(al,th,de,n);
    pdf = twdpdf(x,al,th,de);
    figure;
    subplot(2,1,1);
    plot(x,pdf,'r.');
    title([tit1 ' - Tweedie PDF'],'interpreter','latex','Fontsize',16);
    cdf = twdcdf(x,al,th,de);
    subplot(2,1,2);
    plot(x,cdf,'b.');
    title([tit1 ' - Tweedie CDF, found by integrating the PDF'],'interpreter','latex','Fontsize',16);

    Related Examples

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  • Estimating the empirical CDF by its definition.
  • F(x) = (Number of observations in X<=x)/(Total number of observations).

    % x = vector specifying random variables.
    % y = vector specifying points for which CDF and PDF has to be evaluated.  
    % The CDF is evaluated using the data in the vector x above, 
    % at the points in y.
    close all
    clear all
    n = 500;
    param1 = [-0.0936 , 1.27 *10^(-6)  , 0.6145];
    al = param1(1) ; th = param1(2) ; de = param1(3) ;
    x = twdrnd(al,th,de,n);
    y  = linspace(0,max(x));
    ny = length(y);
    nx = length(x);
    for i = 1:ny
    p = 0;              % True  Probability
    q = 0;              % False Probability
    for j = 1:nx
    if x(j)<=y(i)   % Definition of CDF
    p = p + 1;
    else
    q = q + 1;
    end
    end
    F(i) = p/(p + q);   % Calulating Probability
    end
    plot(y,F,'o');
    % Now a faster and more compact version of the previous definition
    x1  = unique(x); % in principle duplicates should be removed
    nx = length(x1);
    y1  = linspace(0,max(x1));
    ny = length(y1);
    sump = sum(x1<=y1 , 1);
    FF = sump/nx;
    hold on;
    plot(y1,FF,'-r')
    legend('CDF using loops' , 'CDF using vectorization');
    title({'CDF computed using definition' , '$F(x) = (\sum (I(X<=x)))/(n)$'},'interpreter','latex','Fontsize',16);
    % both definitions above are along the matlab fiunction cdfplot(x)

    Input Arguments

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    x — Values at which to evaluate cdf. Array of scalar values.

    To evaluate the pdf at multiple values, specify x using an array.

    Data Types: single| double

    alpha — Distribution parameter value. Non-zero value smaller than 1.

    The location parameter of the Tweedie distribution. Default is alpha = 1.

    Data Types: single| double

    theta — Distribution parameter value. Positive value.

    The dispersion parameter of the Tweedie distribution. Default is theta = 0 (Dirac).

    Data Types: single| double

    delta — Distribution parameter value. Positive value.

    The power parameter of the Tweedie distribution. delta is such that the variance is $var(Y) = \theta * \alpha^\delta$. delta is greater than or equal to one, or less than or equal to zero. Default is delta = 1.

    Interesting special cases are: the normal (delta=0), Poisson (delta=1 with theta=1), gamma (delta=2) and inverse Gaussian (delta=3). Other values of delta lead to cases that cannot be written in closed form, and the computation becomes difficult.

    When 1 < delta < 2, the distribution is continuous for Y>0 and has a positive mass at Y=0. For delta > 2, the distribution is continuous for Y>0.

    Data Types: single| double

    Output Arguments

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    cdf —Cumulative distribution function values. Scalar value or array of scalar values

    pdf values, evaluated at the values in x, returned as a scalar value or an array of scalar values.

    Data Types - Double.

    More About

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    Additional Details

    Detailed information can be found in function twdrnd.

    References

    Tweedie, M. C. K. (1984), An index which distinguishes between some important exponential families, "in Statistics: Applications and New Directions, Proceedings of the Indian Statistical Institute Golden Jubilee International Conference (J.K. Ghosh and J. Roy, eds.), Indian Statistical Institute, Calcutta", pp. 579-604.

    See Also

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