Statistical Parameter Estimation Research Articles

Maximum likelihood estimation of mixed C-vines with application to exchange rates

Multivariate copulas are commonly used in economics, finance and risk management. They allow for very flexible dependency structures, even though they are applied to transformed financial data after marginal time dependencies are removed. This is necessary to facilitate statistical parameter estimation. In this paper we consider a very flexible class of mixed C-vines, which allows the variables to be ordered according to their influence. Vines are built from bivariate copulas only and the term ‘mixed’ refers to allowing the pair-copula family to be chosen individually for each term. In addition there are many C-vine structure specifications possible and therefore we propose a novel data driven sequential selection procedure, which selects both the C-vine structure and its attached pair-copula families with parameters. After the model selection maximum likelihood (ML) estimation of the parameters is facilitated using the sequential estimates as starting values. An extensive simulation study shows a satisfactory performance of ML estimates in small samples. Finally an application involving US-exchange rates demonstrates the need for mixed C-vine models.

Flaw strength distributions and statistical parameters for ceramic fibers: The normal distribution

The present paper investigates large sets of ceramic fibre failure strengths (500 to 1000 data) produced using tensile tests on tows that contained either 500 or 1000 filaments. The probability density function was determined through acoustic emission monitoring which allowed detection and counting of filament fractures. The statistical distribution of filament strengths was described using the normal distribution. The Weibull equation was then fitted to this normal distribution for estimation of statistical parameters. A perfect agreement between both distributions was obtained, and a quite negligible scatter in statistical parameters was observed, as opposed to the wide variability that is reported in the literature. Thus it was concluded that flaw strengths are distributed normally and that the statistical parameters that were derived are the true ones. In a second step, the conventional method of estimation of Weibull parameters was applied to these sets of data and, then, to subsets selected randomly. The influence of other factors involved in the conventional method of determination of statistical parameters is discussed. It is demonstrated that selection of specimens, sample size, and method of construction of so-called Weibull plots are responsible for statistical parameters variability.

The origin and orbit of the old, metal-rich, open cluster NGC 6791

NGC 6791 is a unique stellar system among Galactic open clusters being at the same time one of the oldest open clusters and the most metal rich. Combination of its properties is puzzling and poses question of its origin. One possible scenario is that the cluster formed close to the Galactic Center and later migrated outwards to its current location. In this work we study the cluster's orbit and investigate the possible migration processes which might have displaced NGC 6791 to its present-day position, under the assumption that it actually formed in the inner disk. To this aim we performed integrations of NGC 6791's orbit in a potential consistent with the main Milky Way parameters. In addition to analytical expressions for halo, bulge and disk, we also consider the effect of bar and spiral arm perturbations, which are expected to be very important for the disk dynamical evolution, especially inside the solar circle. Starting from state-of-the art initial conditions for NGC 6791, we calculate 1000 orbits back in time for about 1 Gyr turning on and off different non-axisymmetric components of the global potential. We then compare statistical estimates of the cluster's recent orbital parameters with the orbital parameters of 10^4 test-particles originating close to the Galactic Center (having initial galocentric radii in the range of 3-5 kpc) and undergoing radial migration during 8 Gyr of forward integration. We find that a model which incorporates a strong bar and spiral arm perturbations can indeed be responsible for the migration of NGC 6791 from the inner disk (galocentric radii of 3-5 kpc) to its present-day location. Such a model can provide orbital parameters which are close enough to the observed ones. However, the probability of this scenario as it results from our investigations is very low.

AD Model Builder: using automatic differentiation for statistical inference of highly parameterized complex nonlinear models

Many criteria for statistical parameter estimation, such as maximum likelihood, are formulated as a nonlinear optimization problem. Automatic Differentiation Model Builder (ADMB) is a programming framework based on automatic differentiation, aimed at highly nonlinear models with a large number of parameters. The benefits of using AD are computational efficiency and high numerical accuracy, both crucial in many practical problems. We describe the basic components and the underlying philosophy of ADMB, with an emphasis on functionality found in no other statistical software. One example of such a feature is the generic implementation of Laplace approximation of high-dimensional integrals for use in latent variable models. We also review the literature in which ADMB has been used, and discuss future development of ADMB as an open source project. Overall, the main advantages of ADMB are flexibility, speed, precision, stability and built-in methods to quantify uncertainty.

Statistical inference of in vivo properties of human DNA methyltransferases from double-stranded methylation patterns.

DNA methyltransferases establish methylation patterns in cells and transmit these patterns over cell generations, thereby influencing each cell's epigenetic states. Three primary DNA methyltransferases have been identified in mammals: DNMT1, DNMT3A and DNMT3B. Extensive in vitro studies have investigated key properties of these enzymes, namely their substrate specificity and processivity. Here we study these properties in vivo, by applying novel statistical analysis methods to double-stranded DNA methylation patterns collected using hairpin-bisulfite PCR. Our analysis fits a novel Hidden Markov Model (HMM) to the observed data, allowing for potential bisulfite conversion errors, and yields statistical estimates of parameters that quantify enzyme processivity and substrate specificity. We apply this model to methylation patterns established in vivo at three loci in humans: two densely methylated inactive X (Xi)-linked loci ( and ), and an autosomal locus (), where methylation densities are tissue-specific but moderate. We find strong evidence for a high level of processivity of DNMT1 at and , with the mean association tract length being a few hundred base pairs. Regardless of tissue types, methylation patterns at are dominated by DNMT1 maintenance events, similar to the two Xi-linked loci, but are insufficiently informative regarding processivity to draw any conclusions about processivity at that locus. At all three loci we find that DNMT1 shows a strong preference for adding methyl groups to hemi-methylated CpG sites over unmethylated sites. The data at all three loci also suggest low (possibly 0) association of the de novo methyltransferases, the DNMT3s, and are consequently uninformative about processivity or preference of these enzymes. We also extend our HMM to reanalyze published data on mouse DNMT1 activities in vitro. The results suggest shorter association tracts (and hence weaker processivity), and much longer non-association tracts than human DNMT1 in vivo.

Adaptive blinded sample size adjustment for comparing two normal means—a mostly Bayesian approach

Adaptive sample size redetermination (SSR) for clinical trials consists of examining early subsets of on-trial data to adjust prior estimates of statistical parameters and sample size requirements. Blinded SSR, in particular, while in use already, seems poised to proliferate even further because it obviates many logistical complications of unblinded methods and it generally introduces little or no statistical or operational bias. On the other hand, current blinded SSR methods offer little to no new information about the treatment effect (TE); the obvious resulting problem is that the TE estimate scientists might simply 'plug in' to the sample size formulae could be severely wrong. This paper proposes a blinded SSR method that formally synthesizes sample data with prior knowledge about the TE and the within-treatment variance. It evaluates the method in terms of the type 1 error rate, the bias of the estimated TE, and the average deviation from the targeted power. The method is shown to reduce this average deviation, in comparison with another established method, over a range of situations. The paper illustrates the use of the proposed method with an example.

Likelihood-Based Approach to Multidisciplinary Analysis Under Uncertainty

This paper proposes a new methodology for uncertainty quantification in systems that require multidisciplinary iterative analysis between two or more coupled component models. This methodology is based on computing the probability of satisfying the interdisciplinary compatibility equations, conditioned on specific values of the coupling (or feedback) variables, and this information is used to estimate the probability distributions of the coupling variables. The estimation of the coupling variables is analogous to likelihood-based parameter estimation in statistics and thus leads to the proposed likelihood approach for multidisciplinary analysis (LAMDA). Using the distributions of the feedback variables, the coupling can be removed in any one direction without loss of generality, while still preserving the mathematical relationship between the coupling variables. The calculation of the probability distributions of the coupling variables is theoretically exact and does not require a fully coupled system analysis. The proposed method is illustrated using a mathematical example and an aerospace system application—a fire detection satellite.

Improving data structures in systems operation of air navigation facilities

Application procedures consistent statistical parameter estimation in engineering practice as an example of the process extension service (resource) air navigation facilities. A comparative analysis of the three variants of data structures in systems operation

Static and dynamic flexural strength of 99.5% alumina: Relation to porosity

Advanced ceramic materials are widely implemented in a variety of cutting edge engineering applications. Their dynamic flexural strength is an important property under impact loading conditions. Therefore, this work reports a systematic comparison of static and dynamic flexural strength results for 99.5% commercial alumina, obtained using a recently reported adaptation of the 1-point impact experimental technique. Specimens of the same size and surface roughness conditions were used in this study, in a large sample batch, thus allowing for estimation of basic statistical parameters for the evaluated material. The main observation is that the flexural strength of this material is distinctly rate-sensitive, a point that was explained by fracture surface porosity levels corresponding to the different loading regimes. During static loading the crack apparently selects a weaker path, by connecting between pores, with a resulting lower strength values and rougher fracture surface. Under dynamic loading conditions, the fracture surface is smoother and the crack path connects only the larger pores, accompanied by an increase in dynamic flexural strength.

Limit Theorems for Semi-Markov Queues and Their Applications

We begin by definition of semi-Markov flow and discussion of its properties. Asymptotic behavior of multi-server and single-server queueing systems is studied under assumption of time-compression or service time growth. The results obtained are used for calculation of large systems reliability. Statistical estimates of parameters involved are also provided.

Machine Health Monitoring based on Stationary Wavelet Transform and Fourth-Order Cumulants

Early stage faults detection for machine health monitoring demands high level of fault classification accuracy under poor signal-to-noise ratio (SNR). Vibration signal which is used for signature matching in case of abnormality detection and diagnosis, requires robust tools such as wavelet transform (WT) for time-frequency analysis. WT is specifically used to deal with non- stationary signals. In order to guarantee improved performance under poor SNR, this paper proposes a scheme for feature extraction based on fourth-order cumulant and stationary wavelet transform (FoCSWT). Higher order cumulants have the tendency to mitigate the impact of Gaussian noise. Fourth-order cumulant corresponds to the “peakedness” of the random distribution and the fault detection capability quantifies it as the most dominant cumulant among higher order statistics. Stationary wavelet transform is used to avoid down-sampling on the vibration data prior to feature extraction which gives better estimation of statistical parameters of the data distribution and gives performance enhancement in terms of fault classification accuracy. Simulation studies show that FoCSWT outperforms the existing techniques in terms of fault detection accuracies under poor SNR.

Information-Theoretic Lower Bounds on the Oracle Complexity of Stochastic Convex Optimization

Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardn4516420ess of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We introduce a new notion of discrepancy between functions, and use it to reduce problems of stochastic convex optimization to statistical parameter estimation, which can be lower bounded using information-theoretic methods. Using this approach, we improve upon known results and obtain tight minimax complexity estimates for various function classes.

Performance Bound of Position Estimation in Image Matching

Position estimation in image matching is a fundamental step in computer vision and image processing. To deal with the problem of performance prediction, we formulate it under statistical parameter estimation aspect. The lower bound of position estimation variance is obtained based on Cramer-Rao lower bound (CRLB) theory. This paper analyses the impact of noise to 1-D signal matching, derives the lower bound of variance, and then extends it to 2-D image matching. Furthermore, we derive numerical expression that can be computed from observed data. Finally, we use Monte Carlo simulation method to verify the derived analytical expressions. Experimental results show that the derived CRLB is tight to simulation estimated variance. The CRLB can characterize the performance bound of position estimation in image matching.

Direct Observation of Ferrielectricity at Ferroelastic Domain Boundaries in CaTiO3 by Electron Microscopy

High-resolution aberration-corrected transmission electron microscopy aided by statistical parameter estimation theory is used to quantify localized displacements at a (110) twin boundary in orthorhombic CaTiO(3). The displacements are 3-6 pm for the Ti atoms and confined to a thin layer. This is the first direct observation of the generation of ferroelectricity by interfaces inside this material which opens the door for domain boundary engineering.

Model-based electron microscopy: From images toward precise numbers for unknown structure parameters

Statistical parameter estimation theory is proposed as a method to quantify electron microscopy images. It aims at obtaining precise and accurate values for the unknown structure parameters including, for example, atomic column positions and types. In this theory, observations are purely considered as data planes, from which structure parameters have to be determined using a parametric model describing the images. The method enables us to measure positions of atomic columns with a precision of the order of a few picometers even though the resolution of the electron microscope is one or two orders of magnitude larger. Moreover, small differences in averaged atomic number, which cannot be distinguished visually, can be quantified using high-angle annular dark field scanning transmission electron microscopy images. Finally, it is shown how to optimize the experimental design so as to attain the highest precision. As an example, the optimization of the probe size for nanoparticle radius measurements is considered. It is also shown how to quantitatively balance signal-to-noise ratio and resolution by adjusting the probe size.

Statistical Simulation to Estimate Uncertain Behavioral Parameters of Hybrid Energy-Economy Models

In energy-economy modeling, new hybrid models attempt to combine the technological explicitness of bottom-up models with the macroeconomic feedbacks and statistically estimated behavioral parameters of top-down models. However, statistical estimation of behavioral parameters (portraying firm and household technology choices) with such models is challenged by the number of uncertain variables and the lack of historical data on technologies in terms of capital costs, operating costs, and market shares. Multiple combinations of parameter values might equally explain past technology choices. This paper reports on the application of a Bayesian statistical simulation approach for estimating the most likely values for these key behavioral parameters in order to best explain past technology choices and then simulate policies to influence future technology choices. The method included (1) data collection of key technology market shares, capital costs, and operating costs over the past; (2) backcasting a hybrid energy-economy model over a historical time period; and (3) the application of Markov chain Monte Carlo statistical simulation using the Metropolis–Hastings algorithm as a tool for estimating distributions for key parameters in the model. The results provide a means of indicating the uncertainty bounds around key behavioral parameters when generating forecasts of the effect of certain policies. However, the results also indicate that this approach may have limited applicability, given that future available technologies may differ substantially from past technologies and that it is difficult to separate the effects of parameter uncertainty from model structure uncertainty.

Parameter estimation in a simple stochastic differential equation for phytoplankton modelling

Abstract The use of stochastic differential equations (SDEs) for the simulation of aquatic ecosystems has attracted increasing attention in recent years. The SDE setting also provides the opportunity for statistical estimation of ecosystem parameters. We present an estimation procedure, based on Kalman filtering and likelihood estimation, which has proven useful in other fields of application. The estimation procedure is presented and the development from ordinary differential equations (ODEs) to SDEs is discussed with emphasis on autocorrelated residuals, commonly encountered with ODEs. The estimation procedure is applied to a simple nitrogen-phytoplankton model, with data from a Danish estuary (1988–2006). The resulting SDE is simple enough to have a well-known stationary distribution and this distribution is presented and compared with observed phytoplankton data.

Adaptive recursive M-robust system parameter identification using the QQ-plot approach

A new adaptive algorithm for the robust estimation of parameters of linear dynamic discrete-time systems in the presence of non-Gaussian impulsive noise within a measurement sequence is proposed in this study. Starting from the theory of robust estimation, a simple, adaptive, practically applicable robust approximate maximum likelihood algorithm is derived that, in the cases of contaminated normal distribution of measurement noise, demonstrates a high level of efficiency. The QQ-plot technique, combined with data cleaning based on the robustified winsorisation technique, is used as a framework for the classification of sorted data into the class of regular normally distributed data and the class of irregular data belonging to the contaminating distribution with a variance that is much greater than nominal. The link between the QQ-plot technique and a specific linear regression is established, so that the estimation of statistical parameters of the contaminated measurement distribution is performed using the least-squares technique. Then, the suboptimal maximum likelihood criterion is defined, and the system parameter estimation problem is solved robustly, using the proposed recursive robust parameter estimation scheme. Simulation results illustrate the discussion and show the efficiency of the proposed adaptive recursive parameter estimation algorithm in the presence of glint spikes or outliers.

Three-dimensional atomic imaging of crystalline nanoparticles

Determining the three-dimensional (3D) arrangement of atoms in crystalline nanoparticles is important for nanometre-scale device engineering and also for applications involving nanoparticles, such as optoelectronics or catalysis. A nanoparticle's physical and chemical properties are controlled by its exact 3D morphology, structure and composition. Electron tomography enables the recovery of the shape of a nanoparticle from a series of projection images. Although atomic-resolution electron microscopy has been feasible for nearly four decades, neither electron tomography nor any other experimental technique has yet demonstrated atomic resolution in three dimensions. Here we report the 3D reconstruction of a complex crystalline nanoparticle at atomic resolution. To achieve this, we combined aberration-corrected scanning transmission electron microscopy, statistical parameter estimation theory and discrete tomography. Unlike conventional electron tomography, only two images of the target--a silver nanoparticle embedded in an aluminium matrix--are sufficient for the reconstruction when combined with available knowledge about the particle's crystallographic structure. Additional projections confirm the reliability of the result. The results we present help close the gap between the atomic resolution achievable in two-dimensional electron micrographs and the coarser resolution that has hitherto been obtained by conventional electron tomography.

Risk modelling with the mixed Erlang distribution

AbstractA review of analytical and computational properties of the mixed Erlang distribution is given in the context of risk analysis. Basic distributional properties are discussed, and examples of members of the class are provided. Its use in aggregate claims, stop‐loss analysis, and risk measures is considered, as are applications in ruin theoretic analysis of the surplus process. Statistical estimation of model parameters is then discussed. Copyright © 2010 John Wiley & Sons, Ltd.