Maximum likelihood fit of univariate distributions

Fit of univariate distributions using maximum likelihood for censored or non censored data.

Usage

mledist(data, distr, start = NULL, fix.arg = NULL, optim.method = "default", 
    lower = -Inf, upper = Inf, custom.optim = NULL, weights = NULL, silent = TRUE, 
    gradient = NULL, checkstartfix=FALSE, calcvcov=FALSE, ...)

Arguments

data: A numeric vector for non censored data or a dataframe of two columns respectively named left and right, describing each observed value as an interval for censored data. In that case the left column contains either NA for left censored observations, the left bound of the interval for interval censored observations, or the observed value for non-censored observations. The right column contains either NA for right censored observations, the right bound of the interval for interval censored observations, or the observed value for non-censored observations.
distr: A character string "name" naming a distribution for which the corresponding density function dname and the corresponding distribution function pname must be classically defined.
start: A named list giving the initial values of parameters of the named distribution or a function of data computing initial values and returning a named list. This argument may be omitted (default) for some distributions for which reasonable starting values are computed (see details).
fix.arg: An optional named list giving the values of fixed parameters of the named distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated by this maximum likelihood procedure.
optim.method: "default" (see details) or an optimization method to pass to optim.
lower: Left bounds on the parameters for the "L-BFGS-B" method (see optim).
upper: Right bounds on the parameters for the "L-BFGS-B" method (see optim).
custom.optim: a function carrying the MLE optimisation (see details).
weights: an optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector with strictly positive integers (typically the number of occurences of each observation). If non-NULL, weighted MLE is used, otherwise ordinary MLE.
silent: A logical to remove or show warnings when bootstraping.
gradient: A function to return the gradient of the log-likelihood for the "BFGS", "CG" and "L-BFGS-B" methods. If it is NULL, a finite-difference approximation will be used, see details.
checkstartfix: A logical to test starting and fixed values. Do not change it.
calcvcov: A logical indicating if (asymptotic) covariance matrix is required.
...: further arguments passed to the optim, constrOptim or custom.optim function.

Details

This function is not intended to be called directly but is internally called in fitdist and bootdist when used with the maximum likelihood method and fitdistcens and bootdistcens.

It is assumed that the distr argument specifies the distribution by the probability density function and the cumulative distribution function (d, p). The quantile function and the random generator function (q, r) may be needed by other function such as mmedist, qmedist, mgedist, fitdist,fitdistcens, bootdistcens and bootdist.

For the following named distributions, reasonable starting values will be computed if start is omitted (i.e. NULL) : "norm", "lnorm", "exp" and "pois", "cauchy", "gamma", "logis", "nbinom" (parametrized by mu and size), "geom", "beta", "weibull" from the stats package; "invgamma", "llogis", "invweibull", "pareto1", "pareto", "lgamma", "trgamma", "invtrgamma" from the actuar package. Note that these starting values may not be good enough if the fit is poor. The function uses a closed-form formula to fit the uniform distribution. If start is a list, then it should be a named list with the same names as in the d,p,q,r functions of the chosen distribution. If start is a function of data, then the function should return a named list with the same names as in the d,p,q,r functions of the chosen distribution.

The mledist function allows user to set a fixed values for some parameters. As for start, if fix.arg is a list, then it should be a named list with the same names as in the d,p,q,r functions of the chosen distribution. If fix.arg is a function of data, then the function should return a named list with the same names as in the d,p,q,r functions of the chosen distribution.

When custom.optim=NULL (the default), maximum likelihood estimations of the distribution parameters are computed with the R base optim or constrOptim. If no finite bounds (lower=-Inf and upper=Inf) are supplied, optim is used with the method specified by optim.method. Note that optim.method="default" means optim.method="Nelder-Mead" for distributions with at least two parameters and optim.method="BFGS" for distributions with only one parameter. If finite bounds are supplied (among lower and upper) and gradient != NULL, constrOptim is used. If finite bounds are supplied (among lower and upper) and gradient == NULL, constrOptim is used when optim.method="Nelder-Mead"; optim is used when optim.method="L-BFGS-B" or "Brent"; in other case, an error is raised (same behavior as constrOptim).

When errors are raised by optim, it's a good idea to start by adding traces during the optimization process by adding control=list(trace=1, REPORT=1).

If custom.optim is not NULL, then the user-supplied function is used instead of the R base optim. The custom.optim must have (at least) the following arguments fn for the function to be optimized, par for the initialized parameters. Internally the function to be optimized will also have other arguments, such as obs with observations and ddistname with distribution name for non censored data (Beware of potential conflicts with optional arguments of custom.optim). It is assumed that custom.optim should carry out a MINIMIZATION. Finally, it should return at least the following components par for the estimate, convergence for the convergence code, value for fn(par), hessian, counts for the number of calls (function and gradient) and message (default to NULL) for the error message when custom.optim raises an error, see the returned value of optim. See examples in fitdist and fitdistcens.

Optionally, a vector of weights can be used in the fitting process. By default (when weigths=NULL), ordinary MLE is carried out, otherwise the specified weights are used to balance the log-likelihood contributions. It is not yet possible to take into account weights in functions plotdist, plotdistcens, plot.fitdist, plot.fitdistcens, cdfcomp, cdfcompcens, denscomp, ppcomp, qqcomp, gofstat, descdist, bootdist, bootdistcens and mgedist. (developments planned in the future).

NB: if your data values are particularly small or large, a scaling may be needed before the optimization process. See Example (7).

Value

mledist returns a list with following components,

estimate: the parameter estimates.
convergence: an integer code for the convergence of optim/constrOptim defined as below or defined by the user in the user-supplied optimization function. 0 indicates successful convergence. 1 indicates that the iteration limit of optim has been reached. 10 indicates degeneracy of the Nealder-Mead simplex. 100 indicates that optim encountered an internal error.
value: the minimal value reached for the criterion to minimize.
hessian: a symmetric matrix computed by optim as an estimate of the Hessian at the solution found or computed in the user-supplied optimization function. It is used in fitdist to estimate standard errors.
optim.function: the name of the optimization function used for maximum likelihood.
optim.method: when optim is used, the name of the algorithm used, the field method of the custom.optim function otherwise.
fix.arg: the named list giving the values of parameters of the named distribution that must kept fixed rather than estimated by maximum likelihood or NULL if there are no such parameters.
fix.arg.fun: the function used to set the value of fix.arg or NULL.
weights: the vector of weigths used in the estimation process or NULL.
counts: A two-element integer vector giving the number of calls to the log-likelihood function and its gradient respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to log-likelihood function to compute a finite-difference approximation to the gradient. counts is returned by optim or the user-supplied function or set to NULL.
optim.message: A character string giving any additional information returned by the optimizer, or NULL. To understand exactly the message, see the source code.
loglik: the log-likelihood value.
method: "closed formula" if appropriate otherwise NULL.

References

Venables WN and Ripley BD (2002), Modern applied statistics with S. Springer, New York, pp. 435-446, doi:10.1007/978-0-387-21706-2 .

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34, doi:10.18637/jss.v064.i04 .

Author

Marie-Laure Delignette-Muller and Christophe Dutang.

Examples


# (1) basic fit of a normal distribution with maximum likelihood estimation
#

set.seed(1234)
x1 <- rnorm(n=100)
mledist(x1,"norm")
#> $estimate
#>       mean         sd 
#> -0.1567617  0.9993707 
#> 
#> $convergence
#> [1] 0
#> 
#> $value
#> [1] 1.418309
#> 
#> $hessian
#>         mean       sd
#> mean 1.00126 0.000000
#> sd   0.00000 2.002538
#> 
#> $optim.function
#> [1] "optim"
#> 
#> $optim.method
#> [1] "Nelder-Mead"
#> 
#> $fix.arg
#> NULL
#> 
#> $fix.arg.fun
#> NULL
#> 
#> $weights
#> NULL
#> 
#> $counts
#> function gradient 
#>       43       NA 
#> 
#> $optim.message
#> NULL
#> 
#> $loglik
#> [1] -141.8309
#> 
#> $vcov
#> NULL
#> 

# (2) defining your own distribution functions, here for the Gumbel distribution
# for other distributions, see the CRAN task view dedicated to probability distributions

dgumbel <- function(x,a,b) 1/b*exp((a-x)/b)*exp(-exp((a-x)/b))
mledist(x1,"gumbel",start=list(a=10,b=5))
#> Error in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg,     argddistname, hasnodefaultval): 'start' must specify names which are arguments to 'distr'.

# (3) fit of a discrete distribution (Poisson)
#

set.seed(1234)
x2 <- rpois(n=30,lambda = 2)
mledist(x2,"pois")
#> $estimate
#> lambda 
#>    1.7 
#> 
#> $convergence
#> [1] 0
#> 
#> $value
#> [1] 1.539478
#> 
#> $hessian
#>           lambda
#> lambda 0.5882357
#> 
#> $optim.function
#> [1] "optim"
#> 
#> $optim.method
#> [1] "BFGS"
#> 
#> $fix.arg
#> NULL
#> 
#> $fix.arg.fun
#> NULL
#> 
#> $weights
#> NULL
#> 
#> $counts
#> function gradient 
#>        4        1 
#> 
#> $optim.message
#> NULL
#> 
#> $loglik
#> [1] -46.18434
#> 
#> $vcov
#> NULL
#> 

# (4) fit a finite-support distribution (beta)
#

set.seed(1234)
x3 <- rbeta(n=100,shape1=5, shape2=10)
mledist(x3,"beta")
#> $estimate
#>    shape1    shape2 
#>  4.859798 10.918841 
#> 
#> $convergence
#> [1] 0
#> 
#> $value
#> [1] -0.7833052
#> 
#> $hessian
#>             shape1      shape2
#> shape1  0.16295311 -0.06542753
#> shape2 -0.06542753  0.03047900
#> 
#> $optim.function
#> [1] "optim"
#> 
#> $optim.method
#> [1] "Nelder-Mead"
#> 
#> $fix.arg
#> NULL
#> 
#> $fix.arg.fun
#> NULL
#> 
#> $weights
#> NULL
#> 
#> $counts
#> function gradient 
#>       47       NA 
#> 
#> $optim.message
#> NULL
#> 
#> $loglik
#> [1] 78.33052
#> 
#> $vcov
#> NULL
#> 


# (5) fit frequency distributions on USArrests dataset.
#

x4 <- USArrests$Assault
mledist(x4, "pois")
#> $estimate
#> lambda 
#> 170.76 
#> 
#> $convergence
#> [1] 0
#> 
#> $value
#> [1] 24.2341
#> 
#> $hessian
#>             lambda
#> lambda 0.005856175
#> 
#> $optim.function
#> [1] "optim"
#> 
#> $optim.method
#> [1] "BFGS"
#> 
#> $fix.arg
#> NULL
#> 
#> $fix.arg.fun
#> NULL
#> 
#> $weights
#> NULL
#> 
#> $counts
#> function gradient 
#>        2        1 
#> 
#> $optim.message
#> NULL
#> 
#> $loglik
#> [1] -1211.705
#> 
#> $vcov
#> NULL
#> 
mledist(x4, "nbinom")
#> $estimate
#>       size         mu 
#>   3.822579 170.747853 
#> 
#> $convergence
#> [1] 0
#> 
#> $value
#> [1] 5.806593
#> 
#> $hessian
#>               size            mu
#> size  3.518616e-02 -3.987921e-07
#> mu   -3.987921e-07  1.282598e-04
#> 
#> $optim.function
#> [1] "optim"
#> 
#> $optim.method
#> [1] "Nelder-Mead"
#> 
#> $fix.arg
#> NULL
#> 
#> $fix.arg.fun
#> NULL
#> 
#> $weights
#> NULL
#> 
#> $counts
#> function gradient 
#>       47       NA 
#> 
#> $optim.message
#> NULL
#> 
#> $loglik
#> [1] -290.3297
#> 
#> $vcov
#> NULL
#> 

# (6) fit a continuous distribution (Gumbel) to censored data.
#

data(fluazinam)
log10EC50 <-log10(fluazinam)
# definition of the Gumbel distribution
dgumbel  <-  function(x,a,b) 1/b*exp((a-x)/b)*exp(-exp((a-x)/b))
pgumbel  <-  function(q,a,b) exp(-exp((a-q)/b))
qgumbel  <-  function(p,a,b) a-b*log(-log(p))

mledist(log10EC50,"gumbel",start=list(a=0,b=2),optim.method="Nelder-Mead")
#> Error in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg,     argddistname, hasnodefaultval): 'start' must specify names which are arguments to 'distr'.

# (7) scaling problem
# the simulated dataset (below) has particularly small values, 
# hence without scaling (10^0),
# the optimization raises an error. The for loop shows how scaling by 10^i
# for i=1,...,6 makes the fitting procedure work correctly.

set.seed(1234)
x2 <- rnorm(100, 1e-4, 2e-4)
for(i in 6:0)
    cat(i, try(mledist(x*10^i, "cauchy")$estimate, silent=TRUE), "\n")
#> 6 Error in eval(expr, envir) : object 'x' not found
#>  
#> 5 Error in eval(expr, envir) : object 'x' not found
#>  
#> 4 Error in eval(expr, envir) : object 'x' not found
#>  
#> 3 Error in eval(expr, envir) : object 'x' not found
#>  
#> 2 Error in eval(expr, envir) : object 'x' not found
#>  
#> 1 Error in eval(expr, envir) : object 'x' not found
#>  
#> 0 Error in eval(expr, envir) : object 'x' not found
#>  
        
 
# (17) small example for the zero-modified geometric distribution
#

dzmgeom <- function(x, p1, p2) p1 * (x == 0) + (1-p1)*dgeom(x-1, p2) #pdf
x2 <- c(2,  4,  0, 40,  4, 21,  0,  0,  0,  2,  5,  0,  0, 13,  2) #simulated dataset
initp1 <- function(x) list(p1=mean(x == 0)) #init as MLE
mledist(x2, "zmgeom", fix.arg=initp1, start=list(p2=1/2))
#> Error in checkparamlist(arg_startfix$start.arg, arg_startfix$fix.arg,     argddistname, hasnodefaultval): 'start' must specify names which are arguments to 'distr'.