BMTfit.mqde: Minimum Quantile Distance Fit of the BMT Distribution to...

In BMT: The BMT Distribution

Minimum Quantile Distance Fit of the BMT Distribution to Non-censored Data.

Description

Fit of the BMT distribution to non-censored data by minimum quantile distance (mqde), which can also be called maximum quantile goodness-of-fit.

Usage

BMTfit.mqde(
  data,
  probs = (1:length(data) - 0.5)/length(data),
  qtype = 5,
  dist = "euclidean",
  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,
  weights = NULL,
  silent = TRUE,
  ...
)

Arguments

data	A numeric vector with the observed values for non-censored data.
probs	A numeric vector of the probabilities for which the minimum quantile distance estimation is done. p[k] = (k - 0.5) / n (default).
qtype	The quantile type used by the R quantile function to compute the empirical quantiles. Type 5 (default), i.e. x[k] is both the kth order statistic and the type 5 sample quantile of p[k] = (k - 0.5) / n.
dist	The distance measure between observed and theoretical quantiles to be used. This must be one of "euclidean" (default), "maximum", or "manhattan". Any unambiguous substring can be given.
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. Given the close-form expression of the quantile function, two optimization methods were added when the euclidean distance is selected: Coordinate descend ("CD") and Newton-Rhapson ("NR").
custom.optim	A function carrying the optimization (see the 'details' section of mledist).
weights	an optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector with strictly positive numbers. If non-NULL, weighted mqde is used, otherwise ordinary mqde.
silent	A logical to remove or show warnings when bootstrapping.
...	Further arguments to be passed to generic functions or to the function "mqdedist". See mqdedist for details.

Details

This function is not intended to be called directly but is internally called in BMTfit when used with the minimum quantile distance method.

BMTfit.mqde is based on the function mqdedist but it focuses on the minimum quantile distance parameter estimation for the BMT distribution (see BMT for details about the BMT distribution and mqdedist for details about minimum quantile distance fit of univariate distributions).

Given the close-form expression of the quantile function, two optimization methods were added when the euclidean distance is selected: Coordinate descend ("CD") and Newton-Rhapson ("NR").

Value

BMTfit.mqde 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 value of the corresponding objective function of the estimation method at the estimate.
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.
loglik	the log-likelihood value.
probs	the probability vector on which observed and theoretical quantiles were calculated.
dist	the name of the distance between observed and theoretical quantiles used.
optim.function	the name of the optimization function used for maximum product of spacing.
optim.method	when optim is used, the name of the algorithm used, NULL otherwise.
fix.arg	the named list giving the values of parameters of the named distribution that must 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.
weights	the vector of weights 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.

Author(s)

Camilo Jose Torres-Jimenez [aut,cre] [e-mail]

Source

Based on the function mqdedist which in turn is based on the function mledist 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.

References

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 Also

See BMT for the BMT density, distribution, quantile function and random deviates. See BMTfit.mme, BMTfit.mle, BMTfit.mge, BMTfit.mpse and BMTfit.qme for other estimation methods. See optim and constrOptim for optimization routines. See BMTfit and fitdist for functions that return an object of class "fitdist".

Examples

# (1) basic fit by minimum quantile distance estimation
set.seed(1234)
x1 <- rBMT(n=100, p3=0.25, p4=0.75)
BMTfit.mqde(x1)

# (2) quantile matching is a particular case of minimum quantile distance
BMTfit.mqde(x1, probs=c(0.2,0.4,0.6,0.8), qtype=7)

# (3) maximum or manhattan instead of euclidean distance
BMTfit.mqde(x1, dist="maximum")
BMTfit.mqde(x1, dist="manhattan")

# (4) how to change the optimisation method?
BMTfit.mqde(x1, optim.method="L-BFGS-B") 
BMTfit.mqde(x1, custom.optim="nlminb")

# (5) estimation of the tails weights parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mqde(x1, start=list(p3=0.5, p4=0.5), fix.arg=list(p1=0, p2=1))

# (6) estimation of the asymmetry-steepness parameters of the BMT 
# distribution with domain fixed at [0,1]
BMTfit.mqde(x1, start=list(p3=0, p4=0.5), type.p.3.4 = "a-s", 
            fix.arg=list(p1=0, p2=1))

BMT documentation built on June 8, 2025, 12:44 p.m.