On the convergence of Newton's method when estimating higher dimensional parameters

In this paper, we consider the estimation of a parameter of interest where the estimator is one of the possibly several solutions of a set of nonlinear empirical equations. Since Newton's method is often used in such a setting to obtain a solution, it is important to know whether the so obtained iteration converges to the locally unique consistent root to the aforementioned parameter of interest. Under some conditions, we show that this is eventually the case when starting the iteration from within a ball about the true parameter whose size does not depend on n. Any preliminary almost surely consistent estimate will eventually lie in such a ball and therefore provides a suitable starting point for large enough n. As examples, we will apply our results in the context of M-estimates, kernel density estimates, as well as minimum distance estimates.     

Translational energy release and stereochemistry of steroids. VIII-The mechanism of the elimination of water from metastable ions in epimeric 3-hydroxy steroids of the 5α-series

The mechanism of water elimination from metastable molecular, [M ? CH₃˙]⁺ and [M ? ring D]⁺˙ ions of epimeric 3-hydroxy steroids of the 5α-series has been elucidated. Deuterium labelling, the measurement of the translational energy released during the loss of water, and collision-induced decomposition mass-analysed kinetic energy spectrometry were the techniques used. It was found that the mechanisms of water loss from metastable M⁺˙ and [M ? ring D]⁺˙ ions is different from that from [M ? CH₃˙]⁺ ions.     

Determination of enzyme kinetic parameters of cyclic CMP-specific phosphodiesterase by quantitative fast atom bombardment tandem mass spectrometry

The determination of cytidine 3′,5′-cyclic monophosphate-specific phosphodiesterase activity by means of fast-atom bombardment (FAB) mass Spectrometry with mass-analysed ion kinetic energy (MIKE) spectrum scanning is described. Initial efforts to determine the activity of the enzyme by this method were unsuccessful owing to the obfuscation of sample-related peaks by peaks emanating from the incubation buffer and cation adducts; dilution of buffer and a desalting procedure overcame these difficulties. In the resulting positive-ion FAB mass spectra, characteristic peaks of the enzyme substrate and product could be readily identified and the protonated molecular ions selected for MIKE scanning. By spiking enzyme incubates with known amounts of substrate and product, and measuring peak heights in the MIKE spectra of both spiked and unspiked samples, the substrate/product ratio at the end of a series of phosphodiesterase incubations was determined. From the data obtained, the K_m and V_max of the phosphodiesterase were calculated as 6.08 mM and 11 μmol min^?1 mg^?1, respectively, showing good agreement with the analogous values of 8.06 mM and 5.8 μmol^?1 min^?1 mg^?1 obtained by radioactive assay.     

Loss of H2O from M + t-Butyl Complex Ions of Benzyl Alcohol in Isobutane Chemical Ionization Mass Spectrometry

In isobutane chemical ionization mass spectrometry benzyl alcohol exhibits ions at m/z 147 (‘M + 39’) that arise by a loss of H₂O from [M + C₄H₉]⁺, i.e. ‘M + 57’ complex ions. Electrophilic aromatic substitution of a proton at an ortho-position of neutral C₆H₅CH₂OH with [t-C₄H₉]⁺ and, alternatively, nucleophilic substitution of H₂O at the benzylic carbon in \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm C_6 H_5 CH_2}\mathop {\rm O}\limits^+ {\rm H}_2 $\end{document} with CH₂?C (CH₃)₂ are discussed as possible pathways. Evidence in favor of the latter is derived from the analysis of C₆D₅CH₂OH and C₆H₅CD₂OH for the origin of the H-atoms lost in H₂O. The inferred ion structure of m/z 147 is verified by mass-analyzed ion kinetic energy (MIKE.) measurements of its collision-activated (CA.) decomposition. MIKE./CA. spectra of mass-selected m/z 147 ions, once generated by (CI(i-C₄H₁₀) from benzyl alcohol and, once, from 2-methyl-4-phenyl-2-butanol match closely and, thus, reflect identical ion structures. With reference to the simple genesis of this ion from the latter precursor, the structure in question can be concluded to be \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm C_6 H_5 CH_2 CH_2}\mathop {\rm C}\limits^+ ({\rm CH}_3)_2 $\end{document} .     

Investigation of uremic analytes in hemodialysate and their structural elucidation from accurate mass maps generated by a multi‐dimensional liquid chromatography/mass spectrometry approach

Historically, structural elucidation of unknown analytes by mass spectrometry alone has involved tandem mass spectrometry experiments using electron ionization. Most target molecules for bioanalysis in the metabolome are unsuitable for detection by this previous methodology. Recent publications have used high‐resolution accurate mass analysis using an LTQ‐Orbitrap with the more modern approach of electrospray ionization to identify new metabolites of known metabolic pathways. We have investigated the use of this methodology to build accurate mass fragmentation maps for the structural elucidation of unknown compounds. This has included the development and validation of a novel multi‐dimensional LC/MS/MS methodology to identify known uremic analytes in a clinical hemodialysate sample. Good inter‐ and intra‐day reproducibility of both chromatographic stages with a high degree of mass accuracy and precision was achieved with the multi‐dimensional liquid chromatography/tandem mass spectrometry (LC/MS/MS) system. Fragmentation maps were generated most successfully using collision‐induced dissociation (CID) as, unlike high‐energy CID (HCD), ions formed by this technique could be fragmented further. Structural elucidation is more challenging for large analytes >270 Da and distinguishing between isomers where their initial fragmentation pattern is insufficiently different. For small molecules (<200 Da), where fragmentation data may be obtained without loss of signal intensity, complete structures can be proposed from just the accurate mass fragmentation data. This methodology has led to the discovery of a selection of known uremic analytes and two completely novel moieties with chemical structural assignments made. Copyright © 2009 John Wiley & Sons, Ltd.     

Accurate mass measurement: Terminology and treatment of data

High-resolution mass spectrometry has become ever more accessible with improvements in instrumentation, such as modern FT-ICR and Orbitrap mass spectrometers. This has resulted in an increase in the number of articles submitted for publication quoting accurate mass data. There is a plethora of terms related to accurate mass analysis that are in current usage, many employed incorrectly or inconsistently. This article is based on a set of notes prepared by the authors for research students and staff in our laboratories as a guide to the correct terminology and basic statistical procedures to apply in relation to mass measurement, particularly for accurate mass measurement. It elaborates on the editorial by Gross in 1994 regarding the use of accurate masses for structure confirmation [1]. We have presented and defined the main terms in use with reference to the International Union of Pure and Applied Chemistry (IUPAC) recommendations for nomenclature and symbolism for mass spectrometry. The correct use of statistics and treatment of data is illustrated as a guide to new and existing mass spectrometry users with a series of examples as well as statistical methods to compare different experimental methods and datasets.