Topp-Leone Distribution Family Function

Estimating the parameter of the Topp-Leone distribution by maximum likelihood estimation.

topple(lshape = "logitlink", zero = NULL, gshape = ppoints(8),
       parallel = FALSE, percentiles = 50,
       type.fitted = c("mean", "percentiles", "Qlink"))

Arguments

lshape, gshape

Details at CommonVGAMffArguments. The CIA link is loglink, for shape approaching unity.

zero, parallel

Details at CommonVGAMffArguments.

type.fitted, percentiles

See CommonVGAMffArguments for information. Using "Qlink" is for quantile-links in VGAMextra.

Details

The Topple distribution has a probability density function that can be written $$f(y;s) = 2 s (1 - y) [y (2-y)]^{s-1}$$ for \(0<y<1\) and shape parameter \(0<s<1\). The mean of \(Y\) is \(1 - 4^s [\Gamma(1+s)]^2 / \Gamma(2 + 2s)\) (returned as the fitted values).

Value

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

References

Topp, C. W. and F. C. Leone (1955). A family of J-shaped frequency functions. Journal of the American Statistical Association, 50, 209–219.

Author

T. W. Yee

Note

Fisher-scoring and Newton-Raphson are the same here. A related distribution is the triangle distribution. This VGAM family function handles multiple responses.

See also

Topple, Triangle.

Examples

tdata <- data.frame(y = rtopple(1000, logitlink(1, inverse = TRUE)))
tfit <- vglm(y ~ 1, topple, tdata, trace = TRUE, crit = "coef")
#> Iteration 1: coefficients = 0.77050635
#> Iteration 2: coefficients = 0.77054154
#> Iteration 3: coefficients = 0.77054154
coef(tfit, matrix = TRUE)
#>             logitlink(shape)
#> (Intercept)        0.7705415
Coef(tfit)
#>    shape 
#> 0.683638