twdpdf

TWDPDF computes the probability density function of the Tweedie distribution.

Syntax

  • pdf=twdpdf(x,alpha,theta,delta)example

Description

This function returns the pdf 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

pdf =twdpdf(x, alpha, theta, delta) Example from Barabesi et.

Examples

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  • Example from Barabesi et.
  • al (2016).

    n = 1000;
    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;
    plot(x,pdf,'r.');
    title(tit1);

    Related Examples

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  • Check consistency with R theedie package.
  • Note that the package adopts the parametrization of Jorgensen (1987), which introduces the Tweedie distribution as a special case of the exponential dispersion model.

    % In the R tweedie package, the parameters are:
    % p     = vector of probabilities;
    % n     = the number of observations;
    % xi    = the value of xi such that the variance is $var(Y)=\phi \mu^{xi}$;
    % power = a synonym for xi
    % mu    = the mean
    % phi	= the dispersion
    % # R code
    % power <- 2.5
    % mu    <- 1
    % phi   <- 1
    % y     <- seq(0, 6, length=500)
    % fy    <- dtweedie( y=y, power=power, mu=mu, phi=phi)
    % plot(y, fy, type="l", lwd=2, ylab="Density")
    % # Compare to the saddlepoint density
    % f.saddle <- dtweedie.saddle( y=y, power=power, mu=mu, phi=phi)
    % lines( y, f.saddle, col=2 )
    % legend("topright", col=c(1,2), lwd=c(2,1),
    %     legend=c("Actual","Saddlepoint") )
    % parameter values in Jorgensen parametrization
    power = 2.5;
    mu    = 1 ;
    phi   = 1 ;
    % reparametrization
    alpha = (power-2)/(power-1);
    theta = phi*mu;
    delta = (phi/(power-1))^(1/(power-1));
    % pdf
    y = linspace(0,6,500);
    pdf = twdpdf(y,alpha,theta,delta);
    figure;
    subplot(2,1,1);
    plot(y,pdf,'r.');
    title({'Jorgensen parametrization: power = 2.5, mu = 1, phi = 1' , ...
    're-parametrization: alpha = 0.3333, theta = 1, delta = 0.7631'});
    Y = twdrnd(alpha,theta,delta,500);
    subplot(2,1,2);
    histogram(Y);
    Click here for the graphical output of this example (link to Ro.S.A. website)

    Input Arguments

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    x — Values at which to evaluate pdf. 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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    pdf —Probability density 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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