The asymptotic distributions of incomplete U-statistics

Summary An incomplete U-statistic is obtained by sampling the terms of an U-statistic. This paper derives the asymptotic distribution (if the variance is finite). Depending on the number of sampled terms, the resulting distribution is either the same as for the U-statistic, a normal distribution, or something intermediate. Also the case of a non-random sampling of the terms is treated. As an example, a non-parametric test of the independence of two circular random variables is studied. The results are generalized to generalized U-statistics.     

The Cut Metric,Random Graphs,and Branching Processes

In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model introduced by the present authors in Random Struct. Algorithms 31:3–122 (2007), as well as related results of Bollobás, Borgs, Chayes and Riordan (Ann. Probab. 38:150–183, 2010), all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering (Random Struct. Algorithms, 2010, to appear).     

Sesqui-type branching processes

We consider branching processes consisting of particles (individuals) of two types (type $L$ and type $S$) in which only particles of type $L$ have offspring, proving estimates for the survival probability and the (tail of) the distribution of the total number of particles. Such processes are in some sense closer to single- than to multi-type branching processes. Nonetheless, the second, barren, type complicates the analysis significantly. The results proved here (about point and survival probabilities) are a key ingredient in the analysis of bounded-size Achlioptas processes in a recent paper by the last two authors.     

Precise Logarithmic Asymptotics for the Right Tails of Some Limit Random Variables for Random Trees

For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant’s precise value remains unknown. Research supported by NSF grants DMS-0104167 and DMS-0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

A new approach to the giant component problem

We study the largest component of a random (multi)graph on n vertices with a given degree sequence. We let n → ∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability all the components are small, and other conditions that imply that with high probability there is a giant component and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the results by Molloy and Reed on the size of the largest component in a random graph with a given degree sequence. We further obtain a new sharp result for the giant component just above the threshold, generalizing the case of G(n,p) with np = 1 + ω(n)n^?1/3, where ω(n) → ∞ arbitrarily slowly. Our method is based on the properties of empirical distributions of independent random variables, and leads to simple proofs. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Simultaneous analysis of vocal fold vibration and transglottal airflow: exploring a new experimental setup

The purpose of this study was to develop an analysis system for studying the relationship between vocal fold vibration and the associated transglottal airflow. Recordings of airflow, electroglottography (EGG), oral air pressure, and acoustic signals were performed simultaneously with high-speed imaging at a rate of approximately 1900 frames/s. Inverse filtered airflow is compared with the simultaneous glottal area extracted from the high-speed image sequence. The accuracy of the synchronization between the camera images and the foot pedal synchronization pulse was examined, showing that potential synchronization errors increase with time distance to the synchronization pulse. Therefore, analysis was limited to material near the synchronization pulse. Results corroborate previous predictions that air flow lags behind area, but also they reveal that relationships between these two entities may be complex and apparently varying with phonation mode.     

Comparing closed quotient in children singers' voices as measured by high-speed-imaging, electroglottography, and inverse filtering

The closed quotient, i.e., the ratio between the closed phase and the period, is commonly studied in voice research. However, the term may refer to measures derived from different methods, such as inverse filtering, electroglottography or high-speed digital imaging (HSDI). This investigation compares closed quotient data measured by these three methods in two boy singers. Each singer produced sustained tones on two different pitches and a glissando. Audio, electroglottographic signal (EGG), and HSDI were recorded simultaneously. The audio signal was inverse filtered by means of the decap program; the closed phase was defined as the flat minimum portion of the flow glottogram. Glottal area was automatically measured in the high speed images by the built-in camera software, and the closed phase was defined as the flat minimum portion of the area-signal. The EGG-signal was analyzed in four different ways using the matlab open quotient interface. The closed quotient data taken from the EGG were found to be considerably higher than those obtained from inverse filtering. Also, substantial differences were found between the closed quotient derived from HSDI and those derived from inverse filtering. The findings illustrate the importance of distinguishing between these quotients.     

Novel rapid liquid chromatography tandem masspectrometry method for vemurafenib and metabolites in human plasma,including metabolite concentrations at steady state

A novel, rapid and sensitive liquid chromatography tandem–mass spectrometry method for quantification of vemurafenib in human plasma, that also for the first time allows for metabolite semi‐quantification, was developed and validated to support clinical trials and therapeutic drug monitoring. Vemurafenib was analysed by precipitation with methanol followed by a 1.9 min isocratic liquid chromatography tandem masspectrometry analysis using an Acquity BEH C₁₈ column with methanol and formic acid using isotope labelled internal standards. Analytes were detected in multireaction monitoring mode on a Xevo TQ. Semi‐quantification of vemurafenib metabolites was performed using the same analytical system and sample preparation with gradient elution. The vemurafenib method was successfully validated in the range 0.5–100 μg/mL according to international guidelines. The metabolite method was partially validated owing to the lack of commercially available reference materials. For the first time concentration levels at steady state for melanoma patients treated with vemurafenib is presented. The low abundance of vemurafenib metabolites suggests that they lack clinical significance. Copyright © 2016 John Wiley & Sons, Ltd.     

Multivariate design

In multivariate data analysis such as principal components analysis (PCA) and projections to latent structures (PLS), it is essential that the training set systems (objects) are selected to provide data with substantial information for model parametrization, and to represent properly any future situations where the multilvariate model is used for predictions. In the framework of multivariate projections (PCA, SIMCA and PLS), elementary concepts of statistical design (fractional factorials and composite designs) can be used with the latent variables (PC or PLS scores) as design variables. The plan of action thus becomes: (1) problem formulation (specify aim and model, make a conceptual division of the investigated system into subsystems); (2) collection of multivariate data for each type of subsystems; (3) estimation of the practical dimensionality of the data for each type of subsystems by PC or PLS analysis; (4) use of the PC or PLS scores (t) as design variables in the combination of subsystems to systems in the training set; (5) measurement of responses (Y); (6) analysis of data by PCA or PLS; (7) interpretation of results with possible feedback to steps 1, 2 or 3. The procedures are illustrated by two problems: a structure/activity relationship for a family of peptides, and optimization of an organic synthesis with respect to system variables (solvent, substrate, co-reactant_) and process variables (temperature, reactant concentrations).