Web - based Supplementary Materials for “ Deconvolution Estimation of Mixture Distributions with Boundaries ”

In this section, we discuss the numerical implementation of obtaining our proposed sieve estimators. The estimation procedure is based on the EM algorithm. In particular, we describe the special considerations regarding efficient optimization of the constrained maximization problem in each maximization step. Let’s start from the model (3.4) in Section 3.2 where the target variable X is approximated by replacing its continuous componentXc with X̃c. To simplify the notation, let u = (a1, . . . , aν , x1, . . . , xr) T and let p = (π1, . . . , πν , πν+1θ T ) . Note that p and (π,θ) has 1-1 correspondence, so the estimation of p is equivalent to the estimation of (π,θ). Hence the log-likelihood (3.5) in Section 3.2 can be re-formulated as l(p) = N ∑ i=1 log fY (Yi |p) = N ∑ i=1 log { r+ν ∑ k=1 fY (Yi |X = uk)P (X = uk |p) }