Higher order asymptotic theory for normalizing transformations of maximum likelihood estimators

Suppose thatX _n =(X ₁,...X _n) is a collection ofm-dimensional random vectorsX _i forming a stochastic process with a parameter . Let be the MLE of . We assume that a transformationA( ) of has thek-thorder Edgeworth expansion (k=2,3). IfA extinguishes the terms in the Edgeworth expansion up tok-th-order (k2), then we say thatA is thek-th-order normalizing transformation. In this paper, we elucidate thek-th-order asymptotics of the normalizing transformations. Some conditions forA to be thek-th-order normalizing transformation will be given. Our results are very general, and can be applied to the i.i.d. case, multivariate analysis and time series analysis. Finally, we also study thek-th-order asymptotics of a modified signed log likelihood ratio in terms of the Edgeworth approximation.Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

On the approximation of continuous time threshold ARMA processes

Threshold autoregressive (AR) and autoregressive moving average (ARMA) processes with continuous time parameter have been discussed in several recent papers by Brockwellet al. (1991,Statist. Sinica,1, 401–410), Tong and Yeung (1991,Statist. Sinica,1, 411–430), Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) and Brockwell (1994,J. Statist. Plann. Inference,39, 291–304). A threshold ARMA process with boundary width 2>0 is easy to define in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold ARMA processes with =0 in the important case when only the autoregressive coefficients change with the level of the process. (This of course includes all threshold AR processes with constant scale parameter.) The idea is to express the distributions of the process in terms of the weak solution of a certain stochastic differential equation. It is shown that the joint distributions of this solution with =0 are the weak limits as 0 of the distributions of the solution with >0. The sense in which the approximating sequence of processes used by Brockwell and Hyndman (1992,International Journal Forecasting,8, 157–173) converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron-Martin-Girsanov formula. It is used in particular to fit continuous-time threshold models to the sunspot and Canadian lynx series.Research partially supported by National Science Foundation Research Grants DMS 9105745 and 9243648.     

Radial limit of lacunary Fourier series with coefficients in non-commutative symmetric spaces

Let be a rearrangement invariant space, an arbitrary set and a von Neumann algebra with a semifinite normal faithful trace. It is proved that the associated symmetric space of measurable operators has -RNP if and only if has -RNP extending in this way some previous results by Q. Xu.

     

On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank

We show that the Denjoy rank and the Zalcwasser rank are incomparable. We construct for any countable ordinal differentiable functions and such that the Zalcwasser rank and the Kechris-Woodin rank of are but the Denjoy rank of is 2 and the Denjoy rank and the Kechris-Woodin rank of are but the Zalcwasser rank of is 1. We then derive a theorem that shows the surprising behavior of the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank.

     

Asymptotics for Lp-norms of Fourier series density estimators

LetX ₁, X₂, , be a sequence of independent, identically distributed bounded random variables with a smooth density functionf. We prove that is asymptotically normal, wheref _{m, n} is the Fourier series density estimator offandw is a nonnegative weight function.Communicated by Edward B. Saff.AMS classification: Primary 60F05, 60F25; Secondary 62G05.     

The effect of the nature of the substituent at the nitrogen atom on transformations of 3-bromo-2,2,6,6-tetramethyl-4-piperidinone and its 1-hydroxy and 1-oxyl derivatives under conditions of the Favorsky rearrangement

3-Bromo-2,2,6,6-tetramethyl-4-oxopiperidine-1-oxyl reacts with NH₄OH to give 3-carbamoyl-2,2,5,5-tetramethylpyrrolidine-1-oxyl, a product of the Favorsky rearrangement. 3-Bromo-2,2,6,6-tetramethyl-4-piperidinone is transformed under these conditions into a bicyclic amino ketone, while its 1-hydroxy derivative affords acyclic nitrosoenone. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1189–1191, June, 1997.     

Synthesis and transformations of 2-(5-amino-(mercapto)-1,3,4-thiadiazolylthio)-7-methyl-5-oxo-5H-1,3,4-thiadiazolo[3,2-a]pyrimidines

The reactions of 2-amino-5-mercapto-(or 2,5-dimercapto)-1,3,4-thiadiazoles with 2-bromo-7-methyl-5-oxo-5H-1, 3,4-thiadiazolo[3, 2-a]pyrimidine to give the corresponding sulfides have been studied. The possibility of S-alkylation and addition of quinone at the free mercapto group in the 1,3,4-thiadiazole ring has been shown. The reactions at the amino group with benzoyl chloride and chloroformates have been investigated. The conditions of cyciodehydration at the amino group with ethyl acetoacetate and bromination of the pyrimidine fragment of 7-methyl-5-oxo-5H-1,3,4-thiadiazolo[3,2-a]pyrimidine have been found.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 1954–1957, November, 1993     

二茂镱羧酸衍生物的合成

本文报道了四种新的二茂镱羧酸衍生物的合成:Cp_2YbOCOC(CH_3)_3(1),Cp_2YbOCOCCl_3,(2),Cp_2YbOCOCH_2CH_3(3),Cp_2YbOCOCH_2Cl(4)。这些络合物都经元素分折、红外光谱及质谱鉴定。可能由于这些络合物具有桥式二聚体结构,因而增强了它们对空气的稳定性。     

The rich crystal chemistry of the AMM′(PO4)2 morphotropic series: NaZnAl(PO4)2, the first Na representative with a new structure type

A novel phosphate, sodium zinc aluminium bis(phosphate), NaZnAl(PO₄)₂, was obtained under mild‐temperature hydrothermal conditions at 553 K. The crystal structure has been studied using single‐crystal X‐ray experimental data. The pseudo‐hexagonal phase NaZnAl(PO₄)₂ crystallizes in the monoclinic space group P2₁/c. Its unique crystal structure is based on a three‐dimensional (3D) framework built by Zn‐, Al‐ and P‐centred tetrahedra sharing vertices. Channels parallel to the [101] and [01] directions are limited by six‐ and eight‐membered windows, and incorporate Na atoms. The new compound is discussed as a member of the morphotropic series AMM′PO₄, where A = Na, K, Rb or NH₄, M = Cu, Ni, Co, Fe, Zn or Mg and M′ = Fe, Al or Ga. The title compound is the first Na representative within the series and is characterized by a 3D architecture of tetrahedra populated in an ordered manner by Zn²⁺, Al³⁺ and P⁵⁺ ions.     

Linear Approximation of Basis Set Orbital Exponents for Isoelectronic Series of Atoms

The orbital exponents of Slater type atomic orbitals (AOs) in isoelectronic series of atoms may be approximated by the linear dependence on the nuclear charge using a technique developed for optimization of AO basis sets in Hartree–Fock–Roothaan calculations. This approach yields the analytical Hartree–Fock wave functions for any ion in the isoelectronic atomic series without optimization of orbital exponents. The approximated linear equations for atomic orbital basis sets of B, C, O, and F in the ground state are presented as an example.