Triangle Distribution Family Function
Estimating the parameter of the triangle distribution by maximum likelihood estimation.
triangle(lower = 0, upper = 1,
link = extlogitlink(min = 0, max = 1), itheta = NULL)lower, upper |
lower and upper limits of the distribution. Must be finite. Called A and B respectively below. |
link |
Parameter link function applied to the parameter theta,
which lies in (A,B).
See |
itheta |
Optional initial value for the parameter. The default is to compute the value internally. |
The triangle distribution has a probability density function that consists of two lines joined at theta, which is the location of the mode. The lines intersect the y = 0 axis at A and B. Here, Fisher scoring is used.
On fitting, the extra slot has components called lower
and upper which contains the values of the above arguments
(recycled to the right length).
The fitted values are the mean of the distribution, which is
(A + B + theta)/3.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
The MLE regularity conditions do not hold for this
distribution
(e.g., the first derivative evaluated at the mode
does not exist because it is not continuous)
so that misleading inferences may result, e.g.,
in the summary and vcov of the object.
Additionally, convergence to the MLE often appears to fail.
The response must contain values in (A, B). For most data sets (especially small ones) it is very common for half-stepping to occur.
Arguments lower and upper and link must match.
For example, setting
lower = 0.2 and upper = 4 and
link = extlogitlink(min = 0.2, max = 4.1) will result in an error.
Ideally link = extlogitlink(min = lower, max = upper)
ought to work but it does not (yet)!
Minimal error checking is done for this deficiency.
T. W. Yee
Kotz, S. and van Dorp, J. R. (2004). Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications. Chapter 1. World Scientific: Singapore.
Nguyen, H. D. and McLachlan, G. J. (2016). Maximum likelihood estimation of triangular and polygon distributions. Computational Statistics and Data Analysis, 102, 23–36.
# Example 1
tdata <- data.frame(y = rtriangle(n <- 3000, theta = 3/4))
fit <- vglm(y ~ 1, triangle(link = "identitylink"), data = tdata, trace = TRUE)
coef(fit, matrix = TRUE)
Coef(fit)
head(fit@extra$lower)
head(fitted(fit))
with(tdata, mean(y))
# Example 2; Kotz and van Dorp (2004), p.14
rdata <- data.frame(y = c(0.1, 0.25, 0.3, 0.4, 0.45, 0.6, 0.75, 0.8))
fit <- vglm(y ~ 1, triangle(link = "identitylink"), data = rdata, trace = TRUE,
crit = "coef", maxit = 1000)
Coef(fit) # The MLE is the 3rd order statistic, which is 0.3.
fit <- vglm(y ~ 1, triangle(link = "identitylink"), data = rdata, trace = TRUE,
crit = "coef", maxit = 1001)
Coef(fit) # The MLE is the 3rd order statistic, which is 0.3.Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.