R: Bivariate Gamma: McKay's Distribution

bigamma.mckay {VGAM} | R Documentation

## Bivariate Gamma: McKay's Distribution

### Description

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

### Usage

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

### Arguments

`lscale, lshape1, lshape2` | Link functions applied to the (positive) parameters a, p and q respectively. See `Links` for more choices.
`iscale, ishape1, ishape2` | Optional initial values for a, p and q respectively. The default is to compute them internally.
`imethod, zero` | See `CommonVGAMffArguments`.

### Details

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.

### Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm` and `vgam`.

### Note

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).

### Author(s)

T. W. Yee

### References

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.

### See Also

`gamma2`.

### Examples

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

    colMeans(depvar(fit))  # Check moments
    head(fitted(fit), 1)

[Package VGAM version 1.0-3 Index]