# cauchy: Cauchy Distribution Family Function In VGAM: Vector Generalized Linear and Additive Models

## Description

Estimates either the location parameter or both the location and scale parameters of the Cauchy distribution by maximum likelihood estimation.

## Usage

 ```1 2 3 4 5``` ```cauchy(llocation = "identitylink", lscale = "loglink", imethod = 1, ilocation = NULL, iscale = NULL, gprobs.y = ppoints(19), gscale.mux = exp(-3:3), zero = "scale") cauchy1(scale.arg = 1, llocation = "identitylink", ilocation = NULL, imethod = 1, gprobs.y = ppoints(19), zero = NULL) ```

## Arguments

 `llocation, lscale` Parameter link functions for the location parameter a and the scale parameter b. See `Links` for more choices. `ilocation, iscale` Optional initial value for a and b. By default, an initial value is chosen internally for each. `imethod` Integer, either 1 or 2 or 3. Initial method, three algorithms are implemented. The user should try all possible values to help avoid converging to a local solution. Also, choose the another value if convergence fails, or use `ilocation` and/or `iscale`. `gprobs.y, gscale.mux, zero` See `CommonVGAMffArguments` for information. `scale.arg` Known (positive) scale parameter, called b below.

## Details

The Cauchy distribution has density function

f(y;a,b) = 1 / [pi * b * [1 + ((y-a)/b)^2]]

where y and a are real and finite, and b>0. The distribution is symmetric about a and has a heavy tail. Its median and mode are a, but the mean does not exist. The fitted values are the estimates of a. Fisher scoring is used.

If the scale parameter is known (`cauchy1`) then there may be multiple local maximum likelihood solutions for the location parameter. However, if both location and scale parameters are to be estimated (`cauchy`) then there is a unique maximum likelihood solution provided n > 2 and less than half the data are located at any one point.

## Value

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

## Warning

It is well-known that the Cauchy distribution may have local maximums in its likelihood function; make full use of `imethod`, `ilocation`, `iscale` etc.

## Note

Good initial values are needed. By default `cauchy` searches for a starting value for a and b on a 2-D grid. Likewise, by default, `cauchy1` searches for a starting value for a on a 1-D grid. If convergence to the global maximum is not acheieved then it also pays to select a wide range of initial values via the `ilocation` and/or `iscale` and/or `imethod` arguments.

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.

Barnett, V. D. (1966). Evaluation of the maximum-likehood estimator where the likelihood equation has multiple roots. Biometrika, 53, 151–165.

Copas, J. B. (1975). On the unimodality of the likelihood for the Cauchy distribution. Biometrika, 62, 701–704.

Efron, B. and Hinkley, D. V. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information. Biometrika, 65, 457–481.

`Cauchy`, `cauchit`, `studentt`, `simulate.vlm`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```# Both location and scale parameters unknown set.seed(123) cdata <- data.frame(x2 = runif(nn <- 1000)) cdata <- transform(cdata, loc = exp(1 + 0.5 * x2), scale = exp(1)) cdata <- transform(cdata, y2 = rcauchy(nn, loc, scale)) fit2 <- vglm(y2 ~ x2, cauchy(lloc = "loglink"), data = cdata, trace = TRUE) coef(fit2, matrix = TRUE) head(fitted(fit2)) # Location estimates summary(fit2) # Location parameter unknown cdata <- transform(cdata, scale1 = 0.4) cdata <- transform(cdata, y1 = rcauchy(nn, loc, scale1)) fit1 <- vglm(y1 ~ x2, cauchy1(scale = 0.4), data = cdata, trace = TRUE) coef(fit1, matrix = TRUE) ```