inversegamcdf

inversegampdf

inversegaminv

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

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Syntax

x=inversegaminv(p,a,b)example
x=inversegaminv(p,a,b,nocheck)example

Description

example

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

example

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

Examples

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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:
          0.00

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:
          0.00

Maximum deviation from 0 using fast routine:
          0.00

Input Arguments

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`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

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`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

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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 $1/b$ .

Documentation

inversegaminv

Syntax

Description

Examples

Compare the results using option nocheck=1.

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

Input Arguments

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

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

Optional Arguments

`nocheck` — Check input arguments. Scalar.

Output Arguments

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

More About

Additional Details

References

See Also

Documentation

inversegaminv

Syntax

Description

Examples

Compare the results using option nocheck=1.

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

Input Arguments

p — Probability at which the inverse of the cdf must be evaluated 0 \leq p \leq 10 \leq p \leq 1. Scalar, vector or matrix 3D array of the same size of x and b.

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

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

Optional Arguments

nocheck — Check input arguments. Scalar.

Output Arguments

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

More About

Additional Details

References

See Also

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

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

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

`nocheck` — Check input arguments. Scalar.

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