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

```
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)
```

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.

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

,
and `vgam`

.

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

, `ilocation`

, `iscale`

etc.

The Cauchy distribution has density function $$f(y;a,b) = \left\{ \pi b [1 + ((y-a)/b)^2] \right\}^{-1} $$ 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.

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.

# NOT RUN { # 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) # }

Run the code above in your browser using DataCamp Workspace