McCullagh (1989) Distribution Family Function

Estimates the two parameters of the McCullagh (1989) distribution by maximum likelihood estimation.

Usage

mccullagh89(ltheta = "rhobitlink", lnu = "logofflink(offset = 0.5)",
            itheta = NULL, inu = NULL, zero = NULL)

Arguments

ltheta, lnu

Link functions for the $\theta$ and $\nu$ parameters. See Links for general information.

itheta, inu

Numeric. Optional initial values for $\theta$ and $\nu$. The default is to internally compute them.

zero

See CommonVGAMffArguments for information.

Details

The McCullagh (1989) distribution has density function $$f(y;\theta,\nu) = \frac{ \{ 1-y^2 \}^{\nu-\frac12}} { (1-2\theta y + \theta^2)^{\nu} \mbox{Beta}(\nu+\frac12, \frac12)}$$ where $-1 < y < 1$ and $-1 < \theta < 1$. This distribution is equation (1) in that paper. The parameter $\nu$ satisfies $\nu > -1/2$, therefore the default is to use an log-offset link with offset equal to 0.5, i.e., $\eta_2=\log(\nu+0.5)$. The mean is of $Y$ is $\nu \theta / (1+\nu)$, and these are returned as the fitted values.

This distribution is related to the Leipnik distribution (see Johnson et al. (1995)), is related to ultraspherical functions, and under certain conditions, arises as exit distributions for Brownian motion. Fisher scoring is implemented here and it uses a diagonal matrix so the parameters are globally orthogonal in the Fisher information sense. McCullagh (1989) also states that, to some extent, $\theta$ and $\nu$ have the properties of a location parameter and a precision parameter, respectively.

Value

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

References

McCullagh, P. (1989). Some statistical properties of a family of continuous univariate distributions. Journal of the American Statistical Association, 84, 125–129.

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd edition, Volume 2, New York: Wiley. (pages 612–617).

Author

T. W. Yee

Note

Convergence may be slow or fail unless the initial values are reasonably close. If a failure occurs, try assigning the argument inu and/or itheta. Figure 1 of McCullagh (1989) gives a broad range of densities for different values of $\theta$ and $\nu$, and this could be consulted for obtaining reasonable initial values if all else fails.

Examples

# Limit as theta = 0, nu = Inf:
mdata <- data.frame(y = rnorm(1000, sd = 0.2))
fit <- vglm(y ~ 1, mccullagh89, data = mdata, trace = TRUE)
#> Iteration 1: loglikelihood = -1635.3204
#> Iteration 2: loglikelihood = -1030.6074
#> Iteration 3: loglikelihood = -784.59672
#> Iteration 4: loglikelihood = -498.93554
#> Iteration 5: loglikelihood = -199.53709
#> Iteration 6: loglikelihood = 63.488204
#> Iteration 7: loglikelihood = 183.93181
#> Iteration 8: loglikelihood = 201.97799
#> Iteration 9: loglikelihood = 202.32432
#> Iteration 10: loglikelihood = 202.32445
#> Iteration 11: loglikelihood = 202.32445
head(fitted(fit))
#>             [,1]
#> [1,] 0.006075406
#> [2,] 0.006075406
#> [3,] 0.006075406
#> [4,] 0.006075406
#> [5,] 0.006075406
#> [6,] 0.006075406
with(mdata, mean(y))
#> [1] 0.006081826
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = mccullagh89, data = mdata, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1  0.01318    0.01358   0.971    0.332    
#> (Intercept):2  2.50690    0.04384  57.185   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: rhobitlink(theta), logofflink(nu, offset = 0.5)
#> 
#> Log-likelihood: 202.3244 on 1998 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 11 
#> 
coef(fit, matrix = TRUE)
#>             rhobitlink(theta) logofflink(nu, offset = 0.5)
#> (Intercept)        0.01318364                     2.506896
Coef(fit)
#>        theta           nu 
#>  0.006591724 11.766790927