inversegaminv

inversegaminv computes the inverse of the inverse-gamma cumulative distribution function.

expand all in page

Syntax

Description

example

x =inversegaminv(p, a, b) Compare the results using option nocheck=1.

example

x =inversegaminv(p, a, b, nocheck) Check accuracy of results monitoring $|p-F_{IG} (F_{IG}^{-1}(p))|$.

Examples

expand all

  • Compare the results using option nocheck=1.

    • x=(0:0.0001:0.9999)';
a=[1,2,3,50,100,10000];
b=[1,10,100,0.05,10,800];
Y=zeros(length(x),length(a));
Ychk=Y;
for i=1:length(x)
Y(i,:)=inversegaminv(x(i),a,b);
Ychk(i,:)=inversegaminv(x(i),a,b,1);
end
disp('Maximum absolute difference is:');
disp(max(max(abs(Y-Ychk))));
    Maximum absolute difference is:
   1.1369e-13

  • Check accuracy of results monitoring $|p-F_{IG} (F_{IG}^{-1}(p))|$.

    • a=[1,2,3,50,100,10000];
b=[1,10,100,0.05,10,800];
x=(0:0.0001:0.9999)';
Y=zeros(length(x),length(a));
Ychk=Y;
for i=1:length(x)
Y(i,:)=x(i)-inversegamcdf(inversegaminv(x(i),a,b),a,b);
Ychk(i,:)=x(i)-inversegamcdf(inversegaminv(x(i),a,b,1),a,b,1);
end
disp('Maximum deviation from 0 passing through routine gaminv:');
disp(max(max(abs(Y))));
disp('Maximum deviation from 0 using fast routine:');
disp(max(max(abs(Ychk))));
    Maximum deviation from 0 passing through routine gaminv:
   1.4821e-14

Maximum deviation from 0 using fast routine:
   1.1324e-14

    Input Arguments

    expand all

    p — Probability at which the inverse of the cdf must be evaluated $0 \leq p \leq 1$. Scalar, vector or matrix 3D array of the same size of x and b.

    A scalar input functions as a constant matrix of the same size as the other input.

    See the section called "More About:" for more details about the inverse gamma distribution.

    Data Types: single | double

    a — shape parameter of the inverse-gamma distribution. Scalar, vector or matrix 3D array of the same size of x and b.

    A scalar input functions as a constant matrix of the same size as the other input.

    See the section called "More About:" for more details about the inverse gamma distribution.

    Data Types: single | double

    b — scale parameter b of the inverse-gamma distribution. Scalar, vector or matrix 3D array of the same size of x and a.

    A scalar input functions as a constant matrix of the same size as the other input.

    Unlike the Gamma distribution, which contains a somewhat similar exponential term, $b$ is a scale parameter as the distribution function satisfies:

    \[ f_{IG}(x,a,b)=\frac{f(x/b,a,1)}{b} \]

    See the section called "More About:" for more details about the inverse gamma distribution.

    Data Types: single | double

    Optional Arguments

    nocheck — Check input arguments. Scalar.

    If nocheck is equal to 1 no check is performed and input and the inverse cdf is evaluated directly through MATLAB buit in function gammaincinv else we use MATLAB function gaminv.

    Example: 'nocheck',1

    Data Types: double

    Output Arguments

    expand all

    x —inverse CDF value. Scalar, vector or matrix or 3D array of the same size of input arguments p, a and b

    $p=\int_0^x f_{IG}(t | a,b) dt$ is the inverse of the inverse-gamma cdf with shape parameters in a and scale parameters in b for the corresponding probabilities in p.

    More About

    expand all

    Additional Details

    The cdf of the inverse gamma distribution defined over the support $x>0$ with shape parameter $a$ and scale parameter $b$ is \[ F_{IG}(x, a, b) =\int_0^x t^{-a -1} \exp (-b/t) \frac{b^a}{\Gamma(a)} dt \]

    inversegaminv computes the inverse of the inverse-gamma cdf with shape parameters in a and scale parameters in b for the corresponding probabilities in p. p, a, and b can be vectors, matrices, or multidimensional arrays that all have the same size. A scalar input is expanded to a constant array with the same dimensions as the other inputs. The parameters in a and b must all be positive, and the values in x must lie on the interval $[0, \infty)$.

    Note that $F_{IG}(x,a,b)=\frac{\Gamma(a,b/x)}{\Gamma(\alpha)}$ therefore Therefore, the CDF for an inverse Gamma distribution can be computed using the incomplete gamma function (also called regularized gamma function, i.e. MATLAB function gammainc) of course keeping into account that we need the upper tail.

    The chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution if an uninformative prior is used; and as an analytically tractable conjugate prior if an informative prior is required.

    Relation with the Gamma distribution.

    If $X \sim Gamma(a,b)$ then $\frac{1}{X} \sim$ inverse-gamma distribution with paramters $a$ and $1/b$.

    References

    See Also

    | |

    This page has been automatically generated by our routine publishFS