Finite Number Of Observations Research Articles

A new approach in non-parametric estimation of returns in mean-downside risk portfolio frontier

The DownSide Risk (DSR) model for portfolio optimization allows to overcome the drawbacks of the classical Mean-Variance model concerning the asymmetry of returns and the risk perception of investors. This optimization model deals with a positive definite matrix that is endogenous with respect to the portfolio weights and hence yields to a non standard optimization problem. To bypass this hurdle, Athayde (2001) developed a new recursive minimization procedure that ensures the convergence to the solution. However, when a finite number of observations is available, the portfolio frontier usually exhibits some inflexion points which make this curve not very smooth. In order to overcome these points, Athayde (2003) proposed a mean kernel estimation of returns to get a smoother portfolio frontier. This technique provides an effect similar to the case in which an infinite number of observations is available. In spite of the originality of this approach, the proposed algorithm was not neatly written. Moreover, no application was presented in his paper. Ben Salah et al (2015), taking advantage on the the robustness of the median, replaced the mean estimator in Athayde's model by a nonparametric median estimator of the returns, and gave a tidily and comprehensive version of the former algorithm (of Athayde (2001, 2003)). In all the previous cases, the problem is computationally complex since at each iteration, the returns (for each asset and for the portfolio) need to be re-estimated. Due to the changes in the kernel weights for every time, the portfolio is altered. In this paper, a new method to reduce the number of iterations is proposed. Its principle is to start by estimating non parametrically all the returns for each asset; then, the returns of a given portfolio will be derived from the previous estimated assets returns. Using the DSR criterion and Athayde's algorithm, a smoother portfolio frontier is obtained when short selling is or is not allowed. The proposed approach is applied on the French and Brazilian stock markets.

A parametric uncertainty analysis method for queues with vacations

In this paper, we propose a numerical method, based on Taylor series expansion, for computing the uncertainty in performance measures of vacation queueing model due to epistemic uncertainties in its parameters. Specifically, we consider the M/G/1/N queue with vacations, where we assume that vacation parameter is derived from finite number of observations, they themselves will have epistemic uncertainty associated with them. Using this method, we compute the expected value and the variance of the stationary distribution associated with the M/G/1/N queue with vacations. In order to illustrate the applicability of the proposed method, a series of numerical examples are discussed and the obtained results are compared to the corresponding Monte Carlo simulations ones.

Bayesian Convergence to the Truth and the Metaphysics of Possible Worlds

In a recent paper, Belot argues that Bayesians are epistemologically flawed because they believe with probability 1 that they will learn the truth about observational propositions in the limit. While Belot’s considerations suggest that this result should be interpreted with some care, the concerns he raises can largely be defused by putting convergence to the truth in the context of learning from an arbitrarily large but finite number of observations.

Functional principal component analysis for multivariate multidimensional environmental data

Data with spatio-temporal structure can arise in many contexts, therefore a considerable interest in modelling these data has been generated, but the complexity of spatio-temporal models, together with the size of the dataset, results in a challenging task. The modelization is even more complex in presence of multivariate data. Since some modelling problems are more natural to think through in functional terms, even if only a finite number of observations is available, treating the data as functional can be useful (Berrendero et al. in Comput Stat Data Anal 55:2619–2634, 2011). Although in Ramsay and Silverman (Functional data analysis, 2nd edn. Springer, New York, 2005) the case of multivariate functional data is also contemplated, they do not cope with more than one dimension (only time is considered as covariate) in the domain of the considered functions. In estimating functional data through smoothing methods, a proper framework for incorporating space-time structures can be found in the generalized additive models (GAM), while classical dimension reduction techniques for functional data lead to functional principal component analysis (FPCA). In a previous work Ruggieri et al. (J Appl Stat 40(4):795–807, 2013) extended temporal FPCA, that is FPCA on data modelled as functions of the one-dimensional time, to multivariate (more than one variable, in our case pollutants) context. In this paper the computational aspects of FPCA are extended to more than one dimension: space (long, lat) and/or space-time; moreover, multidimensional (spatial, spatio-temporal) FPCA is extended to multivariate case. In order to provide a generalization of FPCA to multidimensional (spatio-temporal) and simultaneously multivariate data, we link GAM models together with the approach proposed by Ramsay and Silverman (2005). The paper describes all the computational details useful to implement this approach, while its effectiveness will be shown by a multivariate spatio-temporal environmental dataset, whose structure is actually very common in literature.

Bayesian Numerical Homogenization

Numerical homogenization, i.e., the finite-dimensional approximation of solution spaces of PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements. These basis elements are oftentimes found after a laborious process of scientific investigation and plain guesswork. Can this identification problem be facilitated? Is there a general recipe/decision framework for guiding the design of basis elements? We suggest that the answer to the above questions could be positive based on the reformulation of numerical homogenization as a Bayesian inference problem in which a given PDE with rough coefficients (or multiscale operator) is excited with noise (random right-hand side/source term) and one tries to estimate the value of the solution at a given point based on a finite number of observations. We apply this reformulation to the identification of bases for the numerical homogenization of arbitrary integro-differential equations and show that these bases have optimal recovery properties. In particular we show how rough polyharmonic splines can be rediscovered as the optimal solution of a Gaussian filtering problem.

Decomposing the Gini Inequality Index

The aim of this article is to propose a new breakdown of the Gini inequality ratio into three components ( within-group inequality, between-group inequality, and intensity of transvariation between groups to the total inequality index). The between-group inequality concept computes all the differences in salaries between men and women. The main contribution is to show this relationship when dealing with noninteger frequency data appearing on survey data. This article also proves the relationship between the very intuitive Gini concentration ratio and the very commonly used Gini index in the case of discrete distributions with a finite number of observations. The former refers to the relationship between the cumulative proportion of population and income, while the latter is the absolute mean difference divided by twice the arithmetic mean. This article complements the inequality decomposition literature that has stemmed from Gini’s work. The research is applied to analyze gender income inequality in a Spanish region using Spanish Structure of Earnings Survey data from 2010.

Sensitivity analysis of ranked data: from order statistics to quantiles

In this paper we provide the mathematical theory for sensitivity analysis of order statistics of continuous random variables, where the sensitivity is with respect to a distributional parameter. Sensitivity analysis of order statistics over a finite number of observations is discussed before addressing sensitivity analysis of quantiles. In particular, we provide a central limit theorem and related confidence intervals for the proposed sensitivity estimators, and we improve the known convergence results for IPA. Our analysis provides guidelines for sensitivity analysis of order statistic related performance measures such as basic order-statistics or quantiles. Our findings are corroborated by a series of numerical examples from different areas of operations research: stochastic activity networks, queues, reliability, and financial engineering.

A consistent nonparametric test of parametric regression functional form in fixed effects panel data models

We propose a consistent test for a linear functional form against a nonparametric alternative in a fixed effects panel data model. We show that the test has a limiting standard normal distribution under the null hypothesis, and show that the test is a consistent test. We also establish the asymptotic validity of a bootstrap procedure which is used to better approximate the finite sample null distribution of the test statistic. Simulation results show that the proposed test performs well for panel data with a large number of cross-sectional units and a finite number of observations across time.

Testing Measurement Equivalence of Categorical Items’ Threshold/Difficulty Parameters: A Comparison of CCFA and (M)IRT Approaches

Multiple group confirmatory factor analyses and differential item functioning basing on the unidimensional or the multidimensional item response theory were the two most commonly used methods to assess the measurement equivalence of categorical items. Unlike the traditional linear factor analysis, multiple-group categorical confirmatory factor analysis (CCFA) could model the categorical measures with a threshold structure appropriately, which is comparable to the difficulty parameters in the multidimensional IRT [(M)IRT)]. In this study, we compared the multiple-group categorical CFA (CCFA) and (M)IRT in terms of their power to detect violations of measurement invariance (i.e., DIF) with the Monte Carlo method. Moreover, given the limitation of the assumptions under the traditional unidimensional IRT model, this study extended the DIF test method to the (M)IRT model. Simulation studies under both unidimensional and multidimensional conditions were conducted to compare the DIFFTEST method, IRT-LR method (for unidimensional scale), and MIRT-MG (for multidimensional scale) with respect to their power to detect the lack of invariance across groups. Results indicated that the three methods, namely, DIFFTEST, IRT-LR, and MIRT-MG, showed reasonable power to identify the measurement non-equivalence when the difference of threshold was large. For unidimensional scale, the IRT-LR test demonstrated superior power to DIFFTEST. Whereas, for multidimensional scale, the results were not completely consistent across different conditions. The power of MIRT-MG was higher than that of DIFFTEST when test length was long and the correlation between dimensions was high. In contrast, the power of DIFFTEST was higher than that of MIRT-MG when test length was short and the correlations between dimensions were low. For a fixed number of noninvariant items, the power of the DIFFTEST method became smaller as the test length increased, whereas the power of the IRT-LR and MIRT-MG methods became larger as the test length increased. The number of respondents per group (sample size) was found to be one of the most important factors affecting the performance of these three approaches. The power of the DIFFTEST, IRT-LR, and, MIRT-MG methods would increase as the sample size increased. For a finite number of observations, the power of all three methods was larger under the balanced design when the two groups were equal in size than when two groups were unequal in size in the unbalanced design. For the DIFFTEST method, the Type I errors reached the nominal error rate at 5%, while the IRT-LR and MIRT-MG methods produced much lower Type I error rates.

Learning the Dynamics of Objects by Optimal Functional Interpolation

Many areas of science and engineering rely on functional data and their numerical analysis. The need to analyze time-varying functional data raises the general problem of interpolation, that is, how to learn a smooth time evolution from a finite number of observations. Here, we introduce optimal functional interpolation (OFI), a numerical algorithm that interpolates functional data over time. Unlike the usual interpolation or learning algorithms, the OFI algorithm obeys the continuity equation, which describes the transport of some types of conserved quantities, and its implementation shows smooth, continuous flows of quantities. Without the need to take into account equations of motion such as the Navier-Stokes equation or the diffusion equation, OFI is capable of learning the dynamics of objects such as those represented by mass, image intensity, particle concentration, heat, spectral density, and probability density.

Non-commutative large entries for cognitive radio applications

Cognitive radio has been proposed as a solution for the problem of underutilization of the radio spectrum. Indeed, measurements have shown that large portions of frequency bands are not efficiently assigned since large pieces of bandwidth are unused or only partially used. In the last decade, studies in different areas, such as signal processing, random matrix theory, information theory, game theory, etc., have brought us to the current state of cognitive radio research. These theoretical advancements represents a solid base for practical applications and even further developments. However, still open questions need to be answered. In this study, free probability theory, through the free deconvolution technique, is used to attack the huge problem of retrieving useful information from the network with a finite number of observations. Free deconvolution, based on the moments method, has shown to be a helpful approach to this problem. After giving the general idea of free deconvolution for known models, we show how the moments method works in the case where scalar random variables are considered. Since, in general, we have a situation where more complex systems are involved, the parameters of interest are no longer scalar random variables but random vectors and random matrices. Random matrices are non-commutative operators with respect to the matrix product and they can be considered elements of what is called non-commutative probability space. Therefore, we focus on the case where random matrices are considered. Concepts from combinatorics, such as crossing and non-crossing partitions are useful tools to express the moments of Gaussian and Vandermonde matrices, respectively. Our analysis and simulation results show that free deconvolution framework can be used for studying relevant information in cognitive radio such as power detection, users detection, etc.

Introducingmultivator: A Multivariate Emulator

A multivariate generalization of the emulator technique described by Hankin (2005) is presented in which random multivariate functions may be assessed. In the standard univariate case (Oakley 1999), a Gaussian process, a finite number of observations is made; here, observations of different types are considered. The technique has the property that marginal analysis (that is, considering only a single observation type) reduces exactly to the univariate theory. The associated software is used to analyze datasets from the field of climate change.

Exact Synchronization for Finite-State Sources

We analyze how an observer synchronizes to the internal state of a finite-state information source, using the epsilon-machine causal representation. Here, we treat the case of exact synchronization, when it is possible for the observer to synchronize completely after a finite number of observations. The more difficult case of strictly asymptotic synchronization is treated in a sequel. In both cases, we find that an observer, on average, will synchronize to the source state exponentially fast and that, as a result, the average accuracy in an observer's predictions of the source output approaches its optimal level exponentially fast as well. Additionally, we show here how to analytically calculate the synchronization rate for exact epsilon-machines and provide an efficient polynomial-time algorithm to test epsilon-machines for exactness.

Large Deviation Bounds for Functionals of Viterbi Paths

In a number of applications, the underlying stochastic process is modeled as a finite-state discrete-time Markov chain that cannot be observed directly and is represented by an auxiliary process. The maximum a posteriori (MAP) estimator is widely used to estimate states of this hidden Markov model through available observations. The MAP path estimator based on a finite number of observations is calculated by the Viterbi algorithm, and is often referred to as the Viterbi path. It was recently shown in, and, (see also and) that under mild conditions, the sequence of estimators of a given state converges almost surely to a limiting regenerative process as the number of observations approaches infinity. This in particular implies a law of large numbers for some functionals of hidden states and finite Viterbi paths. The aim of this paper is to provide the corresponding large deviation estimates.

Incomplete observations in a linear identification problem

The article considers a highly general linear identification problem for an unknown vector parameter from output observations. The main focus is on sufficiency of a finite number of observations of the output signal for unique estimation of the unknown vector parameter. Two general sufficient conditions are obtained, ensuring unique computability of the unknown vector parameter from finitely many observations of the output signal.

Widely linear prediction for transfer function models based on the infinite past

The problem of widely linear (WL) prediction for both WL ARMA models and WL transfer function models on the basis of infinite past information is studied. A recursive algorithm to obtain a suboptimum predictor for WL ARMA systems is first given. Then this algorithm is used to develop another recursive algorithm which performs WL prediction for transfer function models. The suggested solutions become an alternative to the WL prediction based on a finite number of observations provided the size of the time series is sufficiently large.

Maximally Robust 2-D Channel Estimation for OFDM Systems

Two-dimensional minimum mean square error (MMSE) channel estimation for orthogonal frequency division multiplexing (OFDM) systems is known to perform better than the least squares, as well as the conventional 1-D MMSE estimation, owing to its ability of exploiting both, the time and the frequency correlations among the channel frequency response (CFR) coefficients. Its superior performance comes however at the price of increased requirements-the knowledge of observation noise power and that of the channel frequency, as well as time correlation functions. In practical transmission scenarios, where channel correlation functions are not known or cannot be easily estimated, it is desirable to have an estimator that is robust to mismatches between the assumed and the actual channel correlation function. While such a robust estimator, for the case of an infinite number of observations, is well known for various uncertainty classes, not much attention has been paid to the practical case of a finite number of observations. We derive in this paper, the maximally robust (MR) 2-D channel estimator for the case of a finite number of pilot observations under some realistic constraints on the uncertainty class to which the 2-D channel correlation sequence belongs. We demonstrate that the correlation sequence associated with the MR estimator can be obtained by a simple semi-definite optimization procedure and is interestingly different from the well-known heuristic proposals. Simulation results establish the superiority of the proposed MR estimator over commonly employed heuristic robust estimator by as much as 3 dB in terms of the worst-case estimation MSE and around 1 dB in terms of the average BER performance under different practical transmission scenarios of interest.

Structural Uncertainty Effect on Classical Wing Flutter Characteristics

Flutter-critical velocity is usually estimated from deterministic analyses by assuming that physical and geometrical parameters are perfectly known. However, aleatory and epistemic uncertainties, especially those associated with the definition of material properties, are intrinsically present partly because of the random nature of every physical system and partly because those properties are evaluated from a finite number of observations. The paper focuses on the combination of structural reliability analysis with aeroelastic simulation to give a correct flutter speed evaluation for design purposes. Two typical aeronautical test cases have been considered: (1) an isotropic material structure and (2) a composite material structure. Different computational methodologies (classical and developed by authors) have been coupled with a simple two-dimensional aeroelastic model to study the qualitative consequences of uncertainty in determination of critical flutter speed and to provide a comparison between differ...

Optimal Joint Target Detection and Parameter Estimation by MIMO Radar

We consider multiple-input multiple-output (MIMO) radar systems with widely-spaced antennas. Such antenna configuration facilitates capturing the inherent diversity gain due to independent signal dispersion by the target scatterers. We consider a new MIMO radar framework for detecting a target that lies in an unknown location. This is in contrast with conventional MIMO radars which break the space into small cells and aim at detecting the presence of a target in a specified cell. We treat this problem through offering a novel composite hypothesis testing framework for target detection when (i) one or more parameters of the target are unknown and we are interested in estimating them, and (ii) only a finite number of observations are available. The test offered optimizes a metric which accounts for both detection and estimation accuracies. In this paper as the parameter of interest we focus on the vector of time-delays that the waveforms undergo from being emitted by the transmit antennas until being observed by the receive antennas. The analytical and empirical results establish that for the proposed joint target detection and time-delay estimation framework, MIMO radars exhibit significant gains over phased-array radars for extended targets which consist of multiple independent scatterers. For point targets modeled as single scatterers, however, the detection/estimation accuracies of MIMO and phased-array radars for this specific setup (joint target detection and time-delay estimation) are comparable.

An overcomplete stitching algorithm for time decoding machines

We investigate a class of finite-dimensional time decoding algorithms that: (1) is insensitive with respect to the time-encoding parameters; (2) is highly efficient and stable; and (3) can be implemented in real time. These algorithms are based on the observation that the recovery of time encoded signals given a finite number of observations has the property that the quality of signal recovery is very high in a reduced time range. We show how to obtain a local representation of the time encoded signal in an efficient and stable manner using a Vandermonde formulation of the recovery algorithm. Once the signal values are obtained from a finite number of possibly overlapping observations, the reduced-range segments are stitched together. The signal obtained by segment stitching is subsequently filtered for improved performance in recovery. Finally, we evaluate the complexity of the algorithms and their computational requirements for real-time implementation.