Showing papers on "U-statistic published in 2011"
01 Jan 2011
TL;DR: The phenomenon of self-organized criticality (SOC) can be identified from many observations in the universe, by sampling statistical distributions of physical parameters, such as the distributions of time scales, spatial scales, or energies, for a set of events.
Abstract: The phenomenon of self-organized criticality (SOC) can be identified from many observations in the universe, by sampling statistical distributions of physical parameters, such as the distributions of time scales, spatial scales, or energies, for a set of events. SOC manifests itself in the statistics of nonlinear processes.
382 citations
TL;DR: The lens median (LM) as mentioned in this paper is a multivariate analogue of the univariate median, as the point where the lens median is maximised, which is the vector that is covered by the most number of hyper-lenses formed between any two sample observations.
Abstract: We define the lens depth (LD) function LD(t; F) of a vector t∈R d with respect to a distribution function F to be the probability that t is contained in a random hyper-lens formed by the intersection of two closed balls centred at two i.i.d observations from F. We show that LD is a statistical depth function and explore its properties, including affine invariance, symmetry, maximality at the centre and monotonicity. We define the sample LD and investigate its uniform consistency, asymptotic normality and computational complexity in high-dimensional settings. We define the lens median (LM), a multivariate analogue of the univariate median, as the point where the LD is maximised. The sample LM is the vector that is covered by the most number of hyper-lenses formed between any two sample observations. We state its asymptotic consistency and normality and examine its breakdown point and relative efficiency. The sample LM is robust and efficient for estimating the centre of a unimodal distribution. A compariso...
40 citations
TL;DR: The central limit theorem and the law of the iterated logarithm for U-quantiles are obtained as straightforward corollaries for strongly mixing random variables and functionals of absolutely regular sequences.
35 citations
TL;DR: In this article, the restricted almost unbiased ridge estimator (RAURE) based on the RRE by Sarkar et al. was introduced and compared with the corresponding competitors in literatures.
Abstract: Sarkar (1992) and Kac\i ranlar et al. (1999), respectively, proposed the restricted ridge regression estimator (RRE) and restricted Liu estimator (RLE) to combat the well-known multicollinearity problem in linear regression. In this article, the restricted almost unbiased ridge estimator (RAURE) based on the RRE by Sarkar (1992) and the restricted almost unbiased Liu estimator (RAULE) by Kac\i ranlar et al. (1999) are introduced. The biases and variance matrices of the proposed estimators are derived and compared with the corresponding competitors in literatures. Furthermore, a Monte Carlo evaluation of the estimators is given to illustrate some of the theoretical results.
24 citations
TL;DR: In this paper, the estimators are compared using Asympotic Expected Deficiency (AED) criterion leading to recommendation of uniform minimum variance unbiased estimators over maximum likelihood estimators for some measures.
Abstract: Maximum likelihood and uniform minimum variance unbiased estimators of steady-state probability distribution of system size, probability of at least l customers in the system in steady state, and certain steady-state measures of effectiveness in the M/M/1 queue are obtained/derived based on observations on X, the number of customer arrivals during a service time. The estimators are compared using Asympotic Expected Deficiency (AED) criterion leading to recommendation of uniform minimum variance unbiased estimators over maximum likelihood estimators for some measures.
21 citations
TL;DR: In this article, the authors introduced the restricted almost unbiased ridge regression estimator (RRI) and almost unbiased Liu estimator for the vector of parameters in a multiple linear regression model with linear restrictions.
Abstract: In this paper, the restricted almost unbiased ridge regression estimator and restricted almost unbiased Liu estimator are introduced for the vector of parameters in a multiple linear regression model with linear restrictions. The bias, variance matrices and mean square error (MSE) of the proposed estimators are derived and compared. It is shown that the proposed estimators will have smaller quadratic bias but larger variance than the corresponding competitors in literatures. However, they will respectively outperform the latter according to the MSE criterion under certain conditions. Finally, a simulation study and a numerical example are given to illustrate some of the theoretical results.
21 citations
TL;DR: In this paper, the authors considered the problem of unbiased estimation of a sparse nonrandom vector corrupted by additive white Gaussian noise and derived simple closed-form lower and upper bounds on the variance of LMVU estimators.
Abstract: The problem studied in this paper is unbiased estimation of a sparse nonrandom vector corrupted by additive white Gaussian noise. It is shown that while there are infinitely many unbiased estimators for this problem, none of them has uniformly minimum variance. Therefore, the focus is placed on locally minimum variance unbiased (LMVU) estimators. Simple closed-form lower and upper bounds on the variance of LMVU estimators or, equivalently, on the Barankin bound (BB) are derived. These bounds allow an estimation of the threshold region separating the low-signal-to-noise ratio (SNR) and high-SNR regimes, and they indicate the asymptotic behavior of the BB at high SNR. In addition, numerical lower and upper bounds are derived; these are tighter than the closed-form bounds and thus characterize the BB more accurately. Numerical studies compare the proposed characterizations of the BB with established biased estimation schemes, and demonstrate that while unbiased estimators perform poorly at low SNR, they may perform better than biased estimators at high SNR. An interesting conclusion of this analysis is that the high-SNR behavior of the BB depends solely on the value of the smallest nonzero entry of the sparse vector, and that this type of dependence is also exhibited by the performance of certain practical estimators.
19 citations
TL;DR: In this paper, the authors introduce new tests for normality which use as test statistics weighted L 1 -distances between the standard normal density and local U-statistics based on standardized observations and show that such test statistics converge at the root-n rate and determine their limit distributions as functionals of Gaussian processes.
16 citations
TL;DR: It is shown that the maximum depth estimators for the Gaussian and Gumbel copulas are biased, and they can be corrected and the new estimators are robust against contamination.
14 citations
TL;DR: In this paper, the authors established the asymptotic distribution for a class of multiple change point estimators in the following setup: a finite sequence of independent random variables consists of segments given by a known number of so-called change points such that the underlying distribution differs from segment to segment.
10 citations
TL;DR: In this article, the problem of unbiased estimation of P [X>Y] (=θ) for two independent random variables X and Y each following a two-parameter exponential distribution is considered and necessary and sufficient conditions for the existence of an unbiased estimator of θ are given.
Abstract: The problem considered is that of unbiased estimation of P [X>Y] (=θ) for two independent random variables X and Y each following a two-parameter exponential distribution. We give necessary and sufficient conditions for the existence of an unbiased estimator of θ based on arbitrary sets of order statistics obtained from independent random samples from the two populations and suggest unbiased estimators in some situations where unbiased estimators exist.
01 Jan 2011
TL;DR: In this article, the authors consider unbiased estimation of σ in a σ µ N population, and propose unbiased estimators consist of appropriate multiples of both the sample standard and the sample distribution.
Abstract: We consider unbiased estimation of σ in a () 2 , σ µ N population. Traditional unbiased estimators consist of appropriate multiples of both the sample standard
Journal Article•
TL;DR: In this paper, the authors show that the random group method with reimputation produces asymptotically unbiased and consistent variance estimators for estimated population totals for the case of no nonresponse.
Abstract: Random hot deck imputation is often applied to survey data with nonresponse. One of the popular methods for variance estimation without nonresponse is the random group method, which has to be adjusted when it is applied to imputed data. One such kind of adjustment is reimputing nonrespondents in each random group. We show that the random group method with reimputation produces asymptotically unbiased and consistent variance estimators for estimated population totals. As a special case of our general result, the random group variance estimator for the case of no nonresponse is asymptotically unbiased and consistent, a result that has not been documented although the random group method is frequently used in applications. We also show how to apply a shortcut random group method, which reduces the computational complexity due to reimputation, and establish the asymptotic unbiasedness and consistency of the resulting variance estimators.
TL;DR: In this paper, a class of tests of lack-of-fit of a parametric regression model when design is non-random and uniform on [0, 1] is discussed.
01 Jan 2011
TL;DR: The matrix A is a known n × p model matrix, the vector y is an observable ndimensional random vector, β is a p × 1 vector of unknown parameters, and ε is an unobservable vector of random errors with expectation E(ε) = 0, and covariance matrix cov(�) = σV, where σ > 0 is an unknown constant.
Abstract: where X is a known n × p model matrix, the vector y is an observable ndimensional random vector, β is a p × 1 vector of unknown parameters, and ε is an unobservable vector of random errors with expectation E(ε) = 0, and covariance matrix cov(ε) = σV, where σ > 0 is an unknown constant. The nonnegative definite (possibly singular) matrix V is known. In our considerations σ has no role and hence we may put σ = 1. As regards the notation, we will use the symbolsA,A,A, C (A), C (A), and N (A) to denote, respectively, the transpose, a generalized inverse, the Moore–Penrose inverse, the column space, the orthogonal complement of the column space, and the null space, of the matrix A. By (A : B) we denote the partitioned matrix with A and B as submatrices. By A we denote any matrix satisfying C (A) = N (A) = C (A). Furthermore, we will write PA = AA = A(AA)A to denote the orthogonal projector (with respect to the standard inner product) onto C (A). In particular, we denote H = PX and M = In −H. One choice for X ⊥ is of course the projector M.
Book•
10 Sep 2011
TL;DR: In this paper, a formal indexing of alternative specifications (e.g. shapes of error-distributions) by a nuisance parameter, and adaptation of admissibility and related concepts and Bayes techniques of the Neyman-Pearson and Wald theories are provided.
Abstract: : The present paper is intended to complement research in efficiency-robustness of estimators, by supplying formulations of concepts, techniques, and initial results for optimally efficiencyrobust estimators and tests in several types of problems. The present approach may be described as a formal indexing of alternative specifications (e.g. shapes of error-distributions) by a nuisance parameter, and adaptation of admissibility and related concepts and Bayes techniques of the Neyman-Pearson and Wald theories to the estimation and testing problems thus formulated. Specific problemsfor which new optimal efficiency-robust estimators are given are: linear estimation of location parameters; rank tests and related estimators for two-sample problems; and unbiased estimation. A by-product is a generalization of Stein's characterization of locallybest unbiased estimators to the class of admissible unbiased estimators together with the corresponding complete class theorem. (Author)
Posted Content•
TL;DR: The third constraint of spatial statistics is derived and it is shown that the classic generalized least-squares estimator of an unknown constant mean of the field is only an asymptotic disjunction of the numerical one.
Abstract: We constraint on computer the best linear unbiased generalized statistics of random field for the best linear unbiased generalized statistics of an unknown constant mean of random field and derive the numerical generalized least-squares estimator of an unknown constant mean of random field. We derive the third constraint of spatial statistics and show that the classic generalized least-squares estimator of an unknown constant mean of the field is only an asymptotic disjunction of the numerical one.
TL;DR: In this article, a sufficient condition for the uniqueness of multinomial sequential unbiased estimators is provided generalizing a classical result for binomial samples, and an application to clinical trials is presented.
Abstract: A sufficient condition for the uniqueness of multinomial sequential unbiased estimators is provided generalizing a classical result for binomial samples. Unbiased estimators are applied to infer the parameters of multidimensional or multinomial Random Walks which are observed until they reach a boundary. An application to clinical trials is presented.
Journal Article•
TL;DR: In this article, the distributions of rth order statistic of innid discrete random variables are obtained and the results related to pf and df of minimum and maximum of the random variables were given.
Abstract: In this study, the distributions of rth order statistic of innid discrete random variables are obtained. Then, the results related to pf and df of minimum and maximum of innid discrete random variables are given.
11 Dec 2011
TL;DR: This paper derives Monte Carlo simulation estimators to compute option price sensitivities under the SABR stochastic volatility model and uses the sensitivity of “vol of vol” as a showcase to demonstrate how to use the pathwise method to obtain unbiased estimators for the sensitivities.
Abstract: We derive Monte Carlo simulation estimators to compute option price sensitivities under the SABR stochastic volatility model. As a companion to the exact simulation method developed by Cai, Chen and Song (2011), this paper uses the sensitivity of "vol of vol" as a showcase to demonstrate how to use the pathwise method to obtain unbiased estimators to the price sensitivities under SABR. By appropriately conditioning on the path generated by the volatility, the evolution of the forward price can be represented as noncentral chi-square random variables with stochastic parameters. Combined with the technique of derivative of random variables, we can obtain fast and accurate unbiased estimators for the sensitivities.
01 Jan 2011
TL;DR: A modified U statistic is proposed for tied observations, which outperforms the other estimators by simulation studies and is recommended using the straightforward plug-in estimator for untied data, and using the modified U statistics when there are rounding errors.
Abstract: Association analyses are performed for two types of multivariate time-to-event data: multivariate clustered competing risks data and bivariate recurrent events data. In the first part, we extend the bivariate hazard ratio [Cheng and Fine, 2008] to multivariate competing risks data and show it is equivalent to the cause-specific cross hazard ratio in Cheng et al. [2010]. Two nonparametric approaches are proposed. One extends the plug-in estimator in Cheng and Fine [2008] and the other adapts the pseudo likelihood estimator for bivariate survival data [Clayton, 1978] to multivariate competing risks data. The asymptotic properties are established by using empirical process techniques. We compare the extended plug-in and pseudo likelihood estimators with the existing U statistic Cheng et al. [2010] by simulations and show that the three methods have comparable performance when no tied events exist. However, the plug-in estimator underestimate and the other two overestimate positive associations in the presence of rounding errors. Hence, we propose a modified U statistic for tied observations, which outperforms the other estimators by simulation studies. All methods are applied to the Cache County Study to examine mother-child and sibship associations in dementia among this aging population. The modified U essentially lies between the plug-in estimate and the original U statistic. We therefore recommend using the straightforward plug-in estimator for untied data, and using the modified U statistic when there are rounding errors. In the second part, bivariate recurrent events data are modeled by a compound Poisson process, whose dependence structure is then modeled by a Levy copula. When only the parameter of dependence structure is of primary interest, we proposed two methods to estimate the dependence parameter of the Levy copula. One uses Kendall's tau assuming the Clayton Levy copula while the other uses two-stage strategy to propose a semiparametric estimator. Consistency and asymptotic normality are also established. Simulation studies show that the proposed semi-parametric estimator is less efficient than the full likelihood estimator but superior to the nonparametric one. The proposed methods are also applied to Danish fire data to examine the relationship between loss to a building and loss to its contents.
TL;DR: In this article, non-iterative, distribution-free, and unbiased estimators of variance components by least squares method are derived for multivariate linear mixed model, where a general intercluster variance matrix, a same-member only general inter-response variance matrix and an uncorrelated intra-cluster error structure for each response are assumed.
TL;DR: In this paper, the unbiased estimator of unknown parametric function based on possion's population is discussed, and the expression of two classes of estimable function are given. But, it is found that the estimator cannot be direct constructed with parametric estimator.
Abstract: The unbiased estimator of unknown parametric function based on possion’s population is discussed, and the expression of two classes of estimable function are given. Applying Maclaurin's series, it’s proved that the unbiased estimator of the functions are exist, and utilizing the induction method , derived out the generalized expression of unbiased estimator. By means of comparing the unbiased estimator of the two classes of estimable functions, it’s found that the estimator cannot be direct constructed with parametric estimator.
TL;DR: In this paper, two parametric entropy estimators, the minimum variance unbiased estimator and the maximum likelihood estimator, for the lognormal distribution for a comparison of the properties of the two estimators are derived.
Abstract: This paper proposes two parametric entropy estimators, the minimum variance unbiased estimator and the maximum likelihood estimator, for the lognormal distribution for a comparison of the properties of the two estimators. The variances of both estimators are derived. The influence of the bias of the maximum likelihood estimator on estimation is analytically revealed. The distributions of the proposed estimators obtained by the delta approximation method are also presented. Performance comparisons are made with the two estimators. The following observations are made from the results. The MSE efficacy of the minimum variance unbiased estimator appears consistently high and increases rapidly as the sample size and variance, n and , become simultaneously small. To conclude, the minimum variance unbiased estimator outperforms the maximum likelihood estimator.
TL;DR: The finite-sample efficiency was the first notion of optimality introduced and it is still encountered in introductory statistics texts as discussed by the authors, however, the definition has several drawbacks however, one being that it is restricted to the class of unbiased estimators.
Abstract: Historically, finite-sample efficiency was the first notion of optimality introduced and it is still encountered in introductory statistics texts. The definition has several drawbacks however, one being that it is restricted to the class of unbiased estimators. An example is given to illustrate this.
TL;DR: In this paper, the authors deal with the estimation of the specific connectivity of a stationary random set in IRD and show that the "natural" estimator is only asymptotically unbiased.
Abstract: This paper deals with the estimation of the specific connectivity of a stationary random set in IRd. It turns out that the "natural" estimator is only asymptotically unbiased. The example of a boolean model of hypercubes illustrates the amplitude of the bias produced when the measurement field is relatively small with respect to the range of the random set. For that reason unbiased estimators are desired. Such an estimator can be found in the literature in the case where the measurement field is a right parallelotope. In this paper, this estimator is extended to apply to measurement fields of various shapes, and to possess a smaller variance. Finally an example from quantitative metallography (specific connectivity of a population of sintered bronze particles) is given.