# Minimum Quantile Distance Fit of Univariate Distributions. Fit of univariate distributions for non-censored data using minimum quantile distance estimation (mqde), which can also be called maximum quantile goodness-of-fit estimation. ## Source 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. Functions `checkparam` and `start.arg.default` are needed and were copied from the same package (fitdistrplus version: 1.0-9). ```r mqdedist(data, distr, probs = (1:length(data) - 0.5)/length(data), qtype = 5, dist = "euclidean", start = NULL, fix.arg = NULL, optim.method = "default", lower = -Inf, upper = Inf, custom.optim = NULL, weights = NULL, silent = TRUE, gradient = NULL, ...) ``` ## Arguments - `data`: A numeric vector with the observed values for non-censored data. - `distr`: A character string `"name"` naming a distribution for which the corresponding quantile function `qname` and the corresponding density distribution `dname` must be classically defined. - `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 $k$th 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 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 the 'details' section of `mledist`). - `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. - `optim.method`: `"default"` (see details) or optimization method to pass to `optim`. - `lower`: Left bounds on the parameters for the `"L-BFGS-B"` method (see `optim`) or the `constrOptim` function (as an equivalent linear constraint). - `upper`: Right bounds on the parameters for the `"L-BFGS-B"` method (see `optim`) or the `constrOptim` function (as an equivalent linear constraint). - `custom.optim`: A function carrying the optimization (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 numbers. If non-`NULL`, weighted mqde is used, otherwise ordinary mqde. - `silent`: A logical to remove or show warnings when bootstraping. - `gradient`: A function to return the gradient of the optimization objective function for the `"BFGS"`, `"CG"` and `"L-BFGS-B"` methods. If it is `NULL`, a finite-difference approximation will be used, see `optim`. - ``...``: Further arguments passed to the `optim`, `constrOptim` or `custom.optim` function. ## Returns `mqdedist` returns a list with following components, - **estimate**: the parameter estimates. - **convergence**: an integer code for the convergence of `optim` 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 optimization objective function at the solution found. - **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. - **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. - **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. - **loglik**: the log-likelihood. - **optim.method**: when `optim` is used, the name of the algorithm used, `NULL` otherwise. - **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 optimization 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. ## Details The `mqdedist` function carries out the minimum quantile distance estimation numerically, by minimization of a distance between observed and theoretical quantiles. The optimization process is the same as `mledist`, see the 'details' section of that function. Optionally, a vector of `weights` can be used in the fitting process. By default (when `weigths=NULL`), ordinary mqde is carried out, otherwise the specified weights are used to compute a weighted distance. We believe this function should be added to the package `fitdistrplus`. Until it is accepted and incorporated into that package, it will remain in the package `BMT`. This function is internally called in `BMTfit.mqde`. ## Examples ```r # (1) basic fit of a normal distribution set.seed(1234) x1 <- rnorm(n = 100) mean(x1); sd(x1) mqde1 <- mqdedist(x1, "norm") mqde1$estimate # (2) defining your own distribution functions, here for the Gumbel # distribution for other distributions, see the CRAN task view dedicated # to probability distributions dgumbel <- function(x, a, b) 1/b*exp((a-x)/b)*exp(-exp((a-x)/b)) pgumbel <- function(q, a, b) exp(-exp((a-q)/b)) qgumbel <- function(p, a, b) a-b*log(-log(p)) mqde1 <- mqdedist(x1, "gumbel", start = list(a = 10, b = 5)) mqde1$estimate # (3) fit a discrete distribution (Poisson) set.seed(1234) x2 <- rpois(n = 30, lambda = 2) mqde2 <- mqdedist(x2, "pois") mqde2$estimate # (4) fit a finite-support distribution (beta) set.seed(1234) x3 <- rbeta(n = 100, shape1 = 5, shape2 = 10) mqde3 <- mqdedist(x3, "beta") mqde3$estimate # (5) fit frequency distributions on USArrests dataset. x4 <- USArrests$Assault mqde4pois <- mqdedist(x4, "pois") mqde4pois$estimate mqde4nbinom <- mqdedist(x4, "nbinom") mqde4nbinom$estimate # (6) weighted fit of a normal distribution set.seed(1234) w1 <- runif(100) weighted.mean(x1, w1) mqde1 <- mqdedist(x1, "norm", weights = w1) mqde1$estimate ``` ## References LaRiccia, V. N. (1982). **Asymptotic Properties of Weighted $L^2$ Quantile Distance Estimators**. The Annals of Statistics, 10(2), 621-624. Torres-Jimenez, C. J. (2017, September), **Comparison of estimation methods for the BMT distribution**. ArXiv e-prints. ## See Also `mpsedist`, `mledist`, `mmedist`, `qmedist`, `mgedist`, `optim`, `constrOptim`, and `quantile`. ## Author(s) Camilo Jose Torres-Jimenez [aut,cre] cjtorresj@unal.edu.co

mqdedist function