View source: R/family.bivariate.R

bigamma.mckay | R Documentation |

Estimate the three parameters of McKay's bivariate gamma distribution by maximum likelihood estimation.

bigamma.mckay(lscale = "loglink", lshape1 = "loglink", lshape2 = "loglink", iscale = NULL, ishape1 = NULL, ishape2 = NULL, imethod = 1, zero = "shape")

`lscale, lshape1, lshape2` |
Link functions applied to the (positive)
parameters |

`iscale, ishape1, ishape2` |
Optional initial values for |

`imethod, zero` |
See |

One of the earliest forms of the bivariate gamma distribution has a joint probability density function given by

*
f(y1,y2;a,p,q) = (1/a)^(p+q) y1^(p-1) (y2-y1)^(q-1)
exp(-y2/a) / [gamma(p) gamma(q)] *

for *a > 0*, *p > 0*, *q > 0* and
*0<y1<y2*
(Mckay, 1934).
Here, *gamma* is the gamma
function, as in `gamma`

.
By default, the linear/additive predictors are
*eta1=log(a)*,
*eta2=log(p)*,
*eta3=log(q)*.

The marginal distributions are gamma, with shape parameters
*p* and *p+q* respectively, but they have a
common scale parameter *a*. Pearson's product-moment
correlation coefficient of *y1* and *y2* is
*sqrt(p/(p+q))*. This distribution is also
known as the bivariate Pearson type III distribution. Also,
*Y2 - y1*, conditional on *Y1=y1*,
has a gamma distribution with shape parameter *q*.

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions
such as `vglm`

and `vgam`

.

The response must be a two column matrix where the first column
is *y1* and the second *y2*.
It is necessary that each element of the vectors *y1*
and
*y2-y1* be positive.
Currently, the fitted value is a matrix with two columns;
the first column has values *ap* for the marginal mean of
*y1*,
while the second column
has values *a(p+q)* for the marginal mean of
*y2* (all evaluated at the final iteration).

T. W. Yee

McKay, A. T. (1934).
Sampling from batches.
*Journal of the Royal Statistical Society—Supplement*,
**1**, 207–216.

Kotz, S. and Balakrishnan, N. and Johnson, N. L. (2000).
*Continuous Multivariate Distributions Volume 1:
Models and Applications*,
2nd edition,
New York: Wiley.

Balakrishnan, N. and Lai, C.-D. (2009).
*Continuous Bivariate Distributions*,
2nd edition.
New York: Springer.

`gamma2`

.

shape1 <- exp(1); shape2 <- exp(2); scalepar <- exp(3) mdata <- data.frame(y1 = rgamma(nn <- 1000, shape1, scale = scalepar)) mdata <- transform(mdata, zedd = rgamma(nn, shape2, scale = scalepar)) mdata <- transform(mdata, y2 = y1 + zedd) # Z defined as Y2-y1|Y1=y1 fit <- vglm(cbind(y1, y2) ~ 1, bigamma.mckay, mdata, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) vcov(fit) colMeans(depvar(fit)) # Check moments head(fitted(fit), 1)

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