# Maximum-likelihood Fitting of Univariate Distributions

### Description

Maximum-likelihood fitting of univariate distributions, allowing parameters to be held fixed if desired.

### Usage

1 |

### Arguments

`x` |
A numeric vector of length at least one containing only finite values. |

`densfun` |
Either a character string or a function returning a density evaluated at its first argument. Distributions |

`start` |
A named list giving the parameters to be optimized with initial values. This can be omitted for some of the named distributions and must be for others (see Details). |

`...` |
Additional parameters, either for |

### Details

For the Normal, log-Normal, geometric, exponential and Poisson
distributions the closed-form MLEs (and exact standard errors) are
used, and `start`

should not be supplied.

For all other distributions, direct optimization of the log-likelihood
is performed using `optim`

. The estimated standard
errors are taken from the observed information matrix, calculated by a
numerical approximation. For one-dimensional problems the Nelder-Mead
method is used and for multi-dimensional problems the BFGS method,
unless arguments named `lower`

or `upper`

are supplied (when
`L-BFGS-B`

is used) or `method`

is supplied explicitly.

For the `"t"`

named distribution the density is taken to be the
location-scale family with location `m`

and scale `s`

.

For the following named distributions, reasonable starting values will
be computed if `start`

is omitted or only partially specified:
`"cauchy"`

, `"gamma"`

, `"logistic"`

,
`"negative binomial"`

(parametrized by `mu`

and
`size`

), `"t"`

and `"weibull"`

. Note that these
starting values may not be good enough if the fit is poor: in
particular they are not resistant to outliers unless the fitted
distribution is long-tailed.

There are `print`

, `coef`

, `vcov`

and `logLik`

methods for class `"fitdistr"`

.

### Value

An object of class `"fitdistr"`

, a list with four components,

`estimate` |
the parameter estimates, |

`sd` |
the estimated standard errors, |

`vcov` |
the estimated variance-covariance matrix, and |

`loglik` |
the log-likelihood. |

### Note

Numerical optimization cannot work miracles: please note the comments
in `optim`

on scaling data. If the fitted parameters are
far away from one, consider re-fitting specifying the control
parameter `parscale`

.

### References

Venables, W. N. and Ripley, B. D. (2002)
*Modern Applied Statistics with S.* Fourth edition. Springer.

### 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 | ```
## avoid spurious accuracy
op <- options(digits = 3)
set.seed(123)
x <- rgamma(100, shape = 5, rate = 0.1)
fitdistr(x, "gamma")
## now do this directly with more control.
fitdistr(x, dgamma, list(shape = 1, rate = 0.1), lower = 0.001)
set.seed(123)
x2 <- rt(250, df = 9)
fitdistr(x2, "t", df = 9)
## allow df to vary: not a very good idea!
fitdistr(x2, "t")
## now do fixed-df fit directly with more control.
mydt <- function(x, m, s, df) dt((x-m)/s, df)/s
fitdistr(x2, mydt, list(m = 0, s = 1), df = 9, lower = c(-Inf, 0))
set.seed(123)
x3 <- rweibull(100, shape = 4, scale = 100)
fitdistr(x3, "weibull")
set.seed(123)
x4 <- rnegbin(500, mu = 5, theta = 4)
fitdistr(x4, "Negative Binomial")
options(op)
``` |