Appendix B — Nomenclature

Symbol Meaning
\(\mathbb{E}\) Expected value operator
\(N\) True population size
\(\mathcal{U}\) Finite population with N units
\(y\) Response variable
\(\bx\) Auxiliary variables
\(p\) Number of auxiliary variables
\(S_A\) Nonprobablity sample
\(S_B\) Probability sample
\(\pi_i^B\) Probability of belonging to a probability sample for the given unit
\(\pi_i^A\) Probability of belonging to a non-probability sample for the given unit
\(\dot{\pi}_i^A(\cdot)\) Derivative of \(\pi_i^A\) at point \(\cdot\)
\(d_i^B\) Design weight for the given unit in a probability sample
\(n_A\) Size of non-probability sample
\(n_B\) Size of probability sample
\(\mu_{y}\) Mean of population for response variable
\(R_{i}^A\) An indicator function for nonprobability sample
\(\omega_i\) Frequency weight
\(\ell\) Log-likelihood function for the model
\(\hat{N}\) Point estimate for true population size
\(\text{var}\) Variance operator
\(\text{var}_p\) Design-based variance operator under the probability sampling design for \(S_B\)
\(Diag(\cdot)\) A diagonal matrix constructed from vector \(\cdot\)
\(\btheta\) Set of the parameters to estimate for selection model
\(\bbeta\) Set of the parameters to estimate for outcome model
\(\phi(\cdot)\) Probability density function of the normal distribution in point \(\cdot\)
\(\Phi(\cdot)\) Cumulative distribution function of the normal distribution in point \(\cdot\)
\(\hat{\mu}_{MI}\) Mass imputation estimator of population mean
\(\hat{\mu}_{IPW}\) Inverse probability weighted estimator of population mean
\(\hat{\mu}_{DR}\) Doubly robust estimator of population mean
\(q\) Model for selection mechanism for non-probability sample
\(\times\) A element by element product of vectors
\(\boldsymbol{\eta}\) A vector of linear predictors \((\eta_{i})\)
\(\lambda\) Tuning parameter for penalization function