Multiple systems estimation strategies have recently been applied to quantify hard-to-reach populations, particularly when estimating the number of victims of human trafficking and modern slavery. In such contexts, it is not uncommon to see sparse or even no overlap between some of the lists on which the estimates are based. These create difficulties in model fitting and selection, and we develop inference procedures to address these challenges. The approach is based on Poisson log-linear regression modeling. Issues investigated in detail include taking proper account of data sparsity in the estimation procedure, as well as the existence and identifiability of maximum likelihood estimates. A stepwise method for choosing the most suitable parameters is developed, together with a bootstrap approach to finding confidence intervals for the total population size. We apply the strategy to two empirical data sets of trafficking in US regions, and find that the approach results in stable, reasonable estimates.
By generalising Rudin's construction of an aperiodic sequence, we derive new substitutionbased structures which have purely absolutely continuous diffraction and mixed dynamical spectrum, with absolutely continuous and pure point parts. We discuss several examples, including a construction based on Fourier matrices which yields constant-length substitutions for any length.
Modifying Rudin's original construction of the Rudin-Shapiro sequence, we derive a new substitution-based sequence with purely absolutely continuous diffraction spectrum.
Consider the U -distribution function defined as P (d (X, Y ) ≤ h) = E I {d(X,Y )≤h} , where Y is an independent copy of X and whose empirical counterpart is φ n (h). Note that, thanks to the law of total expectation, Assumption (A-2), the identification constraint E [ψ (X)] = 1, when h is small enough it holds:Thus, to prove the asymptotic behavior of φ n (h) towards φ θ0 (h) is equivalent to show that φ n (h) converges to the U -distribution function E I {d(X,Y )≤h} . Moreover, it is useful to note that, since the monomial behaviour is assumed, φ θ0 (h) is Lipschitz continuous on H and the U -distribution as well. In fact, for h, h ∈ H, there exists a positive constant L that
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