# Mable deconvolution with a known error density Maximum approximate Bernstein/Beta likelihood estimation in additive density deconvolution model with a known error density. ```r mable.decon( y, gn = NULL, ..., M, interval = c(0, 1), IC = c("none", "aic", "hqic", "all"), vanished = TRUE, controls = mable.ctrl(maxit.em = 1e+05, eps.em = 1e-05, maxit.nt = 100, eps.nt = 1e-10), progress = TRUE ) ``` ## Arguments - `y`: vector of observed data values - `gn`: error density function if known, default is NULL if unknown - `...`: additional arguments to be passed to gn - `M`: a vector `(m0, m1)` specifies the set of consective candidate model degrees, `M = m0:m1`. If `gn` is unknown then `M` a 2 x 2 matrix whose rows `(m0,m1)` and `(k0,k1)` specify lower and upper bounds for degrees `m` and `k`, respectively. - `interval`: a finite vector `(a,b)`, the endpoints of supporting/truncation interval if `gn` is known. Otherwise, it is a 2 x 2 matrix whose rows `(a,b)` and `(a1,b1)` specify supporting/truncation intervals of `X` and $\epsilon$, respectively. See Details. - `IC`: information criterion(s) in addition to Bayesian information criterion (BIC). Current choices are "aic" (Akaike information criterion) and/or "qhic" (Hannan–Quinn information criterion). - `vanished`: logical whether the unknown error density vanishes at both end-points of `[a1,b1]` - `controls`: Object of class `mable.ctrl()` specifying iteration limit and other control options. Default is `mable.ctrl`. - `progress`: if `TRUE` a text progressbar is displayed ## Returns A `mable` class object with components, if $g$ is known, * `M` the vector `(m0, m1)`, where `m1` is the last candidate degree when the search stoped * `m` the selected optimal degree `m` * `p` the estimate of `p = (p_0, ..., p_m)`, the coefficients of Bernstein polynomial of degree `m` * `lk` log-likelihoods evaluated at $m \in \{m_0, \ldots, m_1\}$ * `lr` likelihood ratios for change-points evaluated at $m \in \{m_0+1, \ldots, m_1\}$ * `convergence` An integer code. 0 indicates an optimal degree is successfully selected in `M`. 1 indicates that the search stoped at `m1`. * `ic` a list containing the selected information criterion(s) * `pval` the p-values of the change-point tests for choosing optimal model degree * `chpts` the change-points chosen with the given candidate model degrees if $g$ is unknown, * `M` the 2 x 2 matrix with rows `(m0, m1)` and `(k0,k1)` * `nu_aic` the selected optimal degrees `(m,k)` using AIC method * `p_aic` the estimate of `p = (p_0, ..., p_m)`, the coefficients of Bernstein polynomial model for $f$ of degree `m` as in `nu_aic` * `q_aic` the estimate of `q = (q_0, ..., q_k)`, the coefficients of Bernstein polynomial model for $g$ of degree `k` as in `nu_aic` * `nu_bic` the selected optimal degrees `(m,k)` using BIC method * `p_bic` the estimate of `p = (p_0, ..., p_m)`, the coefficients of Bernstein polynomial model for $f$ of degree `m` as in `nu_bic` * `q_bic` the estimate of `q = (q_0, ..., q_k)`, the coefficients of Bernstein polynomial model for $g$ of degree `k` as in `nu_bic` * `lk` matrix of log-likelihoods evaluated at $m \in \{m_0, \ldots, m_1\}$ and $k \in \{k_0, \ldots, k_1\}$ * `aic` a matrix containing the Akaike information criterion(s) at $m \in \{m_0, \ldots, m_1\}$ and $k \in \{k_0, \ldots, k_1\}$ * `bic` a matrix containing the Bayesian information criterion(s) at $m \in \{m_0, \ldots, m_1\}$ and $k \in \{k_0, \ldots, k_1\}$ ## Details Consider the additive measurement error model $Y = X + \epsilon$, where $X$ has an unknown distribution $F$ on a known support `[a,b]`, $\epsilon$ has a known or unknown distribution $G$, and $X$ and $\epsilon$ are independent. We want to estimate density $f = F'$ based on independent observations, $y_i = x_i + \epsilon_i$, $i = 1, \ldots, n$, of $Y$. We approximate $f$ by a Bernstein polynomial model on `[a,b]`. If $g=G'$ is unknown on a known support `[a1,b1]`, then we approximate $g$ by a Bernstein polynomial model on `[a1,b1]`, $a1<0

mable.decon function