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)Parameter link functions for the location parameter \(a\)
and the scale parameter \(b\).
See Links for more choices.
Optional initial value for \(a\) and \(b\). By default, an initial value is chosen internally for each.
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.
Probabilities used to find the respective sample quantiles;
used to compute iscale.
See CommonVGAMffArguments for information.
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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