gauss.quad.prob {statmod} | R Documentation |

## Gaussian Quadrature with Probability Distributions

### Description

Calculate nodes and weights for Gaussian quadrature in terms of probability distributions.

### Usage

gauss.quad.prob(n,dist="uniform",l=0,u=1,mu=0,sigma=1,alpha=1,beta=1)

### Arguments

`n` |
number of nodes and weights |

`dist` |
distribution that Gaussian quadrature is based on, one of `"uniform"` , `"normal"` , `"beta"` or `"gamma"` |

`l` |
lower limit of uniform distribution |

`u` |
upper limit of uniform distribution |

`mu` |
mean of normal distribution |

`sigma` |
standard deviation of normal distribution |

`alpha` |
positive shape parameter for gamma distribution or first shape parameter for beta distribution |

`beta` |
positive scale parameter for gamma distribution or second shape parameter for beta distribution |

### Details

This is a rewriting and simplification of `gauss.quad`

in terms of probability distributions.

The expected value of `f(X)`

is approximated by `sum(w*f(x))`

where `x`

is the vector of nodes and `w`

is the vector of weights. The approximation is exact if `f(x)`

is a polynomial of order no more than `2n-1`

.
The possible choices for the distribution of `X`

are as follows:

Uniform on `(l,u)`

.

Normal with mean `mu`

and standard deviation `sigma`

.

Beta with density `x^(alpha-1)*(1-x)^(beta-1)/B(alpha,beta)`

on `(0,1)`

.

Gamma with density `x^(alpha-1)*exp(-x/beta)/beta^alpha/gamma(alpha)`

.

### Value

A list containing the components

`nodes` |
vector of values at which to evaluate the function |

`weights` |
vector of weights to give the function values |

### Author(s)

Gordon Smyth

### References

Golub, G. H., and Welsch, J. H. (1969). Calculation of Gaussian
quadrature rules. *Mathematics of Computation* **23**, 221-230.

Golub, G. H. (1973). Some modified matrix eigenvalue problems.
*Siam Review* **15**, 318-334.

Smyth, G. K. (1998). Polynomial approximation. In: *Encyclopedia of Biostatistics*, P. Armitage and T. Colton (eds.), Wiley, London, pp. 3425-3429.
http://www.statsci.org/smyth/pubs/poly.ps

Stroud and Secrest (1966). *Gaussian Quadrature Formulas*. Prentice-
Hall, Englewood Cliffs, N.J.

### See Also

`gauss.quad`

, `integrate`

### Examples

out <- gauss.quad.prob(10,"normal")
sum(out$weights * out$nodes^4)
# the 4th moment of the standard normal is 3
out <- gauss.quad.prob(32,"gamma",alpha=5)
sum(out$weights * log(out$nodes))
# the expected value of log(X) where X is gamma is digamma(alpha)

[Package

*statmod* version 1.2.4

Index]