Fit of the BMT Distribution to Non-censored Data.
Fit of the BMT distribution to non-censored data by maximum likelihood (mle), moment matching (mme), quantile matching (qme), maximum goodness-of-fit (mge), also known as minimum distance, maximum product of spacing (mpse), also called maximum spacing, and minimum quantile distance (mqde), which can also be called maximum quantile goodness-of-fit.
Based on the function fitdist of the R package: fitdistrplus
Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.
BMTfit( data, method = c("mle", "mme", "qme", "mge", "mpse", "mqde"), start = list(p3 = 0.5, p4 = 0.5, p1 = min(data) - 0.1, p2 = max(data) + 0.1), fix.arg = NULL, type.p.3.4 = "t w", type.p.1.2 = "c-d", optim.method = "Nelder-Mead", custom.optim = NULL, keepdata = TRUE, keepdata.nb = 100, ... )
data: A numeric vector with the observed values for non-censored data.
method: A character string coding for the fitting method: "mle"
for 'maximum likelihood estimation', "mme" for 'moment matching estimation', "qme" for 'quantile matching estimation', "mge"
for 'maximum goodness-of-fit estimation', "mpse" for 'maximum product of spacing estimation', and "mqde" for 'minimum quantile estimation'.
start: A named list giving the initial values of parameters of the BMT distribution or a function of data computing initial values and returning a named list. (see the 'details' section of mledist).
fix.arg: An optional named list giving the values of fixed parameters of the BMT distribution or a function of data computing (fixed) parameter values and returning a named list. Parameters with fixed value are thus NOT estimated. (see the 'details' section of mledist).
type.p.3.4: Type of parametrization associated to p3 and p4. "t w" means tails weights parametrization (default) and "a-s" means asymmetry-steepness parametrization.
type.p.1.2: Type of parametrization associated to p1 and p2. "c-d" means domain parametrization (default) and "l-s" means location-scale parametrization.
optim.method: "default" (see the 'details' section of mledist) or optimization method to pass to optim.
custom.optim: A function carrying the optimization (see the 'details' section of mledist).
keepdata: A logical. If TRUE, dataset is returned, otherwise only a sample subset is returned.
keepdata.nb: When keepdata=FALSE, the length (>1) of the subset returned.
...: Further arguments to be passed to generic functions, or to one of the functions "BMTfit.mle", "BMTfit.mme", "BMTfit.qme", "BMTfit.mge", "BMTfit.mpse", or "BMTfit.mqde" depending of the chosen method. See BMTfit.mle, BMTfit.mme, BMTfit.qme, BMTfit.mge, BMTfit.mpse, BMTfit.mqde for details on parameter estimation.
fitdist returns an object of class "fitdist" with the following components:
estimate: the parameter estimates.
method: the character string coding for the fitting method : "mle" for 'maximum likelihood estimation', "mme" for 'moment matching estimation', "qme" for 'quantile matching estimation', "mge" for 'maximum goodness-of-fit estimation', "mpse" for 'maximum product of spacing estimation', and "mqde" for 'minimum quantile estimation'.
sd: the estimated standard errors, NA if numerically not computable or NULL if not available.
cor: the estimated correlation matrix, NA if numerically not computable or NULL if not available.
vcov: the estimated variance-covariance matrix, NULL if not available.
loglik: the log-likelihood.
aic: the Akaike information criterion.
bic: the the so-called BIC or SBC (Schwarz Bayesian criterion).
n: the length of the data set.
data: the data set.
distname: the name of the distribution (BMT).
fix.arg: the named list giving the values of parameters of the named distribution that must be kept fixed rather than estimated or NULL
if there are no such parameters.
fix.arg.fun: the function used to set the value of fix.arg or NULL.
discrete: the input argument or the automatic definition by the function to be passed to functions gofstat, plotdist and cdfcomp.
dots: the list of further arguments passed in ... to be used in bootdist in iterative calls to mledist, mmedist, qmedist, mgedist, mpsedist, mqdedist or NULL if no such arguments.
weights: the vector of weights used in the estimation process or NULL.
This function is based on the function fitdist
but it focuses on the parameter estimation for the BMT distribution (see BMT for details). It has six possible fitting methods: maximum likelihood (mle), moment matching (mme), quantile matching (qme), maximum goodness-of-fit (mge), also known as minimum distance, maximum product of spacing (mpse), also called maximum spacing, and minimum quantile distance (mqde), which can also be called maximum quantile goodness-of-fit. These fitting methods are carried out in BMTfit.mle, BMTfit.mme, BMTfit.qme, BMTfit.mge, BMTfit.mpse, and BMTfit.mqde, respectively (see each function for details). BMTfit returns an object of class "fitdist" (see fitdist for details). Therefore, it benefits of all the developed functions and methods for that class (see fitdistrplus for details).
Generic methods of a fitdist object are print, plot, summary, quantile, logLik, vcov
and coef.
# (1) fit of the BMT distribution by maximum likelihood estimation data(groundbeef) serving <- groundbeef$serving fit.mle <- BMTfit(serving) summary(fit.mle) plot(fit.mle) plot(fit.mle, demp = TRUE) plot(fit.mle, histo = FALSE, demp = TRUE) cdfcomp(fit.mle, addlegend=FALSE) denscomp(fit.mle, addlegend=FALSE) ppcomp(fit.mle, addlegend=FALSE) qqcomp(fit.mle, addlegend=FALSE) # (2) Comparison of various estimation methods fit.mme <- BMTfit(serving, method="mme") fit.mpse <- BMTfit(serving, method="mpse") fit.mqde <- BMTfit(serving, method="mqde") summary(fit.mme) summary(fit.mpse) summary(fit.mqde) cdfcomp(list(fit.mle, fit.mme, fit.mpse, fit.mqde), legendtext=c("mle", "mme", "mpse", "mqde")) denscomp(list(fit.mle, fit.mme, fit.mpse, fit.mqde), legendtext=c("mle", "mme", "mpse", "mqde")) qqcomp(list(fit.mle, fit.mme, fit.mpse, fit.mqde), legendtext=c("mle", "mme", "mpse", "mqde")) ppcomp(list(fit.mle, fit.mme, fit.mpse, fit.mqde), legendtext=c("mle", "mme", "mpse", "mqde")) gofstat(list(fit.mle, fit.mme, fit.mpse, fit.mqde), fitnames=c("mle", "mme", "mpse", "mqde")) # (3) how to change the optimisation method? BMTfit(serving, optim.method="Nelder-Mead") BMTfit(serving, optim.method="L-BFGS-B") BMTfit(serving, custom.optim="nlminb") # (4) estimation of the tails weights parameters of the BMT distribution # with domain fixed at [9,201] using Kolmogorov-Smirnov fit.KS <- BMTfit(serving, method="mge", gof="KS", start=list(p3=0.5, p4=0.5), fix.arg=list(p1=9, p2=201)) summary(fit.KS) plot(fit.KS) # (5) estimation of the asymmetry-steepness parameters of the BMT # distribution with domain fixed at [9,201] using minimum quantile distance # with a closed formula (optim.method="CD") fit.mqde.CD <- BMTfit(serving, method="mqde", optim.method="CD", start=list(p3=0.5, p4=0.5), type.p.3.4 = "a-s", fix.arg=list(p1=9, p2=201)) summary(fit.mqde.CD) plot(fit.mqde.CD)
Torres-Jimenez, C. J. (2017, September), Comparison of estimation methods for the BMT distribution. ArXiv e-prints.
Torres-Jimenez, C. J. (2018), The BMT Item Response Theory model: A new skewed distribution family with bounded domain and an IRT model based on it, PhD thesis, Doctorado en ciencias - Estadistica, Universidad Nacional de Colombia, Sede Bogota.
See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.mle, BMTfit.mme, BMTfit.qme, BMTfit.mge, BMTfit.mpse and BMTfit.mqde for details on parameter estimation. See fitdist for details on the object fitdist and its methods print, plot, summary, quantile, logLik, vcov and coef, and fitdistrplus for an overview of the package to which that object belongs to.
Camilo Jose Torres-Jimenez [aut,cre] cjtorresj@unal.edu.co
Useful links