A Semiparametric Estimator of the Crossing Point in the Two‐Sample Linear Shift Function: Application to Crossing Lifetime Distributions

Let X and Y be two random variables with continuous distribution functions F and G. Consider two independent observations X₁, … , X_m from F and Y₁, … , Y_n from G. Moreover, suppose there exists a unique x* such that F(x) > G(x) for x < x* and F(x) < G(x) for x > x* or vice versa. A semiparametric model with a linear shift function (Doksum, 1974) that is equivalent to a location‐scale model (Hsieh, 1995) will be assumed and an empirical process approach (Hsieh, 1995) is used to estimate the parameters of the shift function. Then, the estimated shift function is set to zero, and the solution is defined to be an estimate of the crossing‐point x*. An approximate confidence band of the linear shift function at the crossing‐point x* is also presented, which is inverted to yield an approximate confidence interval for the crossing‐point. Finally, the lifetime of guinea pigs in days observed in a treatment‐control experiment in Bjerkedal (1960) is used to demonstrate our procedure for estimating the crossing‐point. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

The convergence in distribution of some simple epidemics

A group of n susceptible individuals exposed to a contagious disease isconsidered. It is assumed that at each point in time one or more susceptible individuals can contract the disease. The progress of this simple batch epidemic is modeled by a stochastic process X_n(t), t[0, ∞), representing the number of infectiveindividuals at time t. In this paper our analysis is restricted to simple batch epidemics with transition rates given by [α²X_n(t){n −X_n(t) +X_n(0)}]^1/2, t[0, ∞), α(0, ∞). This class of simple batch epidemics generalizes a model used and motivated by McNeil (1972) to describe simple epidemic situations. It is shown for this class of simple batch epidemics, that X_n(t), with suitable standardization, converges in distribution as n→∞ to a normal random variable for all t(0, t₀), and t₀ is evaluated.     

Convergence to Normality of the Asymptotic Quasi‐Score Function on a Linear Model

The asymptotic quasi‐likelihood method is considered for the model y_t = f_t(θ) + M_t, t = 0,1, …,T where f_tθ) is a linear predictable process of the parameter of interest θ, M_t is a martingale difference, and the nature of E(M_t² | ℱ_t–1) is unknown. This paper is concerned with the limiting distribution of the asymptotic quasi‐score function of such a model. Confidence intervals and hypothesis testing of θ is derived from the limiting distribution. Comparison is made between the estimates obtained through this method and those obtained through the least squares method.     

Optical effects on the surrounding structure during quantitative analysis using indocyanine green videoangiography: A phantom vessel study

Various reports have been published regarding quantitative evaluations of intraoperative fluorescent intensity studies using indocyanine green (ICG) with videoangiography (VAG). The effects of scattering and point‐spread functions (PSF) on quantitative ICG‐VAG evaluations have not been investigated. Clinically, when ICG is administered through the peripheral vein, it reaches the tissue intra‐arterially. To achieve more reliable intraoperative quantitative intensity evaluations, we examined the impact of high‐intensity structures on close areas. The study was conducted using a phantom model and surgical fluorescent microscope. A region of interest (ROI) was created for the vessel model and another ROI was created within 3 cm of that. With an ROI of 6.8 mm in the vessel phantom model, 10% intensity was confirmed, even though there was no fluorescent structure. Intensity decreased gradually as the ROI moved further from the vessel model. Our study results suggest that the presence of a high‐intensity structure and the size of the ROI may affect quantitative intensity evaluations using ICG‐VAG. Results of linear regression analysis indicate that the relationship of intensity (Y) and distance (X) is as follows: Y(real/A) = 29 Exp(?0.062X) + 164.3 Exp(?1.81X). The optical effect should be considered when performing an intraoperative intensity study with a surgical microscope.      

Stochastic prey-predator relationships: A random differential equation approach

A stochastic model describing two interacting populations is considered. The model involves a random differential equation of the form dX/dt=A(t)X+Y(t) where the random matrixA and vectorY represent the interactions and growth rates respectively andX is a (random) vector the components of which are the logarithms of the population's sizes. An expression for the solution of the above equation is obtained whence its statistical properties can be determined. Alternatively, a method based on Liouville's theorem is used to obtain the probability distribution of the solution. Application of both methods to simple cases indicates that the random solution is asymptotically stable in the mean even when the solution to the associated deterministic equation is not, viz. in the absence of self interactions.     

The mathematical theory of group selection. I. Full solution of a nonlinear Levins E = E(x) model

The first complete overtime solution is obtained for a group selection model of Levins E = E(x) type with recolonization but no other gene flow between islands. Assuming a subdivided population at carrying capacity, the model describes selection at a biallelic locus (A, a) where a is opposed by Mendelian selection but is favored by a lower rate of extinction of demes having high a frequency. By contrast to the linear diffusion equations encountered in classical mathematical genetics, the PDE governing the dynamics is now nonlinear in the metapopulation gene frequency distribution φ(x, t); furthermore, the initial conditions now heavily influence the equilibrium distribution φ_∞(x). A fully explicit formula (20) expressing this dependence is derived. The results indicate that a fixation is never reached, but (A, a) polymorphism in the metapopulation will result if , where s 1 parametrizes the strength of Mendelian selection, E(x) is the Levins extinction operator, h (typically in the open interval (0, 1)) is the dominance of a, and B is a parameter measuring the flatness of the initial distribution f(x) in the x → 1 limit.     

Asymptotic line-of-descent distributions

Asymptotic distributions are derived for the number of non-mutant ancestors, at time t in the past, of a sample of n from a neutral infinite alleles model. Either the number of non-mutant ancestors L _n (t) has a normal distribution or n-L_n(t) has a Poisson distribution as n , t 0.     

Factors affecting estimates of size at age and growth in grey triggerfish Balistes capriscus from the northern Gulf of Mexico

Growth zones in dorsal spines of grey triggerfish Balistes capriscus from the northern Gulf of Mexico were utilized to estimate growth and examine factors that may affect estimates of size at age. Age was estimated from dorsal‐spine sections of 4687 individuals sampled from U.S. waters during 2003–2013, including both fishery‐independent (n = 1312) and fishery‐dependent (n = 3375) samples. Ninety‐six per cent (n = 4498) of these sections were deemed suitable for ageing; average per cent error between two independent readers was 10·8%. Fork length (L_F) ranged from 65 to 697 mm and age estimates from 0 to 14 years. Both sex and sample source (fishery‐independent v. recreational) significantly affected estimated size at age for 2–6 year‐old fish. Data were pooled between sources to fit sex‐specific von Bertalanffy growth functions. Results for the female model were L_∞ = 387 mm L_F, k = 0·52 year^?1 and t₀ = 0·01 year, while for males L_∞ = 405 mm L_F, k = 0·55 year^?1 and t₀ = 0·02 year. These results were significantly different between sexes and indicate clear sexual dimorphism. Thus, growth should be modelled separately by sex when examining population parameters or conducting stock assessment modelling. The positive bias in estimates of size at age computed for recreational v. fishery‐independent samples also has clear implications for stock assessment as growth functions computed with fishery‐dependent samples would tend to overestimate stock productivity.     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

Basic functions of variability of simple pre-planned movements

A model of a pre-planned single joint movements performed without feedback is considered. Modifications of this movement result from transformation of a trajectory pattern f(t) in space and time. The control system adjusts the movement to concrete external conditions specifying values of the transform parameters before the movement performance. The preplanned movement is considered to be simple one, if the transform can be approximated by an affine transform of the movement space and time. In this case, the trajectory of the movement is x(t) = Af(t/ + s) +p, were A and 1/ are space and time scales, s and p are translations. The variability of movements is described by time profiles of variances and covariances of the trajectory x(t), velocity v(t), and acceleration a(t). It is assumed that the variability is defined only by parameters variations. From this assumption follows the main finding of this work: the variability time profiles can be expanded on a special system of basic functions corresponding to established movement parameters. Particularly, basic functions of variance time profiles, reflecting spatial and temporal scaling, are x ²(t) and t ² v ²(t) for trajectory, v ²(t) and (v(t) + t · a(t))² for velocity, and a ²(t) and (2a(t) +t · j(t))², where j(t) = d³ x(t)/dt ³, for acceleration. The variability of a model of a reaching movement was studied analytically. The model predicts certain peculiarities of the form of time profiles (e.g., the variance time profile of velocity is bi-modal, the one of acceleration is tri-modal, etc.). Experimental measurements confirmed predictions. Their consistence allows them to be considered invariant properties of reaching movement. A conclusion can be made, that reaching movement belongs to the type of simple preplanned movements. For a more complex movement, time profiles of variability are also measured and explained by the model of movements of this type. Thus, a movement can be attributed to the type of simple pre-planned ones by testing its variability.     

A New Functional Equation Analogous to CAUCHT-PEXIDER Functional Equation and its Application

The CAUCHY-PEXIDER functional equation H (x±y)=F(x) G(y) is generalized to the form H ((x^c±y^c)^1/c) = F(x) G(y), c≠0, assuming the function H(x) possesses a measurable majorant on a set of positive measure. The result is used to obtain a characterization of WEIBULL distribution. This functional equation is generalized to functions of vector variables.     

A model of photosystem II for the analysis of fast fluorescence rise in plant leaves

The polyphasic patterns of fluorescence induction rise in pea leaves in vivo and after the treatment with ionophores have been studied using a Plant Efficiency Analyzer. To analyze in detail photosystem II (PS II) electron transfer processes, an extended PS II model was applied, which included the sums of exponential functions to specify explicitly the light-driven formation of the transmembrane electric potential (ΔΨ(t)) as well as pH in the lumen (pH_L(t)) and stroma (pH_S(t)). PS II model parameters and numerical coefficients in ΔΨ(t), pH_L(t), and pH_S(t) were evaluated to fit fluorescence induction data for different experimental conditions: leaf in vivo or after ionophore treatment at low or high light intensity. The model imitated changes in the pattern of fluorescence induction rise due to the elimination of transmembrane potential in the presence of ionophores, when ΔΨ = 0 and pH_L(t), pH_S(t) changed to small extent relative to control values in vivo, with maximum ΔΨ(t) ∼ 90 mV and ΔΨ(t) ∼ 40 mV for the stationary state at ΔpH ≅ 1.8. As the light intensity was increased from 300 to 1200 μmol m⁻² s⁻¹, the heat dissipation rate constants increased threefold for nonradiative recombination of P680⁺Phe⁻ and by ∼30% for P680⁺Q_A⁻. The parameters ΔΨ, pH_S and pH_L were analyzed as factors of PS II redox state populations and fluorescence yield. The kinetic mechanism of fluorescence quenching is discussed, which is related with light-induced lumen acidification, when ⁺Q_A⁻ and P680⁺ recombination probability increases to regulate the Q_A reduction.     

Zum zeitlichen Verlauf von Substratkonzentrationsprofilen in einer Enzymträgerkugel

In this paper the time evolution of substrate concentration profiles in spherical particles with homogeneous enzyme distribution is investigated. It turns out that the concentration S ? S(x, t) is monotone increasing in the time variable t against the uniquely defined steady state ? ? ?(x). Explizit formulas are derived for the estimation of the difference between the steady state ? and the presteady state S, that hold for all t ≧ 0, i.c., it is not only determined the order of magnitude of the convergence for sufficiently large t.     

Effects of experimental warming on stomatal traits in leaves of maize (Zea may L.)

We examined the warming effects on the stomatal frequency, stomatal aperture size and shape, and their spatial distribution pattern of maize (Zea may L.) leaves using a light microscope, an electron scanning microscope, and geostatistic techniques. A field manipulative experiment was conducted to elevate canopy temperature by 2.08°C, on average. We found that experimental warming had little effect on stomatal density, but significantly increased stomatal index due to the reduction in the number of epidermal cells under the warming treatment. Warming also significantly decreased stomatal aperture length and increased stomatal aperture width. As a result, warming significantly increased the average stomatal aperture area and stomatal aperture circumference. In addition, warming dramatically changed the stomatal spatial distribution pattern with a substantial increase in the average nearest neighbor distance between stomata on both adaxial and abaxial surfaces. The spatial distribution pattern of stomata was scale dependent with regular patterns at small scales and random patterns at larger scales on both leaf surfaces. Warming caused the stomatal distribution to become more regular on both leaf surfaces with smaller L(t) values (Ripley's K‐function, L(t) is an expectation of zero for any value of t) in the warming plots than the control plots.     

On the difference between gauss-markov and least squares estimates

For the usual full rank univariate least squares regression model y = XB + e, E(e) = 0, E(ee) = A, the equality of the estimates occurs when B-B^* = (XA^?1X)^?1XA^-1y-(XX)^?1Xy = 0. A necessary and sufficient condition for this equality is that A has some N - k + 1 roots equal where N is the rank of A and k is the rank of X.     

Applying Canonical Analysis to a Kind of Generalized Regression

This paper is concerned with the generalized model E(φ(Y)! X) =φ(X) involving the transformations on both the predictor vector X and the response variable Y. For this purpose, Taylor expansions and canonical analysis are applied. For optimizing the expansions, it is shown by a simulation study that not only prediction error, the combination of model error and noise error, is an important index, but the distribution of the residuals and the t-values of the coefficients also must be considered. Furthermore, the results of penicillin titrition show that the practical situations often need to be considered in selecting an appropriate model for a real-life problem.     

Age and growth of Carcharhinus leucas in the northern Gulf of Mexico: incorporating variability in size at birth

Age and growth rates of bull shark Carcharhinus leucas[n = 255; 555–2230 mm fork length (L_F)] from the northern Gulf of Mexico were estimated from ring counts on vertebral sections collected from fishery‐dependent and ‐independent surveys. Two growth models were fitted to observed data: the von Bertalanffy growth model (VBGM) with t₀ as the third parameter and a modified version of the VBGM using a fixed size‐at‐birth intercept as the third parameter. To address the variability in size‐at‐birth, a Monte Carlo simulation was incorporated into the size‐at‐birth intercept. The sex‐specific growth models were not significantly different, allowing a sexes combined model to be generated. The traditional VBGM predicted a theoretical maximum size (L_∞) of 3007·1 mm L_F, a growth coefficient (K) of 0·042 year^?1 and a theoretical age at zero length (t₀) of –6·844 years. The modified VBGM with a fixed size‐at‐birth intercept of 565 mm L_F predicted an L_∞ of 2289·2 mm L_F and a K value of 0·089 year^?1. When comparing model estimates to previously published information, the traditional VBGM predicted a significantly lower theoretical maximum size and a higher growth coefficient than those produced using data collected during the 1980s. Overall, results obtained using the VBGM with a fixed size‐at‐birth produced more biologically realistic parameters than that of the VBGM with t₀. The Monte‐Carlo simulation incorporating variability in size‐at‐birth produced similar results to the VBGM using a fixed size‐at‐birth. This study provides the first attempt to incorporate variability at size‐at‐birth and provide measurements of variability around the individual parameter estimates for an elasmobranch.     

Validated age,growth and maturity of the bonnethead Sphyrna tiburo in the western North Atlantic Ocean

The age, growth and maturity of bonnetheads Sphyrna tiburo inhabiting the estuarine and coastal waters of the western North Atlantic Ocean (WNA) from Onslow Bay, North Carolina, south to West Palm Beach, Florida, were examined. Vertebrae were collected and aged from 329 females and 217 males ranging in size from 262 to 1043 mm and 245 to 825 mm fork length, L_F, respectively. Sex‐specific von Bertalanffy growth curves were fitted to length‐at‐age data. Female von Bertalanffy parameters were L_∞ = 1036 mm L_F, k = 0·18, t₀ = ?1·64 and L₀ = 272 mm L_F. Males reached a smaller theoretical asymptotic length and had a higher growth coefficient (L_∞ = 782 mm L_F, k = 0·29, t₀ = ?1·43 and L₀ = 266 mm L_F). Maximum observed age was 17·9 years for females and 16·0 years for males. Annual deposition of growth increments was verified by marginal increment analysis and validated for age classes 2·5+ to 10·5+ years through recapture of 13 oxytetracycline‐injected specimens at liberty in the wild for 1–4 years. Length (L_F50) and age (A₅₀) at 50% maturity were 819 mm and 6·7 years for females, and 618 mm and 3·9 years for males. Both female and male S. tiburo in the WNA had a significantly higher maximum observed age, L_F50, A₅₀ and L_∞, and a significantly lower k and estimated L₀ than evident in the Gulf of Mexico (GOM). These significant differences in life‐history parameters, as well as evidence from tagging and genetic studies, suggest that S. tiburo in the WNA and GOM should be considered separate stocks.