waldff.RdEstimates the parameter of the standard Wald distribution by maximum likelihood estimation.
waldff(llambda = "loglink", ilambda = NULL)See CommonVGAMffArguments for information.
The standard Wald distribution is a special case of the inverse Gaussian distribution with \(\mu=1\). It has a density that can be written as $$f(y;\lambda) = \sqrt{\lambda/(2\pi y^3)} \; \exp\left(-\lambda (y-1)^2/(2 y)\right)$$ where \(y>0\) and \(\lambda>0\). The mean of \(Y\) is \(1\) (returned as the fitted values) and its variance is \(1/\lambda\). By default, \(\eta=\log(\lambda)\).
An object of class "vglmff"
(see vglmff-class).
The object is used by modelling functions
such as vglm,
and vgam.
Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd edition, Volume 1, New York: Wiley.
The VGAM family function inv.gaussianff
estimates the location parameter \(\mu\) too.
wdata <- data.frame(y = rinv.gaussian(1000, mu = 1, exp(1)))
wfit <- vglm(y ~ 1, waldff(ilambda = 0.2), wdata, trace = TRUE)
#> Iteration 1: loglikelihood = -1138.3796
#> Iteration 2: loglikelihood = -858.26854
#> Iteration 3: loglikelihood = -743.30758
#> Iteration 4: loglikelihood = -727.87989
#> Iteration 5: loglikelihood = -727.63692
#> Iteration 6: loglikelihood = -727.63686
#> Iteration 7: loglikelihood = -727.63686
coef(wfit, matrix = TRUE)
#> loglink(lambda)
#> (Intercept) 0.9366431
Coef(wfit)
#> lambda
#> 2.551402
summary(wfit)
#>
#> Call:
#> vglm(formula = y ~ 1, family = waldff(ilambda = 0.2), data = wdata,
#> trace = TRUE)
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> (Intercept) 0.93664 0.04472 20.94 <2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Name of linear predictor: loglink(lambda)
#>
#> Log-likelihood: -727.6369 on 999 degrees of freedom
#>
#> Number of Fisher scoring iterations: 7
#>
#> No Hauck-Donner effect found in any of the estimates
#>