# exponential: Exponential Distribution In VGAM: Vector Generalized Linear and Additive Models

## Description

Maximum likelihood estimation for the exponential distribution.

## Usage

 ```1 2 3 4``` ```exponential(link = "loglink", location = 0, expected = TRUE, type.fitted = c("mean", "percentiles", "Qlink"), percentiles = 50, ishrinkage = 0.95, parallel = FALSE, zero = NULL) ```

## Arguments

 `link` Parameter link function applied to the positive parameter rate. See `Links` for more choices. `location` Numeric of length 1, the known location parameter, A, say. `expected` Logical. If `TRUE` Fisher scoring is used, otherwise Newton-Raphson. The latter is usually faster. `ishrinkage, parallel, zero` See `CommonVGAMffArguments` for information. `type.fitted, percentiles` See `CommonVGAMffArguments` for information.

## Details

The family function assumes the response Y has density

f(y) = rate * exp(-rate * (y-A))

for y > A, where A is the known location parameter. By default, A=0. Then E(Y) = A + 1/rate and Var(Y) = 1/rate^2.

## Value

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

## Note

Suppose A = 0. For a fixed time interval, the number of events is Poisson with mean rate if the time between events has a geometric distribution with mean 1/rate. The argument `rate` in `exponential` is the same as `rexp` etc. The argument `lambda` in `rpois` is somewhat the same as `rate` here.

T. W. Yee

## References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

`amlexponential`, `gpd`, `laplace`, `expgeometric`, `explogff`, `poissonff`, `mix2exp`, `freund61`, `simulate.vlm`, `Exponential`.

## 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``` ```edata <- data.frame(x2 = runif(nn <- 100) - 0.5) edata <- transform(edata, x3 = runif(nn) - 0.5) edata <- transform(edata, eta = 0.2 - 0.7 * x2 + 1.9 * x3) edata <- transform(edata, rate = exp(eta)) edata <- transform(edata, y = rexp(nn, rate = rate)) with(edata, stem(y)) fit.slow <- vglm(y ~ x2 + x3, exponential, data = edata, trace = TRUE) fit.fast <- vglm(y ~ x2 + x3, exponential(exp = FALSE), data = edata, trace = TRUE, crit = "coef") coef(fit.slow, mat = TRUE) summary(fit.slow) # Compare results with a GPD. Has a threshold. threshold <- 0.5 gdata <- data.frame(y1 = threshold + rexp(n = 3000, rate = exp(1.5))) fit.exp <- vglm(y1 ~ 1, exponential(location = threshold), data = gdata) coef(fit.exp, matrix = TRUE) Coef(fit.exp) logLik(fit.exp) fit.gpd <- vglm(y1 ~ 1, gpd(threshold = threshold), data = gdata) coef(fit.gpd, matrix = TRUE) Coef(fit.gpd) logLik(fit.gpd) ```

### Example output

```Loading required package: stats4

The decimal point is at the |

0 | 0000000111111111122222222223333334444445555555566666677778888899
1 | 000111122222577888899
2 | 234566678
3 | 068
4 | 2
5 |
6 | 15

VGLM    linear loop  1 :  deviance = 111.13669
VGLM    linear loop  2 :  deviance = 106.29222
VGLM    linear loop  3 :  deviance = 106.0252
VGLM    linear loop  4 :  deviance = 106.01939
VGLM    linear loop  5 :  deviance = 106.01927
VGLM    linear loop  6 :  deviance = 106.01927
VGLM    linear loop  7 :  deviance = 106.01927
VGLM    linear loop  1 :  coefficients =
0.42193601, -1.22620368,  2.44490723
VGLM    linear loop  2 :  coefficients =
0.2016349, -1.1146637,  2.0719245
VGLM    linear loop  3 :  coefficients =
0.16456383, -1.08207141,  1.96645208
VGLM    linear loop  4 :  coefficients =
0.16335207, -1.08083309,  1.96244178
VGLM    linear loop  5 :  coefficients =
0.16335066, -1.08083166,  1.96243708
VGLM    linear loop  6 :  coefficients =
0.16335066, -1.08083166,  1.96243708
loge(rate)
(Intercept)  0.1633506
x2          -1.0808266
x3           1.9624262

Call:
vglm(formula = y ~ x2 + x3, family = exponential, data = edata,
trace = TRUE)

Pearson residuals:
Min      1Q Median     3Q    Max
loge(rate) -2.691 -0.4981 0.2029 0.7055 0.9864

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)   0.1634     0.1002   1.630  0.10305
x2           -1.0808     0.3557  -3.039  0.00238 **
x3            1.9624     0.3328   5.896 3.73e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Number of linear predictors:  1

Name of linear predictor: loge(rate)

Residual deviance: 106.0193 on 97 degrees of freedom

Log-likelihood: -83.7397 on 97 degrees of freedom

Number of iterations: 7
loge(rate)
(Intercept)   1.496169
rate
4.464551
 1488.506
loge(scale) logoff(shape, offset = 0.5)
(Intercept)   -1.487505                  -0.7106998
scale        shape
0.225935730 -0.008699738
 1488.613
```

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