Minimax kernels for density estimation with biased data

This paper considers the asymptotic properties of two kernel estimates % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\], which have been proposed by Bhattacharyya et al. (1988, Comm. Statist. Theory Methods, A17, 3629–3644) and Jones (1991, Biometrika, 78, 511–519), respectively, for estimating the underlying density f at a point under a general selection biased model. The asymptotic optimality of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]is measured by the corresponding asymptotic minimax mean squared errors under a compactly supported Lipschitz continuous family of the underlying densities. It is shown that, in general, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]is a superior local estimate than % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]in the sense that the asymptotic minimax risk of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]is lower than that of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]. The minimax kernels and bandwidths of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]are computed explicity and shown to have simple forms and depend on the weight functions of the model.     

Revised molecular constants and term values for the XΠ and BΣ states of OH

An improved set of molecular constants and term values are given for the X²Π (v = 0–13) and B²Σ⁺ (v = 0 and 1) states of the OH radical. They are derived from a fit of previously published laboratory data and additional lines taken from infrared solar spectra recorded on orbit.     

THE BIOGENETIC LAW AND ITS INFLUENCE ON THEORIES OF LEARNING MATHEMATICS

Reference to the history of mathematics has been widely used in discussions of the development of curricula, the problems students have in learning mathematics, and the development of concepts in the individual. This paper examines the background to the "Biogenetic Law" [1] which has influenced so much thinking in educational theory, and the use of the "principle of parallelism" where individual development is claimed to mirror the historical development of the subject matter. Interpretations of the history of mathematics which are used to justify these claims are examined, and the universality of the supposed concepts is challenged. Questions are raised about some of the fundamental tenets of Piagetian epistemology.     

Bis­[μ‐3{5}‐(2‐pyridyl)­pyrazolido]­bis­[(aceto­nitrile)copper(II)] diperchlorate

The title compound, [Cu₂(C₈H₆N₃)₂(C₂H₃N)₂](ClO₄)₂, con­tains a centrosymmetric dinuclear dication, with square‐pyrami­dal Cu^II centres linked by two 1,2‐pyrazolide bridging groups. There is a weak apical interaction between each Cu ion and a perchlorate O atom.     

[2,6‐Bis(3,5‐dimethylpyrazol‐1‐ylmethyl)pyridine]iodocopper(I) dichloromethane solvate

The title compound, [CuI(C₁₇H₂₁N₅)]·CH₂Cl₂, contains a tetracoordinate Cu^I centre with an unusual distorted tetrahedral stereochemistry, which has also been observed in other Cu^I complexes containing this tridentate ligand. This distortion is probably a result of intermolecular steric contacts between the I^? ligand and a neighbouring CH₂Cl₂ mol­ecule.     

2,6‐Bis‐(3‐trifluoromethylpyrazol‐1‐yl)pyridine

The title compound, C₁₃H₇F₆N₅, is one of a series of hindered tris‐imine ligands for meridonial co­ordination to transition metals. The mol­ecule has crystallographic C₂ symmetry, the pyrazole and pyridine rings adopting a near‐coplanar transoid conformation.     

Quantitation of the nearest-neighbour effects of amino acid side-chains that restrict conformational freedom of the polypeptide chain using reversed-phase liquid chromatography of synthetic model peptides with L- and D-amino acid substitutions

Side-chain backbone interactions (or "effects") between nearest neighbours may severely restrict the conformations accessible to a polypeptide chain and thus represent the first step in protein folding. We have quantified nearest-neighbour effects (i to i+1) in peptides through reversed-phase liquid chromatography (RP-HPLC) of model synthetic peptides, where L- and D-amino acids were substituted at the N-terminal end of the peptide sequence, adjacent to a L-Leu residue. These nearest-neighbour effects (expressed as the difference in retention times of L- and D-peptide diastereomers at pHs 2 and 7) were frequently dramatic, depending on the type of side-chain adjacent to the L-Leu residue, albeit such effects were independent of mobile phase conditions. No nearest-neighbour effects were observed when residue i is adjacent to a Gly residue. Calculation of minimum energy conformations of selected peptides supported the view that, whether a L- or D-amino acid is substituted adjacent to L-Leu, its orientation relative to this bulky Leu side-chain represents the most energetically favourable configuration. We believe that such energetically favourable, and different, configurations of L- and D-peptide diastereomers affect their respective interactions with a hydrophobic stationary phase, which are thus quantified by different RP-HPLC retention times. Side-chain hydrophilicity/hydrophobicity coefficients were generated in the presence of these nearest-neighbour effects and, despite the relative difference in such coefficients generated from peptides substituted with L- or D-amino acids, the relative difference in hydrophilicity/hydrophobicity between different amino acids in the L- or D-series is maintained. Overall, our results demonstrate that such nearest-neighbour effects can clearly restrict conformational space of an amino acid side-chain in a polypeptide chain.     

Non-empirical LCAO-MO-SCF calculations of the electronic structure of SiF2

Non-empirical LCAO-MO-SCF calculations on SiF₂ using two Gaussian basis sets are reported. The larger basis set gives a calculated geometry in good agreement with experiment. The effect on the energy and population analysis of optimization of the Si 3d exponent was investigated. 3d orbitals are found to be much less important in the bonding than in the isoelectronic molecule SO₂.     

New Synthetic Methodology and Biologically Active Substances

The η⁵-cyclopentadienyl-η²-propenealkylnickel complexes 4–9 (alkyl  CH₃, CD₃, CH₂SiMe₃, CH₂ CH₃, CH₂ CH₂ CH₃ and CH(CH₃)₂) have been prepared by treating nickelocene (1) with the appropriate organomagnesium halides 2a–2f and propene at ?20 to ?10°C. Temperature dependent ¹H-NMR spectra result from rotation of the propene molecule around the nickel—olefin axis; in the case of 4 and 8, two rotamers a and b can be distinguished below ca. ?60°C. The decomposition pathways for 4 and 8 are discussed.     

Insights into the Retention Mechanism for Small Neutral Compounds on Silica-Based Phenyl Phases in Reversed-Phase Liquid Chromatography

The system constants of the solvation parameter model are used to prepare system maps for the retention of small neutral compounds on phenylhexylsiloxane- and pentafluorophenylpropylsiloxane-bonded superficially porous silica stationary phases (Kinetex Phenyl-Hexyl and Kinetex F5) for aqueous mobile phases containing 10–70% (v/v) methanol or acetonitrile. Electrostatic interactions (cation exchange) are important for the retention of weak bases for acetonitrile–water mobile phases, but virtually absent for the same compounds for methanol–water mobile phases. The selectivity of the Kinetex Phenyl-Hexyl stationary phase for small neutral compounds is similar to an octadecylsiloxane-bonded silica stationary phase with similar morphology Kinetex C-18 for both methanol–water and acetonitrile–water mobile phase compositions. The Kinetex Phenyl-Hexyl and XBridge Phenyl stationary phases with the same topology but different morphology are selectivity equivalent, confirming that solvation of the interphase region can be effective at dampening selectivity differences for modern stationary phases. Small selectivity differences observed for XTerra Phenyl (different morphology and topology) confirm previous reports that the length and type of space arm for phenylalkylsiloxane-bonded silica stationary phases can result in small changes in selectivity. The pentafluorophenylpropylsiloxane-bonded silica stationary phase (Kinetex F5) has similar separation properties to the phenylhexylsiloxane-bonded silica stationary phases, but is not selectivity equivalent. However, for method development purposes, the scope to vary separations from an octadecylsiloxane-bonded silica stationary phase (Kinetex C-18) to “phenyl phase” of the types studied here is limited for small neutral compounds. In addition, selectivity differences for the above stationary phases are enhanced by methanol–water and largely suppressed by acetonitrile–water mobile phases. For bases, larger selectivity differences are possible for the above stationary phases if electrostatic interactions are exploited, especially for acetonitrile-containing mobile phases.