Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem

We consider the statistical non-linear inverse problem of recovering the absorption term f>0 in the heat equation with given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u_f. We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.

Continuous series of symmetric peak profile functions determined by standard deviation and kurtosis

A mathematical system for modeling the effects of symmetrized instrumental aberrations has been developed. The system is composed of the truncated Gaussian, sheared Gaussian, and Rosin-Rammler-type functions. The shape of the function can uniquely be determined by the standard deviation and kurtosis. A practical method to evaluate the convolution with the Lorentzian function and results of application to the analysis of experimental powder diffraction data are briefly described.

The supernova remnant populations of the galaxies NGC 45, NGC 55, NGC 1313, NGC 7793: luminosity and excitation functions

ABSTRACT We present a systematic study of the supernova remnant (SNR) populations in the nearby galaxies NGC 45, NGC 55, NGC 1313, and NGC 7793 based on deep H $\rm {\alpha }$ and [S ii] imaging. We find 42 candidate and 51 possible candidate SNRs based on the [S ii]/H $\rm {\alpha }$>0.4 criterion, 81 of which are new identifications. We derive the H $\rm {\alpha }$ and the joint [S ii]–H $\rm {\alpha }$ luminosity functions after accounting for incompleteness effects. We find that the H $\rm {\alpha }$ luminosity function of the overall sample is described with a skewed Gaussian with a mean equal to $\rm \log (L_{H\alpha }/10^{36}\, erg\, s^{-1})=0.07$ and $\rm \sigma (\log (L_{H\alpha }/10^{36}\, erg\, s^{-1}))=0.58$. The joint [S ii]–H $\rm {\alpha }$ function is parametrized by a skewed Gaussian along the log([S ii]$\rm /10^{36}\, erg\, s^{-1}) = 0.88 \times \log (L_{H\alpha }/10^{36}\, erg\, s^{-1}) - 0.06$ line and a truncated Gaussian with $\rm \mu (\log (L_{[S\, II]}/10^{36})) = 0.024$ and $\rm \sigma (\log (L_{[S\, II]}/10^{36})) = 0.14$, on its vertical direction. We also define the excitation function as the number density of SNRs as a function of their [S ii]/H $\rm {\alpha }$ ratios. This function is represented by a truncated Gaussian with a mean at −0.014. We find a sub-linear [S ii]–H $\rm {\alpha }$ relation indicating lower excitation for the more luminous objects.

Modelling the patient accrual with truncated Gaussian mixture distribution for the accurate estimation of sample size in survival trials

In survival trials with fixed trial length, the patient accrual rate has a significant impact on the sample size estimation or equivalently, on the power of trials. A larger sample size is required for the staggered patient entry. During enrollment, the patient accrual rate changes with the recruitment publicity effect, disease incidence and many other factors and fluctuations of the accrual rate occur frequently. However, the existing accrual models are either over-simplified for the constant rate assumption or complicated in calculation for the subdivision of the accrual period. A more flexible accrual model is required to represent the fluctuant patient accrual rate for accurate sample size estimation. In this paper, inspired by the flexibility of the Gaussian mixture distribution in approximating continuous densities, we propose the truncated Gaussian mixture distribution accrual model to represent different variations of accrual rate by different parameter configurations. The sample size calculation formula and the parameter setting of the proposed accrual model are discussed further.

A simple method for amateur investors to analyze covered call options

We describe a simple method which amateur investors can use to analyze covered calls. The most basic version is based on the formula for the expectation of a truncated Gaussian distribution, and it can be generalized to accommodate other assumptions. This approach might be especially considered during a time of market overvaluation, such as the present. During such times, investors should shift their preferences toward writing deep-in-the-money covered calls, which provide a greater margin of safety while monetizing the (probably optimistic) expectations of other market participants regarding future returns.