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

```
cauchy(llocation = "identitylink", lscale = "loge",
ilocation = NULL, iscale = NULL,
iprobs = seq(0.2, 0.8, by = 0.2),
imethod = 1, nsimEIM = NULL, zero = "scale")
cauchy1(scale.arg = 1, llocation = "identitylink",
ilocation = NULL, imethod = 1)
```

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`

.

iprobs

Probabilities used to find the respective sample quantiles;
used to compute `iscale`

.

zero, nsimEIM

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 the default but if `nsimEIM`

is specified then
Fisher scoring with simulation 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 = "loge"), 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) # }

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