R: Triangle Distribution Family Function

triangle {VGAM} | R Documentation

## Triangle Distribution Family Function

### Description

Estimating the parameter of the triangle distribution by maximum likelihood estimation.

### Usage

    triangle(lower = 0, upper = 1,
             link = extlogitlink(min = 0, max = 1), itheta = NULL)

### Arguments

`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 `Links` for more choices. The default constrains the estimate to lie in the interval.
`itheta` | Optional initial value for the parameter. The default is to compute the value internally.

### Details

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.

### Value

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

### Warning

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.

### Note

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.

### Author(s)

T. W. Yee

### References

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.

### See Also

`Triangle`, `Topple`, `simulate.vlm`.

### Examples

    # 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.

[Package VGAM version 1.1-3 Index]