Maximum-likelihood Fitting of Univariate Distributions

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

fitdistr(x, densfun, start, ...)

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 "beta", "cauchy", "chi-squared", "exponential", "gamma", "geometric", "log-normal", "lognormal", "logistic", "negative binomial", "normal", "Poisson", "t" and "weibull" are recognised, case being ignored.
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 densfun or for optim. In particular, it can be used to specify bounds via lower or upper or both. If arguments of densfun (or the density function corresponding to a character-string specification) are included they will be held fixed.

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.

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".

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

## avoid spurious accuracy
op <- options(digits = 3)
set.seed(123)
x <- rgamma(100, shape = 5, rate = 0.1)
fitdistr(x, "gamma")
#>    shape     rate 
#>   6.4870   0.1365 
#>  (0.8946) (0.0196)
## now do this directly with more control.
fitdistr(x, dgamma, list(shape = 1, rate = 0.1), lower = 0.001)
#>    shape     rate 
#>   6.4869   0.1365 
#>  (0.8944) (0.0196)

set.seed(123)
x2 <- rt(250, df = 9)
fitdistr(x2, "t", df = 9)
#>       m         s   
#>   -0.0107    1.0441 
#>  ( 0.0722) ( 0.0543)
## allow df to vary: not a very good idea!
fitdistr(x2, "t")
#> Warning: NaNs produced
#>       m          s          df   
#>   -0.00965    1.00617    6.62729 
#>  ( 0.07147) ( 0.07707) ( 2.71033)
## 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))
#>       m         s   
#>   -0.0107    1.0441 
#>  ( 0.0722) ( 0.0543)

set.seed(123)
x3 <- rweibull(100, shape = 4, scale = 100)
fitdistr(x3, "weibull")
#>     shape     scale 
#>     4.080    99.984 
#>  (  0.313) (  2.582)

set.seed(123)
x4 <- rnegbin(500, mu = 5, theta = 4)
fitdistr(x4, "Negative Binomial")
#>    size     mu  
#>   4.216   4.945 
#>  (0.504) (0.147)
options(op)