Asymptotic Statistics Research Articles

Polarization mode dispersion properties of constantly spun randomly birefringent fibers.

We calculate the asymptotic first-order polarization mode dispersion statistics of constantly spun fibers. We show that in the long length regime the mean differential group delay of constantly spun fibers grows linearly with distance at the same rate as for unspun fibers. Conversely, the short length regime of constantly spun fibers may be much longer, even for limited spin rates.

A "rayleigh-ness" test for DS/SS code acquisition

Testing for statistical distribution has received increasing attention in signal processing for communications. Noncoherent receivers, based on correlation magnitude, are employed when the actual carrier phase of the received samples is unknown. This paper presents a new test, based on high-order statistics, to decide whether a real positive white series is a realization of a Rayleigh-distributed random process. Such a "Rayleigh-ness" test is based on a testing variable that measures the "Ricianity" of the series under investigation. That is, it estimates the possible presence of the mean of the complex Gaussian model generating both Rayleigh and Rice distributions. The asymptotic testing statistics have been derived as explicit functions of the higher order moments of the noise-plus-interference distribution. The performance of the Rayleigh-ness test has been analyzed in comparison with a conventional power detector. The devised test has application to noncoherent initial synchronization of the chip offset (code acquisition) in a symbol-length spreading sequence of direct-sequence code-division multiple-access systems. In such a case, the noise-plus-interference variance under the out-of-sync condition is much larger than the effective variance in the in-sync case. The obtained results evidence the robustness of the Rayleighness test, avoiding the severe performance degradation in the presence of large Gaussian and non-Gaussian noise, as well as multiuser interference in a downlink scenario.

Universal power-law decay in Hamiltonian systems?

The understanding of the asymptotic decay of correlations and of the distribution of Poincar\'e recurrence times $P(t)$ has been a major challenge in the field of Hamiltonian chaos for more than two decades. In a recent Letter, Chirikov and Shepelyansky claimed the universal decay $P(t) \sim t^{-3}$ for Hamiltonian systems. Their reasoning is based on renormalization arguments and numerical findings for the sticking of chaotic trajectories near a critical golden torus in the standard map. We performed extensive numerics and find clear deviations from the predicted asymptotic exponent of the decay of $P(t)$. We thereby demonstrate that even in the supposedly simple case, when a critical golden torus is present, the fundamental question of asymptotic statistics in Hamiltonian systems remains unsolved.

New statistical investigations of the ornstein-uhlenbeck process

An asymptotic analysis is presented for estimation in the three-parameter Ornstein-Uhlenbeck process, where the parameters are the local mean, the drift, and the variance. We are interested in the case when the damping parameter (λ, or λ T = κ) is nearly zero. The asymptotic sufficient statistics can be related to noncentral χ 1 2 distribution. The maximum likelihood estimate of the parameter vector is a solution of a rather complicated system of equations. We describe the methods for solving maximum-likelihood equations. Classical and robust estimators are determined for parameters. It is shown that the lower confidence limit of the drift (or damping) parameter is equal to zero with positive probability when it is near to zero.

Aircraft‐coating weathering studies by analytical methods

AbstractWe predicted the life of an aircraft coating through the characterization of slightly weathered samples. To accomplish this, we completed accelerated weathering of a flat aircraft coating in a xenon‐arc weather‐o‐meter for 0, 500, 1000, 1500, 2000, 2500, and 3000 h. Equatorial mount with mirrors for acceleration with aqua (EMMAQUA) exposures equivalent to 0, 6, 12, 18, 24, 30, 36, 42, and 48 months were also completed in this work. To determine coating‐film degradation, we used electron spectroscopy for chemical analysis, also known as XPS, and L*a*b* color measurement. Nonlinear regression estimations were performed on selected measurements from the test data. Two models were used. The first was the Weibull‐type model. The second was a Gauss‐type model that had also been used in previous work on decay processes. The second model generally provided a better fit as determined by asymptotic R2 statistics. Secondary‐ion mass spectroscopy data did not produce data consistent with decay functions. The models better described the data from color change measurements and XPS. Fourier transform infrared data fit the models but not as well as XPS and color measurement. Furthermore, data from EMMAQUA exposures fit better than data from the xenon‐arc weather‐o‐meter. In the model development work, we used single points for both measurement and exposure. The models can be refined with a high degree of predictive precision if replicates of 10 measurements and replicates of 5 exposures are performed; we plan to report these findings in the future. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 86: 294–313, 2002

Book reviews: Asymptotic methodsin probability and statistics with applications by Balkrishnan, N., Ibragimov, I.A and Nevzorov, V.B.

Book reviews: Asymptotic methodsin probability and statistics with applications by Balkrishnan, N., Ibragimov, I.A and Nevzorov, V.B.

Asymptotic Statistics of Zeroes for the Lamé Ensemble

The Lame polynomials naturally arise when separating variables in Laplace's equation in elliptic coordinates. The products of these polynomials form a class of spherical harmonics, which are joint eigenfunctions of a quantum completely integrable (QCI) system of commuting, second-order differential operators P0=Δ, P1,…,PN−1 acting on C∞(?N). These operators naturally depend on parameters and thus constitute an ensemble. In this paper, we compute the limiting level-spacings distributions for the zeroes of the Lame polynomials in various thermodynamic, asymptotic regimes. We give results both in the mean and pointwise, for an asymptotically full set of values of the parameters.

Limit theorems for a class of tests of gradual changes

Abstract We study a class of asymptotic tests for detecting various types of gradual changes in a location model. The results of Jaruskova (1998. J. Statist. Plann. Inference 70, 263–276) and Huskova (1998. In: Szyszkowicz, B. (Ed.), Asymptotic Methods in Probability and Statistics. Elsevier, Amsterdam, pp. 577–584.) are extended to other families of change alternatives. Under the null hypothesis of “no change”, limiting extreme value distributions of Gumbel type are derived for some asymptotic log-likelihood ratio test statistics. Consistency under certain alternatives is also discussed.

Statistical analysis of some second-order methods for blind channel identification/equalization with respect to channel undermodeling

Many second-order approaches have been proposed for blind FIR channel identification in single-input/multi-output context. In practical conditions, the measured impulse responses usually possess "small" leading and trailing terms, the second-order statistics are estimated from finite sample size, and there is additive white noise. This paper, based on a functional methodology, develops a statistical performance analysis of any second-order approach under these practical conditions. We study two channel models. In the first model, the channel tails are considered to be deterministic. We derive expressions for the asymptotic bias and covariance matrix (when the sample size tends to /spl infin/) of the mth-order estimated significant part of the impulse response. In the second model, the tails are treated as zero mean Gaussian random variables. Expressions for the asymptotic covariance matrix of the estimated significant part of the impulse response are then derived when the sample size tends to /spl infin/, and the variance of the tails tends to 0. Furthermore, some asymptotic statistics are given for the estimated zero-forcing equalizer, the combined channel-equalizer impulse response, and some byproducts, such as the open eye measure. This allows one to assess the influence of the limited sample size and the size of the tails, respectively, on the performance of identification and equalization of the algorithms under study. Closed-form expressions of these statistics are given for the least-squares, the subspace, the linear prediction, and the outer-product decomposition (OPD) methods, as examples. Finally, the accuracy of the asymptotic analysis is checked by numerical simulations; the results are found to be valid in a very large domain of the sample size and the size of the tails.

Simple Robust Testing of Regression Hypotheses

IN THIS PAPER WE CONSIDER the problem of hypothesis testing in models with errors that have serial correlation or heteroskedasticity of unknown form. This situation is often encountered in regression models applied to economic time series data. It is a classic textbook result that while ordinary least squares (OLS) estimates of regression parameters remain consistent and asymptotically normal when errors are heteroskedastic or autocorrelated (provided usual regularity conditions hold and no lagged dependent variables are in the model), standard tests are no longer valid. If the true form of serial correlation/heteroskedasticity were known, then generalized least squares (GLS) provides efficient estimates and standard inference can be conducted on the GLS transformed model. But, in practice the form of serial correlation/heteroskedasticity is often unknown, and this has led to the development of techniques that permit valid asymptotic inference without having to specify a model of the serial correlation or heteroskedasticity. The most common approach is to estimate the variance-covariance matrix of the OLS estimates nonparametrically using spectral methods (heteroskedasticity and autocorrelation consistent (HAC) estimators) and construct standard tests using the asymptotic normality of the OLS estimates. HAC estimators have been extensively analyzed in the econometrics literature and important contributions are given by Andrews (1991), Andrews and Monahan (1992), Gallant (1987), Hansen (1992), Newey and West (1987), Robinson (1991, 1998), and White (1984) among others. The benefit of HAC estimator tests is asymptotically valid inference that is robust to general forms of serial correlation/heteroskedasticity in the errors. We propose an alternative method of constructing robust test statistics. We apply a nonsingular data dependent stochastic transformation to the OLS estimates. The asymptotic distribution of the transformed estimates does not depend on nuisance parameters. Then, test statistics that are asymptotically invariant to nuisance parameters (asymptotic pivotal statistics) are constructed. The resulting test statistics have nonstandard asymptotic distributions that only depend on the number of restrictions being tested, and critical values are easy to simulate using standard techniques. The main advantage of our approach compared to standard approaches is that estimates of the variance-covariance matrix are not explicitly required to construct the tests. This is potentially important for two reasons. First, with the exception of the estimator proposed by Robinson (1998),2 consistent nonparametric estimates of variance-covariance matrices in models with serial

Tests for Equivalence or Noninferiority Between Two Proportions

Bioequivalence between two treatments or two drugs is often assessed by comparing the two proportions (success rate or eradication rate) of binomial outcomes when the conventional pharmacokinetic parameters are inadequate for the assessment. Setting the equivalence limits can be based on one of the three measures: difference, ratio, or odds ratio between the two binomial probabilities. This paper reviews the existing asymptotic test statistics for comparing two independent binomial probabilities in terms of the three measures in the context of equivalence or noninferiority testing. The actual type I error and power of the asymptotic tests are evaluated by enumerating the exact probabilities in the rejection region. The results show that to establish an equivalence between two treatments with an equivalence limit of 20% in difference, a sample size of at least 50 per treatment is needed. When the sample size is sufficient, the actual type I error rate is close to the nominal level (slightly above the nomin...

Book Reviews

Journal of the Royal Statistical Society: Series D (The Statistician)Volume 49, Issue 1 p. 115-133 Book Reviews First published: 28 June 2008 https://doi.org/10.1111/1467-9884.00226AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Collins, S. Constrained Markov Decision Processes Brooks, S. Encyclopedia of Biostatistics Auton, T. Comparative Statistical Inference Penenberg, D. N. Applied Multivariate Statistics in Geohydrology and Related Sciences Sapatinas, T. Model Selection and Inference: a Practical Information-theoretic Approach Mayo, O. Roy Geary 1896–1983, Irish Statistician: Centenary Lecture by John E. Spencer and Associated Papers Collop, M. Computer Assisted Survey Information Collection Woodward, P. W. Fractional Factorial Plans Reed, M. Maximum Entropy and Bayesian Methods Robertson, C. Neural Networks: an Introductory Guide for Social Scientists Hemmett, L. Exploring Social Issues using SPSS1 for WindowsTM Coolen, F. Nonparametric Statistical Methods Auton, T. Statistical Strategies for Small Sample Research Espejo, M. R. The Multivariate Social Scientist: Introductory Statistics using Generalized Linear Models Thas, O. Elements of Large-sample Theory Minkova, L. Stochastic Processes for Insurance and Finance Gani, J. Random Fields and Stochastic Partial Differential Equations Skouras, K. Asymptotic Statistics Wang, D. Population Data at a Glance: Shaded Contour Maps of Demographic Surfaces over Age and Time Ugarte, M. D. Models for Discrete Data Volume49, Issue1April 2000Pages 115-133 RelatedInformation

Portfolio optimization via stochastic programming: Methods of output analysis

Solutions of portfolio optimization problems are often influenced by errors or misspecifications due to approximation, estimation and incomplete information. Selected methods for analysis of results obtained by solving stochastic programs are presented and their scope illustrated on generic examples – the Markowitz model, a multiperiod bond portfolio management problem and a general strategic investment problem. The approaches are based on asymptotic and robust statistics, on the moment problem and on results of parametric optimization.

A Stochastic Tokunaga Model for stream networks

The Tokunaga cyclic model describes average network topology. A stochastic generalization is proposed. The stochastic model assumes that actual tributary numbers are random realizations from a negative binomial distribution whose mean is defined by the Tokunaga parameters ϵ1 and K. These parameters can be interpreted as representing the effects of regional controls. Upon these regional controls is superimposed an inherent spatial variability in network topology. A third parameter α characterizes this spatial variability. When α becomes large, the negative binomial model approaches a Poisson model. A goodness‐of‐fit test based on a χ2 test statistic is developed, and an inference framework for estimation of parameters and stream‐related statistics is described. This methodology is illustrated on tributary data from three catchments, one of the order of 5 and two of the order of 8. It is shown that the stochastic Tokunaga model using the negative binomial distribution is not inconsistent with the tributary data, whereas the Poisson model is unambiguously rejected by the data. Monte Carlo Bayesian methods are used to evaluate the uncertainty in the Tokunaga parameters and in stream number related statistics such as the bifurcation ratio. It is shown that tributary data from the order‐5 network provide little power for discriminating between model hypotheses. The tributary data for the two order‐8 basins are significantly different from the asymptotic stream number statistics predicted by Shreve's random network model. Finally, the problem of space filling or preservation of nontopological properties is considered in the context of the stochastic Tokunaga model.

Asymptotic Statistics of Poincaré Recurrences in Hamiltonian Systems with Divided Phase Space

By different methods we show that for dynamical chaos in the standard map with critical golden curve, the Poincar\'e recurrences $P(\ensuremath{\tau})$ and correlations $C(\ensuremath{\tau})$ decay asymptotically in time as $P\ensuremath{\propto}C/\ensuremath{\tau}\ensuremath{\propto}1/{\ensuremath{\tau}}^{3}$. It is also explained why this asymptotic behavior starts only at very large times. We argue that the same exponent $p\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3$ should be also valid for a general chaos border.

A Bayesian approach to the spatial representation of market structure from consumer choice data

This paper is concerned with the spatial representation of market structure calibrated on actual or intended choice data. Previous models developed for that purpose accommodate consumer heterogeneity by estimating parameters for each consumer, typically using the method of maximum likelihood. This approach to heterogeneity avoids assuming any particular distribution of the individual level parameters across the sample, but leads to problems related to the consistency of the estimates, sufficient degrees of freedom, and the validity of asymptotic standard errors and test statistics. Of greater concern is the assumption of independence of the choice observations within the same individual. This assumption is necessary in a maximum likelihood (MLE) framework to make the estimation computationally feasible. However, the marketing and psychology literature (cf. Manrai, 1995; Tversky and Simonson, 1993; Kim et al., forth coming) demonstrates that dependencies among choice alternatives may exist, and negligence to take such covariance into account may lead to inconsistent estimates, reduced predictive validity, and incorrect managerial action. We develop a new multidimensional scaling (MDS) model that estimates spatial market structures from pick-any/ J choice data, provides for individual level parameters, and allows for correlations among the choice alternatives across individuals. We provide a Bayesian estimation method that overcomes the traditional problems associated with estimating models with such correlated alternatives. We provide an application to pick-any/ J data for various brands of portable telephones. In a comparative analysis, we show that the proposed model outperforms one in which the utilities are assumed to be uncorrelated across choice alternatives.

Level Statistics of Quantum Dots Coupled to Reservoirs

We study the effect of electron tunneling on the level statistics of quantum dots. While the coupling between individual levels and the electron reservoir leads predominantly to the expected level broadening, the indirect coupling of adjacent levels via the reservoir results in new asymptotic level statistics and broadening. These features, which are attributed to renormalized spectral weights rather than to renormalized eigenvalues of the Hamiltonian, are observable in the asymptotic frequency dependence in an absorption experiment.

Compact Matrix Expressions for Generalized Wald Tests of Equality of Moment Vectors

Asymptotic chi-squared test statistics for testing the equality of moment vectors are developed. The test statistics proposed are generalized Wald test statistics that specialize for different settings by inserting an appropriate asymptotic variance matrix of sample moments. Scaled test statistics are also considered for dealing with nonstandard conditions. The specialization will be carried out for testing the equality of multinomial populations, and the equality of variance and correlation matrices for both normal and nonnormal data. When testing the equality of correlation matrices, a scaled version of the normal theory chi-squared statistic is proven to be an asymptotically exact chi-squared statistic in the case of elliptical data.

Two results on multiple Stratonovich integrals

Formulae connecting the multiple Stratonovich integrals with single Ogawa and Stratonovich integrals are derived. Multiple Riemann-Stieltjes integrals with respect to certain smooth approximations of the Wiener process are considered and it is shown that these integrals converge to multiple Stratonovich integrals as the approximation converges to the Wiener process. In an important work Hu and Meyer (1987) introduced a new multi- ple stochastic integral with respect to a Wiener process, called the multiple Startonovich integral (MSI), which is in general different from the usually studied multiple Wiener-Ito integral. However, Hu and Meyer only offered some rather informal definitions and proofs. Johnson and Kallianpur (1993) gave a rigor- ous definition for the MSI (the term MSI was not used in their work), denoted in this work by δp(·), and gave necessary and sufficient conditions for its exis- tence. Recently, multiple Stratonovich integrals have been applied to problems in asymptotic statistics and nonlinear filtering (cf. Budhiraja and Kallianpur (1995, 1996)). In this work we study some properties of multiple Stratonovich integrals which also give some justification for the appearance of the name 'Stratonovich' in the integral. We recall that one of the important properties of multiple Wiener-Itoi nte- grals of symmetric kernels is that they can be expressed as iterated (indefinite) Ito-integrals, by which we mean that if fp ∈ L 2(0, 1) p (the class of real val- ued, square integrable, symmetric functions defined on (0, 1) p ) then the multiple

Do currency futures prices follow random walks?

This paper examines the random walk process for four currency futures prices for the period 1977–1987 by using the variance ratio test. The random walk hypothesis is tested through asymptotic standardized statistics as well as by computing the significance level based on the bootstrap method. Both long time-series prices and individual contract prices for four currency futures, the British pound, the German mark, the Japanese yen, and the Swiss franc are analyzed. The results provide little evidence against the random walk null hypothesis, though non-randomness is documented in the Japanese yen. Additionally, the currency futures markets apparently become more efficient as markets mature over time.