We propose a new inferential framework for constructing confidence regions and testing hypotheses in statistical models specified by a system of high dimensional estimating equations. We construct an influence function by projecting the fitted estimating equations to a sparse direction obtained by solving a large-scale linear program. Our main theoretical contribution is to establish a unified Z-estimation theory of confidence regions for high dimensional problems. Different from existing methods, all of which require the specification of the likelihood or pseudo-likelihood, our framework is likelihood-free. As a result, our approach provides valid inference for a broad class of high dimensional constrained estimating equation problems, which are not covered by existing methods. Such examples include, noisy compressed sensing, instrumental variable regression, undirected graphical models, discriminant analysis and vector autoregressive models. We present detailed theoretical results for all these examples. Finally, we conduct thorough numerical simulations, and a real dataset analysis to back up the developed theoretical results.

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The relationship between tourism and economic growth has created a large body of literature investigating the hypotheses of tourism-led economic growth (TLEGH) and economy-driven tourism growth (EDTGH). In this article, we use mixed-frequency Granger causality tests to investigate the relationship between the two types of growth in Hong Kong from 1974 to 2016. Our analysis reveals the following empirical regularities. First, the hidden short-run causality of TLEGH is detected, and EDTGH is proved in the short run and also in the long run when Granger causality tests are performed in a mixed-frequency framework. Second, mixed-frequency Granger tests demonstrate more power in testing the TLEGH and EDTGH via the rejection frequencies (bootstrap p value). Finally, rolling Granger causality tests reveal an unstable relationship between tourism and economic growth in both magnitude and direction, and the relationship is highly economic- and tourism-event-dependent.

Cation−π interactions play a significant role in a host of processes eminently relevant to biology. However, polarization effects arising from the interaction of cations with aromatic moieties have long been recognized to be inadequately described by pairwise additive force fields. In the present work, we address this longstanding shortcoming through the nonbonded FIX (NBFIX) feature of the CHARMM36 force field, modifying pair-specific Lennard–Jones (LJ) parameters, while circumventing the limitations of the Lorentz–Berthelot combination rules. The potentials of mean force (PMFs) characterizing prototypical cation−π interactions in aqueous solutions are first determined using a hybrid quantum mechanical/molecular mechanics (QM/MM) strategy in conjunction with an importance-sampling algorithm. The LJ parameters describing the cation−π pairs are then optimized to match the QM/MM PMFs. The standard binding free energies of nine cation−π complexes, i.e., toluene, para-cresol, and 3-methyl-indole interacting with either ammonium, guanidinium, or tetramethylammonium, determined with this new set of parameters agree well with the experimental measurements. Additional simulations were carried out on three different classes of biological objects featuring cation−π interactions, including five individual proteins, three protein–ligand complexes, and two protein–protein complexes. Our results indicate that the description of cation−π interactions is overall improved using NBFIX corrections, compared with the standard pairwise additive force field. Moreover, an accurate binding free energy calculation for a protein–ligand complex containing cation−π interactions (2BOK) shows that using the new parameters, the experimental binding affinity can be reproduced quantitatively. Put together, the present work suggests that the NBFIX parameters optimized here can be broadly utilized in the simulation of proteins in an aqueous solution to enhance the representation of cation−π interactions, at no additional computational cost.