Minimum Discrepancy Research Articles

Asymptotic normality of test statistics under alternative hypotheses

The aim of this paper is to present a framework for asymptotic analysis of likelihood ratio and minimum discrepancy test statistics. First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models. In particular, these asymptotics give sufficient and in a sense necessary conditions for asymptotic normality of test statistics under alternative hypotheses. Second order asymptotics, and their implications for bias corrections, are also discussed in a somewhat informal manner. As an example, asymptotics of test statistics in the analysis of covariance structures are discussed in detail.

Empirical Likelihood Estimators for Stochastic Discount Factors

Hansen and Jagannathan (HJ, 1991) provided bounds on the volatility of Stochastic Discount Factors (SDF) that proved extremely useful to diagnose and test asset pricing models. This nonparametric bound reects a duality between the meanstandard deviation frontier for SDFs and the mean-variance frontier for portfolios of asset returns. We extend this fundamental contribution by proposing information bounds that minimize general convex functions of SDFs directly taking into account higher moments of returns. These Minimum Discrepancy bounds reect a duality with nding the optimal portfolio of asset returns with a general HARA utility function. The maximum utility portfolio implies SDF estimators that are based on implied probabilities associated with the class of Generalized Empirical Likelihood estimators. We analyze the implications of these information bounds for the pricing of size portfolios and the performance evaluation of hedge funds.

Mathematical model of heating a prism with boundary conditions of the 3rd kind

This paper describes the procedure of computational determination of the temperature field of a prismatic workpiece heated in a continuous furnace with account for the temperature dependence of the thermal diffusivity. For a numerical solution of the two-dimensional heat conduction equation with boundary conditions of the 3rd kind, an implicit scheme has been used. The calculated time dependences of the temperature for three characteristic points of the cross-section of the prismatic steel workpiece have been compared to the experimental data. The heat transfer coefficients at which the experimental data and the calculated values have a minimum discrepancy have been determined.

The measurement and magnitude of awareness difficulties after traumatic brain injury: A longitudinal study

Previous research suggests that reduced self-awareness is common following traumatic brain injury (TBI). However, few studies have examined the magnitude of this problem in a sample representative of hospitalized individuals. In this longitudinal study, individuals with complicated mild to severe TBIs and their significant others (SO) were evaluated at 1 and 12 months postinjury on the Sickness Impact Profile. Awareness was measured by comparing the level of injury-related problems reported by a person with TBI and their SO. Overall, individuals with TBI did not report fewer difficulties than their SO. In contrast, they frequently reported more injury-related difficulties than their SO. As there is no commonly or universally accepted definition for differential awareness, the magnitude of underreporting and over-reporting problems is presented using four different cutoff scores. A minimum discrepancy is proposed for defining awareness difficulties that is based on the standard error of measurement of the test-retest difference of the measure. Reduced self-awareness was inconsistent across both time and functional domains. These results suggest that reduced self-awareness is not the norm at 1 or 12 months postinjury and highlight the need for a more standardized approach to the measurement and classification of self-awareness.

A new shooting method for quasi-boundary regularization of backward heat conduction problems

A quasi-boundary regularization leads to a two-point boundary value problem of the backward heat conduction equation. The ill-posed problem is analyzed by using the semi-discretization numerical schemes. Then the resulting ordinary differential equations in the discretized space are numerically integrated towards the time direction by the Lie-group shooting method to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G( T) and the formation of a generalized mid-point Lie group element G( r). Then, by imposing G ( T ) = G ( r ) we can search for the missing initial conditions through a minimum discrepancy of the targets in terms of the weighting factor r ∈ ( 0 , 1 ) . Several numerical examples were worked out to persuade that this novel approach has good efficiency and accuracy. Although the final temperature is almost undetectable and/or is disturbed by large noise, the Lie group shooting method is stable to recover the initial temperature very well.

A robust inverse regression estimator

A family of dimension reduction methods was developed by Cook and Ni [Sufficient dimension reduction via inverse regression: a minimum discrepancy approach. J. Amer. Statist. Assoc. 100, 410–428.] via minimizing a quadratic objective function. Its optimal member called the inverse regression estimator (IRE) was proposed. However, its calculation involves higher order moments of the predictors. In this article, we propose a robust version of the IRE that only uses second moments of the predictor for estimation and inference, leading to better small sample results.

Optimal sufficient dimension reduction in regressions with categorical predictors

Though partial sliced inverse regression (partial SIR: Chiaromonte et al. [2002. Sufficient dimension reduction in regressions with categorical predictors. Ann. Statist. 30, 475–497]) extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors, its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice. When this condition fails, partial SIR may provide misleading results. In this article, we propose a new estimation method via a minimum discrepancy approach without this restriction. Our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi-squared distribution. It also gives us the ability to test predictor effects. An asymptotic chi-squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained.

The Lie-group shooting method for boundary layer equations in fluid mechanics

In this paper, we propose a Lie-group shooting method to tackle two famous boundary layer equations in fluid mechanics, namely, the Falkner-Skan and the Blasius equations. We can employ this method to find unknown initial conditions. The pivotal point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing initial conditions through a minimum discrepancy of the target in terms of the weighting factor r ɛ (0,1). It is the first time that we can apply the Lie-group shooting method to solve the boundary layer equations. Numerical examples are worked out to persuade that this novel approach has good efficiency and accuracy with a fast convergence speed by searching r with the minimum norm to fit two targets.

Multivariate hybrid modelling of air-gauge sensor

The quality of measurements depends directly on the quality of the measurement system model. With this concern, novel hybrid modelling techniques have been formulated for model performance enhancement. Air-gauge focus system sensor modelling has been accomplished by theoretical and empirical data integration. These modelling techniques combine a priori knowledge in theoretical model elaboration, and to attain the enhanced levels of adequacy, accuracy and precision they approximate the exact unknown model, simultaneously by available theoretical and appropriate polynomial empirical functions. For such a hybrid model the solution of two approaches by means of linear transformation and successive linear and nonlinear transformations have been developed. The validations of elaborated air-gauge sensor models revealed that sensor hybrid model solving by successive linear and nonlinear transformations permits us to attain minimum discrepancy with empirical evidence for the whole region of interest for model predictor variables.

The Relationship Between Discrepancies Defined on a Domain and on its Subset

The discrepancy is an important optimality criterion for experimental designs. Sometimes for practical reasons one may choose a design on some finite subset of the original experimental domain. This article addresses the question of whether minimum discrepancy designs are the same for the discrepancy defined on the original experimental domain and the discrepancy defined on a subset. Under certain non-trivial conditions they are shown to be equivalent. Examples are given to show when these conditions apply.

Nonextensive entropies from information content and stabilities

We consider a family of nonextensive entropies derived from information contents, which are a basic ingredient in information theory. To give a deeper insight into a certain class of stability of generalized entropies and into recent discussion on it, we present a criterion for stability, which is satisfied by a minimum discrepancy against infinitesimal shift of distributions.

Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory

We describe and develop a close relationship between two problems that have customarily been regarded as distinct: that of maximizing entropy, and that of minimizing worst-case expected loss. Using a formulation grounded in the equilibrium theory of zero-sum games between Decision Maker and Nature, these two problems are shown to be dual to each other, the solution to each providing that to the other. Although Topsøe described this connection for the Shannon entropy over 20 years ago, it does not appear to be widely known even in that important special case. We here generalize this theory to apply to arbitrary decision problems and loss functions. We indicate how an appropriate generalized definition of entropy can be associated with such a problem, and we show that, subject to certain regularity conditions, the above-mentioned duality continues to apply in this extended context. This simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust Bayes acts. We also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions. This leads to an extension, to arbitrary discrepancies, of a well-known minimax theorem for the case of Kullback–Leibler divergence (the “redundancy-capacity theorem” of information theory). For the important case of families of distributions having certain mean values specified, we develop simple sufficient conditions and methods for identifying the desired solutions. We use this theory to introduce a new concept of “generalized exponential family” linked to the specific decision problem under consideration, and we demonstrate that this shares many of the properties of standard exponential families. Finally, we show that the existence of an equilibrium in our game can be rephrased in terms of a “Pythagorean property” of the related divergence, thus generalizing previously announced results for Kullback–Leibler and Bregman divergences.

Parameter identification of concrete lining for a subway tunnel

The key factor in filling the gap between the real and predicted behaviour of structures is the use of field measurements not only to monitor the stability of structures, but also to re-evaluate the input data of the parameters used in the numerical analysis. Based on this re-evaluation of the input data, the original model is continuously modified to achieve minimum discrepancy between the behaviour of the model and the real structure. This can be done by using the back analysis procedures. Till now most of the researchers concentrate on the parameter identification of the soil constants only and consider the constants of the supporting elements (concrete or shotcrete) as fixed known values although the true constants of the supporting elements may be different than their assumed values. In a previous paper, the parameter identification of the concrete constants has been done and checked with some theoretical examples. In this study, the parameter identification of the constants of the tunnel concrete lining had been achieved. A two-dimensional element, called BEAM 6 element, is used to simulate the tunnel concrete lining while another element, called, LST element (Linearly Varying Strain Triangular element), is used to simulate the surrounding soil. The back analysis was done based on the dual boundary control method. The parameter identification has been applied on two real tunnels. The first application has been done for the soil constants only while the identification of the constants of the tunnel shotcrete lining simultaneously with the soil parameters had been applied on the second tunnel. Although more than one region of concrete are used, the parameters of the concrete lining have been identified simultaneously with the soil parameters. The obtained displacements are compared with the measured ones and the obtained results have been discussed. The main points can be summarized as: Although only parts of the measurements were used, satisfactory results were obtained at most measuring points. The calculated curve of displacements matches the measured curve very well, except for the lower region of measurements and some points in the middle part. The difference between the measured and the calculated displacements for the lower region may be due to unknown loading conditions of air pressure and water level. Another reason of this difference may be due to the accuracy of the sliding micrometers in this region because the measurements are very small. The forces of shotcrete lining in cases of identification of shotcrete parameters simultaneously with the soil parameters are higher than those in cases of identification of soil parameters with fixed shotcrete parameters. Therefore, it is preferable to identify the shotcrete parameters simultaneously with the soil parameters to ensure the safety in tunnelling process. Although the tunnel is still safe, the predicted factor of safety is reduced. Hence, the back analysis helps in making a reliable assessment of tunnelling safety. The identification of soil parameters only and considering the shotcrete parameters as constant parameters gives displacements far from the observed displacements while the identification of all soil layers simultaneously with the shotcrete parameters has a big effect on the calculated displacements. This application can be considered as a successful application of the parameter identification of shotcrete lining although the poor available measurements data. (A). Reprinted with permission from Elsevier. For the covering abstract see ITRD E124500.

Comparison of three different interface mass transfer models used in the experimental measurement of solvent diffusivity in heavy oil

Design and modeling of solvent-based heavy oil recovery processes require the knowledge of solvent diffusivity in heavy oil. The solvent diffusivity in heavy oil at constant temperature can be measured by using the so-called pressure decay method. In this method, the solvent diffusivity is determined by monitoring the decaying pressure of solvent phase while it is in contact with heavy oil in a closed high-pressure cell. This paper examines the effects of three different boundary conditions at the solvent–heavy oil interface on the determined diffusion coefficient. The so-called equilibrium, quasi-equilibrium and non-equilibrium boundary conditions are applied to represent three rather different interface mass transfer models. An analytical solution to the bulk diffusion equation subjected to each boundary condition is presented to determine the concentration distribution of solvent in the heavy oil. With the determined solvent concentration distribution in the heavy oil, pressure in the solvent phase is then predicted by utilizing the solvent mass conservation and the equation of state for a real gas. Therefore, an average solvent diffusivity, i.e., independent of solvent concentration, is determined by matching the numerically predicted pressures with the experimentally measured data. Experimental data for two different systems, CO 2-heavy oil and CH 4-heavy oil, are analyzed and the diffusion coefficients are compared with the previously published values. Based on the minimum discrepancy between the predicted and measured pressure data, it is found that the non-equilibrium BC is more applicable to CO 2-heavy oil system and that the equilibrium BC is most suitable for CH 4-heavy oil system. Thus it is concluded that the obtained solvent diffusivity is sensitive to the boundary condition applied at the solvent–heavy oil interface. Hence, a proper boundary condition for each solvent–heavy oil system should be used to determine the solvent diffusivity in heavy oil.

Amplitude death through a Hopf bifurcation in coupled electrochemical oscillators: experiments and simulations.

Amplitude death was observed in experiments with two coupled periodic electrochemical oscillators without time delay. Simulation results confirmed that the observed amplitude death was obtained via a Hopf bifurcation. The two oscillators must have a minimum discrepancy and both be near their individual Hopf bifurcations for amplitude death to occur. Phase drift (coexisting with unstable asymmetric phase-locked solutions), amplitude death, and in-phase synchronization were observed in both the experiments and simulations as coupling strength was increased. In addition, the simulations showed that a stable asymmetric phase-locked solution exists between the phase drift and amplitude death regions.

Magnetic structure of Er5Ge3 at 4.2 K

A symmetry analysis of the possible magnetic structures of Er5Ge3 in the ground state is performed using the results of measurements of elastic magnetic neutron scattering at 4.2 K. It is shown that the minimum discrepancy factor R m ≈9.5% corresponds to a modulated collinear magnetic structure in which the magnetic moments of erbium atoms are oriented along the a3 axis of the unit cell of the crystal structure and induce an antiferromagnetic longitudinal spin wave (AFLSW). The magnetic structure is characterized by the wave vector k=2π(0, 0, µ /a3) (where µ≈0.293) and the modulation period λ≈3.413a3. The magnetic ordering temperature T N ≈38 K is determined from the temperature dependence of the intensity of magnetic reflections.

Permutations with Low Discrepancy Consecutive k-sums

Consider the permutation π=( π 1,…, π n ) of 1,2,…, n as being placed on a circle with indices taken modulo n. For given k⩽ n there are n sums of k consecutive entries. We say the maximum difference of any consecutive k-sum from the average k-sum is the discrepancy of the permutation. We seek a permutation of minimum discrepancy. We find that in general the discrepancy is small, never more than k+6, independent of n. For g= gcd( n, k)>1, we show that the discrepancy is ⩽ 7 2 . For g=1 it is more complicated. Our constructions show that the discrepancy never exceeds k/2 by more than 9 for large n, while it is at least k/2 for infinitely many n. We also give an analysis for the easier case of linear permutations, where we view the permutation as written on a line. The analogous discrepancy is at most 2 for all n, k.

Analysis of the influence of germanium dead layer on detector calibration simulation for environmental radioactive samples using the Monte Carlo method

Germanium crystals have a dead layer that causes a decrease in efficiency, since the layer is not useful for detection, but strongly attenuates photons. The thickness of this inactive layer is not well known due to the existence of a transition zone where photons are increasingly absorbed. Therefore, using data provided by manufacturers in the detector simulation model, some strong discrepancies appear between calculated and measured efficiencies. The Monte Carlo method is applied to simulate the calibration of a HP Ge detector in order to determine the total inactive germanium layer thickness and the active volume that are needed in order to obtain the minimum discrepancy between estimated and experimental efficiency. Calculations and measurements were performed for all of the radionuclides included in a standard calibration gamma cocktail solution. A Marinelli beaker was considered for this analysis, as it is one of the most commonly used sample container for environmental radioactivity measurements. Results indicated that a good agreement between calculated and measured efficiencies is obtained using a value for the inactive germanium layer thickness equal to approximately twice the value provided by the detector manufacturer. For all energy peaks included in the calibration, the best agreement with experimental efficiency was found using a combination of a small thickness of the inactive germanium layer and a small detection active volume.

Sufficient dimensions reduction in regressions with categorical predictors

Though partial sliced inverse regression (partial SIR: Chiaromonte et al. [2002. Sufficient dimension reduction in regressions with categorical predictors. Ann. Statist. 30, 475–497]) extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors, its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice. When this condition fails, partial SIR may provide misleading results. In this article, we propose a new estimation method via a minimum discrepancy approach without this restriction. Our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi-squared distribution. It also gives us the ability to test predictor effects. An asymptotic chi-squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained.

Single column discrepancy and dynamic max-mini optimizations for quickly finding the most parsimonious evolutionary trees.

In the maximum parsimony (MP) method, the tree requiring the minimum number of changes (discrepancy) to explain the given set of DNA or amino acid sequences is chosen to represent their evolutionary relationships. To find the MP tree, the branch-and-bound algorithm is normally used. For a partial phylogenetic-tree (one that has a subset of the organisms) the traditional algorithm assigns a cost equal to the discrepancy of the partial phylogenetic-tree. We propose a single column discrepancy heuristic which increases this cost by predicting a minimum additional discrepancy needed to attach the sequences yet to be added to the partial phylogenetic-tree. A dynamic Max-mini order of sequence addition is also proposed to quickly terminate branch-and-bound search paths that are guaranteed to lead to suboptimal solutions. We studied the running time of 47 problems generated from 17 data sets. The use of single column discrepancy heuristic speeded up the computation to 2.4-fold for static and 18.2-fold for dynamic search order. The improvement appeared to increase exponentially with the number of sequences. The proposed strategies are also likely to be useful in speeding up the MP tree search using heuristic searches that are based on branch-and-bound-like algorithms. s.kumar@asu.edu