Structural studies on the interactions of a P2N tridentate ligand with copper(I) silver(I) and S : a dissertation presented in partial fulfilment of the degree of Master of Philosophy at Massey University

This thesis presents a study of the coordination chemistry, chemical reactivity, spectroscopy, structure and bonding of the hybrid polydentate ligand 2-(diphenylphosphino)-N-[2-(diphenylphosphino)benzylidene]benzeneamine (PNCP) with copper(I), silver(I) and sulfur. The hybrid polydentate (PNCP) ligand contains two inequivalent phosphorus (soft) and one nitrogen (hard) donor atoms, Chapter One is a brief overview of tertiary phosphines used as monodentate, bidentate, tridentate and polydentate ligands with transition metals. In Chapter Two, the preparation structure and characterisation of PNCP have been studied. Reactions of PNCP with sulphur have been investigated and a small site selectivity for one of the P atoms noted. Experiments have also included selective synthesis of the unsymmetrical mono-sulphide tertiary phosphine ligands SPNCP, PNCPS and of the di-sulfide SPNCPS ligand, as well as a study on the molecular structure of the 3-coordinate complex, [Cu(SPNCPS)]CIO₄. In Chapter Three the preparation of a series of copper(I) complexes of the general formula [Cu(PNCP)ClO₄] and [Cu(PNCP)L]ClO₄ (L- ligands containing S, N donor atoms) have been reported. The crystal structure of [Cu(PNCP)ClO₄] has been determined, and shows PNCP acts as a tridentate ligand coordinated to copper(I) via two phosphorus and one nitrogen donor atoms. The copper(I) atom has a distorted tetrahedral environment with two short Cu-P bonds and a slightly long Cu-N bond. In Chapter Four, studies on the preparation of the mononuclear complex [Ag(PNCP)ClO₄] and the dinuclear complex [Ag(PNCP)(SCN)]₂ are presented. Both complexes were characterized by a variety of physicochemical techniques. The tridentate behaviour of PNCP in the complex [Ag(PNCP)ClO₄] was established but the Ag-N bond was long and weak. In the complex [Ag(PNCP)(SCN)]₂ the Ag-N bond not exist and PNCP acts as a bidentate ligand

Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression

We study the problem of nonparametric regression when the regressor is endogenous, which is an important nonparametric instrumental variables (NPIV) regression in econometrics and a difficult ill-posed inverse problem with unknown operator in statistics. We first establish a general upper bound on the sup-norm (uniform) convergence rate of a sieve estimator, allowing for endogenous regressors and weakly dependent data. This result leads to the optimal sup-norm convergence rates for spline and wavelet least squares regression estimators under weakly dependent data and heavy-tailed error terms. This upper bound also yields the sup-norm convergence rates for sieve NPIV estimators under i.i.d. data: the rates coincide with the known optimal $L^2$-norm rates for severely ill-posed problems, and are power of $\log(n)$ slower than the optimal $L^2$-norm rates for mildly ill-posed problems. We then establish the minimax risk lower bound in sup-norm loss, which coincides with our upper bounds on sup-norm rates for the spline and wavelet sieve NPIV estimators. This sup-norm rate optimality provides another justification for the wide application of sieve NPIV estimators. Useful results on weakly-dependent random matrices are also provided

On The Dependence Structure of Wavelet Coefficients for Spherical Random Fields

We consider the correlation structure of the random coefficients for a wide class of wavelet systems on the sphere (Mexican needlets) which were recently introduced in the literature by Geller and Mayeli (2007). We provide necessary and sufficient conditions for these coefficients to be asymptotic uncorrelated in the real and in the frequency domain. Here, the asymptotic theory is developed in the high resolution sense. Statistical applications are also discussed, in particular with reference to the analysis of cosmological data.Comment: Revised version for Stochastic Processes and their Application

High-Frequency Tail Index Estimation by Nearly Tight Frames

This work develops the asymptotic properties (weak consistency and Gaussianity), in the high-frequency limit, of approximate maximum likelihood estimators for the spectral parameters of Gaussian and isotropic spherical random fields. The procedure we used exploits the so-called mexican needlet construction by Geller and Mayeli in [Geller, Mayeli (2009)]. Furthermore, we propose a plug-in procedure to optimize the precision of the estimators in terms of asymptotic variance.Comment: 38 page

Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators Under Weak Dependence and Weak Conditions

We show that spline and wavelet series regression estimators for weakly dependent regressors attain the optimal uniform (i.e. sup-norm) convergence rate $(n/\log n)^{-p/(2p+d)}$ of Stone (1982), where $d$ is the number of regressors and $p$ is the smoothness of the regression function. The optimal rate is achieved even for heavy-tailed martingale difference errors with finite $(2+(d/p))$th absolute moment for $d/p<2$. We also establish the asymptotic normality of t statistics for possibly nonlinear, irregular functionals of the conditional mean function under weak conditions. The results are proved by deriving a new exponential inequality for sums of weakly dependent random matrices, which is of independent interest.Comment: forthcoming in Journal of Econometric

On rate optimality for ill-posed inverse problems in econometrics

In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases. We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model, can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.Comment: 27 page