## Description

Density, distribution function, quantile function and random generation for the Singh-Maddala distribution with shape parameters `a` and `q`, and scale parameter `scale`.

## Usage

 ```1 2 3 4``` ```dsinmad(x, scale = 1, shape1.a, shape3.q, log = FALSE) psinmad(q, scale = 1, shape1.a, shape3.q, lower.tail = TRUE, log.p = FALSE) qsinmad(p, scale = 1, shape1.a, shape3.q, lower.tail = TRUE, log.p = FALSE) rsinmad(n, scale = 1, shape1.a, shape3.q) ```

## Arguments

 `x, q` vector of quantiles. `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required. `shape1.a, shape3.q` shape parameters. `scale` scale parameter. `log` Logical. If `log = TRUE` then the logarithm of the density is returned. `lower.tail, log.p` Same meaning as in `pnorm` or `qnorm`.

## Details

See `sinmad`, which is the VGAM family function for estimating the parameters by maximum likelihood estimation.

## Value

`dsinmad` gives the density, `psinmad` gives the distribution function, `qsinmad` gives the quantile function, and `rsinmad` generates random deviates.

## Note

The Singh-Maddala distribution is a special case of the 4-parameter generalized beta II distribution.

## Author(s)

T. W. Yee and Kai Huang

## References

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

`sinmad`, `genbetaII`.

## Examples

 ```1 2 3 4 5 6``` ```sdata <- data.frame(y = rsinmad(n = 3000, scale = exp(2), shape1 = exp(1), shape3 = exp(1))) fit <- vglm(y ~ 1, sinmad(lss = FALSE, ishape1.a = 2.1), data = sdata, trace = TRUE, crit = "coef") coef(fit, matrix = TRUE) Coef(fit) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  coefficients =
0.91103039, 2.10456733, 1.14372699
VGLM    linear loop  2 :  coefficients =
0.97011750, 1.98191515, 0.97959993
VGLM    linear loop  3 :  coefficients =
0.97492756, 1.98040389, 0.97192427
VGLM    linear loop  4 :  coefficients =
0.97495815, 1.98033074, 0.97176565
VGLM    linear loop  5 :  coefficients =
0.97496144, 1.98032034, 0.97174638
VGLM    linear loop  6 :  coefficients =
0.97496192, 1.98031883, 0.97174358
VGLM    linear loop  7 :  coefficients =
0.97496199, 1.98031861, 0.97174317
VGLM    linear loop  8 :  coefficients =
0.97496200, 1.98031857, 0.97174311
loge(shape1.a) loge(scale) loge(shape3.q)
(Intercept)       0.974962    1.980319      0.9717431
shape1.a    scale shape3.q
2.651066 7.245051 2.642547
```

VGAM documentation built on Jan. 16, 2021, 5:21 p.m.