Log-gamma Distribution Family Function

Estimation of the parameter of the standard and nonstandard log-gamma distribution.

Usage

lgamma1(lshape = "loglink", ishape = NULL)
lgamma3(llocation = "identitylink", lscale = "loglink",
   lshape = "loglink", ilocation = NULL, iscale = NULL, ishape = 1,
   zero = c("scale", "shape"))

Arguments

llocation, lscale

Parameter link function applied to the location parameter $a$ and the positive scale parameter $b$. See Links for more choices.

lshape

Parameter link function applied to the positive shape parameter $k$. See Links for more choices.

ishape

Initial value for $k$. If given, it must be positive. If failure to converge occurs, try some other value. The default means an initial value is determined internally.

ilocation, iscale

Initial value for $a$ and $b$. The defaults mean an initial value is determined internally for each.

zero

An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,3}. The default value means none are modelled as intercept-only terms. See CommonVGAMffArguments for more information.

Details

The probability density function of the standard log-gamma distribution is given by $$f(y;k)=\exp[ky - \exp(y)] / \Gamma(k),$$ for parameter $k>0$ and all real $y$. The mean of $Y$ is digamma(k) (returned as the fitted values) and its variance is trigamma(k).

For the non-standard log-gamma distribution, one replaces $y$ by $(y-a)/b$, where $a$ is the location parameter and $b$ is the positive scale parameter. Then the density function is $$f(y)=\exp[k(y-a)/b - \exp((y-a)/b)] / (b \, \Gamma(k)).$$ The mean and variance of $Y$ are a + b*digamma(k) (returned as the fitted values) and b^2 * trigamma(k), respectively.

Value

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

References

Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applications, pages 48–49, London: Imperial College Press.

Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd edition, Volume 2, p.89, New York: Wiley.

Author

T. W. Yee

Note

The standard log-gamma distribution can be viewed as a generalization of the standard type 1 extreme value density: when $k = 1$ the distribution of $-Y$ is the standard type 1 extreme value distribution.

The standard log-gamma distribution is fitted with lgamma1 and the non-standard (3-parameter) log-gamma distribution is fitted with lgamma3.

Examples

ldata <- data.frame(y = rlgamma(100, shape = exp(1)))
fit <- vglm(y ~ 1, lgamma1, ldata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = 0.89067759
#> Iteration 2: coefficients = 0.92236538
#> Iteration 3: coefficients = 0.9224699
#> Iteration 4: coefficients = 0.9224699
summary(fit)
#> 
#> Call:
#> vglm(formula = y ~ 1, family = lgamma1, data = ldata, trace = TRUE, 
#>     crit = "coef")
#> 
#> Coefficients: 
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  0.92247    0.05698   16.19   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Name of linear predictor: loglink(shape) 
#> 
#> Log-likelihood: -113.6391 on 99 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 4 
#> 
coef(fit, matrix = TRUE)
#>             loglink(shape)
#> (Intercept)      0.9224699
Coef(fit)
#>    shape 
#> 2.515496 

ldata <- data.frame(x2 = runif(nn <- 5000))  # Another example
ldata <- transform(ldata, loc = -1 + 2 * x2, Scale = exp(1))
ldata <- transform(ldata, y = rlgamma(nn, loc, sc = Scale, sh = exp(0)))
fit2 <- vglm(y ~ x2, lgamma3, data = ldata, trace = TRUE, crit = "c")
#> Iteration 1: coefficients = 
#> -1.45936415,  1.16212337,  0.19172302,  1.72745129
#> Iteration 2: coefficients = 
#> -1.267101646,  1.056979534,  0.082088825,  2.049779064
#> Iteration 3: coefficients = 
#> -1.059054370,  1.006670755,  0.011251028,  2.076384759
#> Iteration 4: coefficients = 
#> -1.0422567728,  1.0005824994,  0.0027422466,  2.0815295492
#> Iteration 5: coefficients = 
#> -1.0401193218,  1.0000875843,  0.0019974207,  2.0821083940
#> Iteration 6: coefficients = 
#> -1.0399548089,  1.0000525843,  0.0019432071,  2.0821525944
#> Iteration 7: coefficients = 
#> -1.0399429064,  1.0000500765,  0.0019393177,  2.0821556926
#> Iteration 8: coefficients = 
#> -1.0399420578,  1.0000498977,  0.0019390402,  2.0821559151
#> Iteration 9: coefficients = 
#> -1.0399419972,  1.0000498849,  0.0019390204,  2.0821559309
#> Iteration 10: coefficients = 
#> -1.039941993,  1.000049884,  0.001939019,  2.082155932
#> Iteration 11: coefficients = 
#> -1.0399419926,  1.0000498839,  0.0019390189,  2.0821559322
coef(fit2, matrix = TRUE)
#>              location loglink(scale) loglink(shape)
#> (Intercept) -1.039942        1.00005    0.001939019
#> x2           2.082156        0.00000    0.000000000