## Description

Maximum likelihood estimation of the 3-parameter Singh-Maddala distribution.

## Usage

 ```1 2 3 4 5``` ```sinmad(lscale = "loglink", lshape1.a = "loglink", lshape3.q = "loglink", iscale = NULL, ishape1.a = NULL, ishape3.q = NULL, imethod = 1, lss = TRUE, gscale = exp(-5:5), gshape1.a = exp(-5:5), gshape3.q = exp(-5:5), probs.y = c(0.25, 0.5, 0.75), zero = "shape") ```

## Arguments

 `lss` See `CommonVGAMffArguments` for important information. `lshape1.a, lscale, lshape3.q` Parameter link functions applied to the (positive) parameters a, `scale`, and q. See `Links` for more choices. `iscale, ishape1.a, ishape3.q, imethod, zero` See `CommonVGAMffArguments` for information. For `imethod = 2` a good initial value for `ishape3.q` is needed to obtain good estimates for the other parameters. `gscale, gshape1.a, gshape3.q` See `CommonVGAMffArguments` for information. `probs.y` See `CommonVGAMffArguments` for information.

## Details

The 3-parameter Singh-Maddala distribution is the 4-parameter generalized beta II distribution with shape parameter p=1. It is known under various other names, such as the Burr XII (or just the Burr distribution), Pareto IV, beta-P, and generalized log-logistic distribution. More details can be found in Kleiber and Kotz (2003).

Some distributions which are special cases of the 3-parameter Singh-Maddala are the Lomax (a=1), Fisk (q=1), and paralogistic (a=q).

f(y) = aq y^(a-1) / [b^a (1 + (y/b)^a)^(1+q)]

for a > 0, b > 0, q > 0, y >= 0. Here, b is the scale parameter `scale`, and the others are shape parameters. The cumulative distribution function is

F(y) = 1 - [1 + (y/b)^a]^(-q).

The mean is

E(Y) = b gamma(1 + 1/a) gamma(q - 1/a) / gamma(q)

provided -a < 1 < aq; these are returned as the fitted values. This family function handles multiple responses.

## Value

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

## Note

See the notes in `genbetaII`.

T. W. Yee

## References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

`Sinmad`, `genbetaII`, `betaII`, `dagum`, `fisk`, `inv.lomax`, `lomax`, `paralogistic`, `inv.paralogistic`, `simulate.vlm`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27``` ```sdata <- data.frame(y = rsinmad(n = 1000, shape1 = exp(1), scale = exp(2), shape3 = exp(0))) fit <- vglm(y ~ 1, sinmad(lss = FALSE), data = sdata, trace = TRUE) fit <- vglm(y ~ 1, sinmad(lss = FALSE, ishape1.a = exp(1)), data = sdata, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) summary(fit) # Harder problem (has the shape3.q parameter going to infinity) set.seed(3) sdata <- data.frame(y1 = rbeta(1000, 6, 6)) # hist(with(sdata, y1)) if (FALSE) { # These struggle fit1 <- vglm(y1 ~ 1, sinmad(lss = FALSE), data = sdata, trace = TRUE) fit1 <- vglm(y1 ~ 1, sinmad(lss = FALSE), data = sdata, trace = TRUE, crit = "coef") Coef(fit1) } # Try this remedy: fit2 <- vglm(y1 ~ 1, data = sdata, trace = TRUE, stepsize = 0.05, maxit = 99, sinmad(lss = FALSE, ishape3.q = 3, lshape3.q = "logloglink")) coef(fit2, matrix = TRUE) Coef(fit2) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = -2944.9284
VGLM    linear loop  2 :  loglikelihood = -2944.9212
VGLM    linear loop  3 :  loglikelihood = -2944.9212
VGLM    linear loop  1 :  loglikelihood = -2944.9284
VGLM    linear loop  2 :  loglikelihood = -2944.9212
VGLM    linear loop  3 :  loglikelihood = -2944.9212
(Intercept)          1.062291       1.953652       -0.05679876
shape1.a     scale  shape3.q
2.8929907 7.0544043 0.9447842

Call:
vglm(formula = y ~ 1, family = sinmad(lss = FALSE, ishape1.a = exp(1)),
data = sdata, trace = TRUE)

Pearson residuals:
Min      1Q   Median     3Q    Max
loglink(shape1.a) -6.228 -0.3902  0.34219 0.7442 0.8747
loglink(scale)    -2.069 -0.9770 -0.05003 1.0063 1.4397
loglink(shape3.q) -9.256 -0.1508  0.06832 0.3409 4.4608

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1  1.06229    0.04872  21.803   <2e-16 ***
(Intercept):2  1.95365    0.06171  31.659   <2e-16 ***
(Intercept):3 -0.05680    0.11296  -0.503    0.615
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-likelihood: -2944.921 on 2997 degrees of freedom

Number of Fisher scoring iterations: 3

No Hauck-Donner effect found in any of the estimates

Taking a modified step...
VGLM    linear loop  2 :  loglikelihood = -1609.9604
Taking a modified step....
VGLM    linear loop  3 :  loglikelihood = -1585.3258
Taking a modified step.....
VGLM    linear loop  4 :  loglikelihood = -1576.4835
Taking a modified step......
VGLM    linear loop  5 :  loglikelihood = -1575.826
Taking a modified step.........
VGLM    linear loop  6 :  loglikelihood = -1575.2006
Taking a modified step..........
VGLM    linear loop  7 :  loglikelihood = -1575.089
Taking a modified step............
VGLM    linear loop  8 :  loglikelihood = -1575.0724
Taking a modified step...............
VGLM    linear loop  9 :  loglikelihood = -1575.061
Taking a modified step................
VGLM    linear loop  10 :  loglikelihood = -1575.0576
Taking a modified step.................
Warning message:
In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
iterations terminated because half-step sizes are very small