共查询到20条相似文献,搜索用时 46 毫秒
1.
Let {Xn,n?1} be a sequence of independent identically distributed random variables, taking nonnegative integer values. An observation Xn is a tie for the maximum if Xn=max{X1,…,Xn-1}. In this paper, we obtain weak and strong laws of large numbers and central limit theorems for the cumulative number of ties for the maximum among the first n observations. 相似文献
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Consider the model where there are I independent multivariate normal treatment populations with p×1 mean vectors μi, i=1,…,I, and covariance matrix Σ. Independently the (I+1)st population corresponds to a control and it too is multivariate normal with mean vector μI+1 and covariance matrix Σ. Now consider the following two multiple testing problems. 相似文献
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For the class of stationary Gaussian long memory processes, we study some properties of the least-squares predictor of Xn+1 based on (Xn,…,X1). The predictor is obtained by projecting Xn+1 onto the finite past and the coefficients of the predictor are estimated on the same realisation. First we prove moment bounds for the inverse of the empirical covariance matrix. Then we deduce an asymptotic expression of the mean-squared error. In particular we give a relation between the number of terms used to estimate the coefficients and the number of past terms used for prediction, which ensures the L2- sense convergence of the predictor. Finally we prove a central limit theorem when our predictor converges to the best linear predictor based on all the past. 相似文献
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We consider density estimation for a smooth stationary process Xt, t∈R, based on a discrete sample Yi=XΔi, i=0,…,n=T/Δ. By a suitable interpolation scheme of order p , we augment data to form an approximation Xp,t, t∈[0,T], of the continuous-time process and base our density estimate on the augmented sample path. Our results show that this can improve the rate of convergence (measured in terms of n) of the density estimate. Among other things, this implies that recording n observations using a small Δ can be more efficient than recording n independent observations. 相似文献
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Consider a sequence of dependent random variables X1,X2,…,Xn, where X1 has distribution F (or probability measure P ), and the distribution of Xi+1 given X1,…,Xi and other covariates and environmental factors depends on F and the previous data, i=1,…,n-1. General repair models give rise to such random variables as the failure times of an item subject to repair. There exist nonparametric non-Bayes methods of estimating F in the literature, for instance, Whitaker and Samaniego [1989. Estimating the reliability of systems subject to imperfect repair. J. Amer. Statist. Assoc. 84, 301–309], Hollander et al. [1992. Nonparametric methods for imperfect repair models. Ann. Statist. 20, 879–896] and Dorado et al. [1997. Nonparametric estimation for a general repair model. Ann. Statist. 25, 1140–1160], etc. Typically these methods apply only to special repair models and also require repair data on N independent items until exactly only one item is left awaiting a “perfect repair”. 相似文献
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Consider a sequence of independent and identically distributed random variables {Xi,i?1} with a common absolutely continuous distribution function F . Let X1:n?X2:n???Xn:n be the order statistics of {X1,X2,…,Xn} and {Yl,l?1} be the sequence of record values generated by {Xi,i?1}. In this work, the conditional distribution of Yl given Xn:n is established. Some characterizations of F based on record values and Xn:n are then given. 相似文献
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José Antonio Moler Fernando Plo Miguel San Miguel 《Journal of statistical planning and inference》2007
We study a randomized adaptive design to assign one of the L treatments to patients who arrive sequentially by means of an urn model. At each stage n, a reward is distributed between treatments. The treatment applied is rewarded according to its response, 0?Yn?1, and 1-Yn is distributed among the other treatments according to their performance until stage n-1. Patients can be classified in K+1 levels and we assume that the effect of this level in the response to the treatments is linear. We study the asymptotic behavior of the design when the ordinary least square estimators are used as a measure of performance until stage n-1. 相似文献
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Let X=(X1,X2,…,Xn) be an exchangeable random vector, and denote X1:i=min{X1,X2,…,Xi} and Xi:i=max{X1,X2,…,Xi}, 1?i?n. These order statistics represent the lifetimes of the series and the parallel systems, respectively, with component lifetimes Xi. In this paper we obtain conditions under which X1:i (or Xi:i) decreases (increases) in i in the likelihood ratio (lr) order. An even more general result involving general (that is, not necessary exchangeable) random vectors is also derived for general series (or parallel) systems. We show that the series (parallel) systems are not necessarily lr-ordered even if the components are independent. 相似文献
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Joseph P.S. Kung Anna de Mier Xinyu Sun Catherine Yan 《Journal of statistical planning and inference》2009
We consider paths in the plane with (1,0), (0,1), and (a,b)-steps that start at the origin, end at height n, and stay strictly to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most b/a, then the ordinary generating function for the number of such paths ending at height n is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or “umbral” generating function, in which the power zn is replaced by a power series of the form znφn(z), where φn(0)=1. Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic. We give several concrete examples, including an alternative way to solve the tennis ball problem. 相似文献
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Let π1,π2,…,πp be p independent Poisson populations with means λ1,…,λp, respectively. Let {X1,…,Xp} denote the set of observations, where Xi is from πi. Suppose a subset of populations is selected using Gupta and Huang's (1975) selection rule which selects πi if and only if Xi+1?cX(1), where X(1)=max{X1,…,Xp}, and 0<c<1. In this paper, the simultaneous estimation of the Poisson means associated with the selected populations is considered for the k-normalized squared error loss function. It is shown that the natural estimator is positively biased. Also, a class of estimators that are better than the natural estimator is obtained by solving certain difference inequalities over the sample space. A class of estimators which dominate the UMVUE is also obtained. Monte carlo simulations are used to assess the percentage improvements and an application to a real-life example is also discussed. 相似文献
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Consider a mixture problem consisting of k classes. Suppose we observe an s-dimensional random vector X whose distribution is specified by the relations P(X∈A|Y=i)=Pi(A), where Y is an unobserved class identifier defined on {1,…,k}, having distribution P(Y=i)=pi. Assuming the distributions Pi having a common covariance matrix, elegant identities are presented that connect the matrix of Fisher information in Y on the parameters p1,…,pk, the matrix of linear information in X, and the Mahalanobis distances between the pairs of P 's. Since the parameters are not free, the information matrices are singular and the technique of generalized inverses is used. A matrix extension of the Mahalanobis distance and its invariant forms are introduced that are of interest in their own right. In terms of parameter estimation, the results provide an independent of the parameter upper bound for the loss of accuracy by esimating p1,…,pk from a sample of X′s, as compared with the ideal estimator based on a random sample of Y′s. 相似文献
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This paper introduces a median estimator of the logistic regression parameters. It is defined as the classical L1-estimator applied to continuous data Z1,…,Zn obtained by a statistical smoothing of the original binary logistic regression observations Y1,…,Yn. Consistency and asymptotic normality of this estimator are proved. A method called enhancement is introduced which in some cases increases the efficiency of this estimator. Sensitivity to contaminations and leverage points is studied by simulations and compared in this manner with the sensitivity of some robust estimators previously introduced to the logistic regression. The new estimator appears to be more robust for larger sample sizes and higher levels of contamination. 相似文献
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We consider a linear regression model with regression parameter β=(β1,…,βp) and independent and identically N(0,σ2) distributed errors. Suppose that the parameter of interest is θ=aTβ where a is a specified vector. Define the parameter τ=cTβ-t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that τ=0. We present a new frequentist 1-α confidence interval for θ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about τ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×2 factorial experiment with 20 replicates. Suppose that the parameter of interest θ is a specified simple effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θ that utilizes this prior information. 相似文献
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For a random sample of size n from an absolutely continuous random vector (X,Y), let Yi:n be ith Y-order statistic and Y[j:n] be the Y-concomitant of Xj:n. We determine the joint pdf of Yi:n and Y[j:n] for all i,j=1 to n, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for Y[j:n]. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of Yi:n and Y[j:n]. For the bivariate normal case, we compute the expectations of the product of Yi:n and Y[i:n] for n=2 to 8 for selected values of the correlation coefficient and illustrate their uses. 相似文献