# paralogistic: Paralogistic Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

Maximum likelihood estimation of the 2-parameter paralogistic distribution.

## Usage

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

## Arguments

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

## Details

The 2-parameter paralogistic distribution is the 4-parameter generalized beta II distribution with shape parameter p=1 and a=q. It is the 3-parameter Singh-Maddala distribution with a=q. More details can be found in Kleiber and Kotz (2003).

The 2-parameter paralogistic has density

f(y) = a^2 y^(a-1) / [b^a (1 + (y/b)^a)^(1+a)]

for a > 0, b > 0, y >= 0. Here, b is the scale parameter `scale`, and a is the shape parameter. The mean is

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

provided a > 1; 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.

`Paralogistic`, `sinmad`, `genbetaII`, `betaII`, `dagum`, `fisk`, `inv.lomax`, `lomax`, `inv.paralogistic`.

## Examples

 ```1 2 3 4 5 6 7``` ```pdata <- data.frame(y = rparalogistic(n = 3000, exp(1), scale = exp(1))) fit <- vglm(y ~ 1, paralogistic(lss = FALSE), data = pdata, trace = TRUE) fit <- vglm(y ~ 1, paralogistic(ishape1.a = 2.3, iscale = 5), data = pdata, trace = TRUE) coef(fit, matrix = TRUE) Coef(fit) summary(fit) ```

### Example output

```Loading required package: stats4
VGLM    linear loop  1 :  loglikelihood = -3616.63447
VGLM    linear loop  2 :  loglikelihood = -3616.47705
VGLM    linear loop  3 :  loglikelihood = -3616.47344
VGLM    linear loop  4 :  loglikelihood = -3616.47344
Taking a modified step.
VGLM    linear loop  4 :  loglikelihood = -3616.47339
VGLM    linear loop  5 :  loglikelihood = -3616.47342
Taking a modified step....................
Warning messages:
1: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
iterations terminated because half-step sizes are very small
2: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
some quantities such as z, residuals, SEs may be inaccurate due to convergence at a half-step
VGLM    linear loop  1 :  loglikelihood = -4733.7391
VGLM    linear loop  2 :  loglikelihood = -4061.44235
VGLM    linear loop  3 :  loglikelihood = -3650.61063
VGLM    linear loop  4 :  loglikelihood = -3631.24299
VGLM    linear loop  5 :  loglikelihood = -3616.97948
VGLM    linear loop  6 :  loglikelihood = -3616.55755
VGLM    linear loop  7 :  loglikelihood = -3616.47613
VGLM    linear loop  8 :  loglikelihood = -3616.47341
VGLM    linear loop  9 :  loglikelihood = -3616.47343
Taking a modified step.
VGLM    linear loop  9 :  loglikelihood = -3616.47339
VGLM    linear loop  10 :  loglikelihood = -3616.47342
Taking a modified step....................
Warning messages:
1: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
iterations terminated because half-step sizes are very small
2: In vglm.fitter(x = x, y = y, w = w, offset = offset, Xm2 = Xm2,  :
some quantities such as z, residuals, SEs may be inaccurate due to convergence at a half-step
loge(scale) loge(shape1.a)
(Intercept)   0.9881697       1.021211
scale shape1.a
2.686313 2.776556

Call:
vglm(formula = y ~ 1, family = paralogistic(ishape1.a = 2.3,
iscale = 5), data = pdata, trace = TRUE)

Pearson residuals:
Min      1Q  Median     3Q    Max
loge(scale)    -0.9544 -0.8029 -0.3514 0.4935 4.6489
loge(shape1.a) -6.6837 -0.3512  0.3532 0.7234 0.8357

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1  0.98817    0.00953  103.69   <2e-16 ***
(Intercept):2  1.02121    0.01466   69.66   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  2

Names of linear predictors: loge(scale), loge(shape1.a)

Log-likelihood: -3616.473 on 5998 degrees of freedom

Number of iterations: 10
```

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