Generalized Fractional Calculus for Gompertz-Type Models

This paper focuses on the construction of deterministic and stochastic extensions of the Gompertz curve by means of generalized fractional derivatives induced by complete Bernstein functions. Precisely, we first introduce a class of linear stochastic equations involving a generalized fractional integral and we study the properties of its solutions. This is done by proving the existence and uniqueness of Gaussian solutions of such equations via a fixed point argument and then by showing that, under suitable conditions, the expected value of the solution solves a generalized fractional linear equation. Regularity of the absolute p-moment functions is proved by using generalized Grönwall inequalities. Deterministic generalized fractional Gompertz curves are introduced by means of Caputo-type generalized fractional derivatives, possibly with respect to other functions. Their stochastic counterparts are then constructed by using the previously considered integral equations to define a rate process and a generalization of lognormal distributions to ensure that the median of the newly constructed process coincides with the deterministic curve.

Moment Functions on Groups

In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed.

Moment functions and exponential monomials on commutative hypergroups

The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions.

Theoretical investigation of the electronic structure and spectra of sulfur monoiodide cation, SI+

A manifold of singlet, triplet, and quintet electronic states of the sulfur monoiodide cation (SI+) correlating with the two lowest-lying dissociation channels is characterized theoretically at a high level of theoretical treatment (SA-CASSCF/MRCI+Q/aug-cc-pV5Z) for the first time. Potential energy curves, also including the effect of spin-orbit couplings, are constructed and the associated spectroscopic parameters and dissociation energies determined. As to the molecular polarity, we computed the dipole moment as a function of the internuclear distance and the associated vibrationally averaged dipole moments. Transition dipole moment functions were also constructed, and transition probabilities, as expressed by the Einstein coefficients for spontaneous emission, were evaluated for selected pairs of states that we identify as more easily accessible to experimental investigation. An analysis of the bonding in this system is also presented. Together with previous studies on neutral and cationic sulfur-monohalides, one has a comprehensive view of this series of molecules.

Improved stand structure characterization from nested plot designs in the Spanish National Forest Inventory

National forest inventories, in which trees are often mapped within the plots, provide a tool for the quantification of large-scale forest structure since they cover all forest areas. Many National Forest Inventories follow a nested design in order to reduce the sampling effort for smaller trees. We propose and test a methodology that allows the spatial pattern of trees, species mingling and size differentiation to be characterized using the nearest neighbour indices and second-order moment functions from nested plot data. The nearest neighbour indices and second-order moment functions for the actual distribution are compared with simulations of the appropriate null model: spatial randomness for spatial pattern characterization or spatial independence for species mingling and size differentiation. The proposed method consists of constraining the null model to fit the nested plot design. For the purposes of the study, we simulated 120 plots and used 26 real plots located in pure and mixed stands in Central Spain, for which a complete census with detailed information about trees was available. The nested design used in the Spanish National Forest Inventory (SNFI) plots was simulated to test the performance, taking the complete census as reference. Despite of the limited accuracy for some structural measures, the proposed method based on nested design data performed better for most of the nearest neighbour indices and second-order moment functions than the strategy currently used in the SNFI for structure assessment in a subsample of SNFI plots, consisting of mapping the 20 trees closest to the plot centre. Nearest neighbour indices provided greater accuracy for species mingling assessment than second-order moment functions, whereas the opposite occurred when describing spatial pattern and size differentiation. The methodology proposed provides the first insight into the characterization of forest structure in nested designs although more evaluations are required for different forest types.

EFFICIENT TWO-STEP GENERALIZED EMPIRICAL LIKELIHOOD ESTIMATION AND TESTS WITH MARTINGALE DIFFERENCES

This paper considers two-step generalized empirical likelihood (GEL) estimation and tests with martingale differences when there is a computationally simple $\sqrt n-$ consistent estimator of nuisance parameters or the nuisance parameters can be eliminated with an estimating function of parameters of interest. As an initial estimate might have asymptotic impact on final estimates, we propose general C(α)-type transformed moments to eliminate the impact, and use them in the GEL framework to construct estimation and tests robust to initial estimates. This two-step approach can save computational burden as the numbers of moments and parameters are reduced. A properly constructed two-step GEL (TGEL) estimator of parameters of interest is asymptotically as efficient as the corresponding joint GEL estimator. TGEL removes several higher-order bias terms of a corresponding two-step generalized method of moments. Our moment functions at the true parameters are martingales, thus they cover some spatial and time series models. We investigate tests for parameter restrictions in the TGEL framework, which are locally as powerful as those in the joint GEL framework when the two-step estimator is efficient.

Long-range potentials and dipole moments of the CO electronic states converging to the ground dissociation limit

The potential-energy and dipole-moment functions for six electronic states are obtained both analytically, in the framework of long-range perturbation theory, and numerically, by using first-principles methods.

LIKELIHOOD INFERENCE ON SEMIPARAMETRIC MODELS WITH GENERATED REGRESSORS

Hahn and Ridder (2013, Econometrica 81, 315–340) formulated influence functions of semiparametric three-step estimators where generated regressors are computed in the first step. This class of estimators covers several important examples for empirical analysis, such as production function estimators by Olley and Pakes (1996, Econometrica 64, 1263–1297) and propensity score matching estimators for treatment effects by Heckman, Ichimura, and Todd (1998, Review of Economic Studies 65, 261–294). The present article studies a nonparametric likelihood-based inference method for the parameters in such three-step estimation problems. In particular, we apply the general empirical likelihood theory of Bravo, Escanciano, and van Keilegom (2018, Annals of Statistics, forthcoming) to modify semiparametric moment functions to account for influences from plug-in estimates into the above important setup, and show that the resulting likelihood ratio statistic becomes asymptotically pivotal without undersmoothing in the first and second step nonparametric estimates.