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1.
Let X1,…,X7 be i.i.d. random variables with a common continuous distribution F, Two parameters, μ(F) = P(X1 < X5 and X1+X4 < X2+X3) and λ(F) = P(X1+X4 < X2+X3 and X1+X7 < X5+X6), which appear in the moments of some rank statistics have been studied by several authors. It is shown that the existing lower bound, 3/10 ≤ μ(F) can be improved to 3/10 < μ(F) and that no further improvement is possible. It is also shown that the existing upper bounds μ(F) ≤ (21/2+6)/24 ≈ 0.30893 and λ(F) ≤ 7/24 ≈ 0.29167 can be improved to [14+(2/3)1/2]/48 ≈ 0.30868 and {7 ? [1 ? (2/3)1/2]2/4}/24 ≈ 0.29132.  相似文献   

2.
Let X, Y and Z be independent random variables with common unknown distribution F. Using the Dirichlet process prior for F and squared erro loss function, the Bayes and empirical Bayes estimators of the parameters λ(F). the probability that Z > X + Y, are derived. The limiting Bayes estimator of λ(F) under some conditions on the parameter of the process is shown to be asymptotically normal. The aysmptotic optimality of the empirical Bayes estimator of λ(F) is established. When X, Y and Z have support on the positive real line, these results are derived for randomly right censored data. This problem relates to testing whether than used discussed by Hollander and Proshcan (1972) and Chen, Hollander and Langberg (1983).  相似文献   

3.
For polynomial regression over spherical regions the d-th order Ds-optimal designs for the λ-th order models are derived for 1 ≤ λ ≤ d ≤ 4. Efficiencies of these designs with respect to the λ-th order D-optimal designs are obtained. The effects of estimating addtional parameters due to an m-th order model (d ≥ m >>λ) on the efficiencies are investigated.  相似文献   

4.
Improved lower bounds are obtained for two parameters λ(F) and μ(F) which occur in the variances of linear rank statistics studied by Doksum, Hollander, and Lehmann. Simplified algorithms are developed for efficient estimation of the parameters. The tighter bounds are used to investigate a slightly conservative approach which eliminates lengthy estimation procedures altoget-her, her, with great computational savings.  相似文献   

5.
Shiue and Bain proposed an approximate F statistic for testing equality of two gamma distribution scale parameters in presence of a common and unknown shape parameter. By generalizing Shiue and Bain's statistic we develop a new statistic for testing equality of L >= 2 gamma distribution scale parameters. We derive the distribution of the new statistic ESP for L = 2 and equal sample size situation. For other situations distribution of ESP is not known and test based on the ESP statistic has to be performed by using simulated critical values. We also derive a C(α) statistic CML and develop a likelihood ratio statistic, LR, two modified likelihood ratio statistics M and MLB and a quadratic statistic Q. The distribution of each of the statistics CML, LR, M, MLB and Q is asymptotically chi-square with L - 1 degrees of freedom. We then conducted a monte-carlo simulation study to compare the perfor- mance of the statistics ESP, LR, M, MLB, CML and Q in terms of size and power. The statistics LR, M, MLB and Q are in general liberal and do not show power advantage over other statistics. The statistic CML, based on its asymptotic chi-square distribution, in general, holds nominal level well. It is most powerful or nearly most powerful in most situations and is simple to use. Hence, we recommend the statistic CML for use in general. For better power the statistic ESP, based on its empirical distribution, is recommended for the special situation for which there is evidence in the data that λ1 < … < λL and n1 < … < nL, where λ1 …, λL are the scale parameters and n1,…, nL are the sample sizes.  相似文献   

6.
Given λ∈(0-,l), let xλ(F) denote the unique λ-quantile of the distribution F. A distribution-free median-unbiased estimator of xλ(F) is explicitly constructed  相似文献   

7.
Robustness of group divisible (GD) designs is investigated, when one block is lost, in terms of efficiency of the residual design. The exact evaluation of the efficiency can be made for singular GD and semi-regular GD designs as ell as regular GD designs with λ1 = 0. In a regular GD design with λ1 > 0, the efficiency may depend upon the lost block and sharp upper and lower bounds on the efficiency are presented. The investigation shows that GD designs are fairly robust in terms of efficiency. As a special case, we can also show the robustness of balanced incomplete block design when one block is lost.  相似文献   

8.
Given an inverse Gaussian distribution I(.μ,a2μ) with known coefficient of variation a, the hypothesis HO: .μ <ce:glyph name="dbnd6"/> μo is tested against H1: μ <ce:glyph name="dbnd6"/> μ1 using the sequential probability ratio test. The maximum of the expected sample number is shown to occur when μ is approximately equal to the geometric mean of μoand μ1 and it is shown that this maximum value depends on .μo and μ1 only through their ratio. It is observed that the test can be used to discriminate between two one-sided hypotheses.  相似文献   

9.
A single unit system supported by N-l inactive standbys and a repair facility is considered when λ,μ are unknown, λand μbeing failure and repair rates of the unit respectively. Three sampling schemes are considered to obtain moment estimators λ?:and μ ?:when the performance of the unit have to be observed in the system only. Asymptotic variances of the estimates are supplied.  相似文献   

10.
We study the r-content Δ of the r -simplex generated by r+ 1 independent random points in R”. Each random point Zj is isotropic and distributed according to λ||Zj||2 ~ beta-type-2(n/2, v), λ, v > 0. We provide an asymptotic normality result which is analogous to the conjecture made by Miles (1971). A method is introduced to work out the exact density of W = (rλ)r(r!Δ)2/(r + |)r+l and hence that of Δ. The distribution of W is also related to some hypothesis-testing problems in multivariate analysis. Furthermore, by using this method, the distribution of W or Δ can easily be simulated.  相似文献   

11.
The CUSUM control chart proposed by Page is a widely used in monitoring the quality of manufacturing processes. The Shiryayev-Roberts (S-R) control chart due to Shiryayev (1963) and Roberts (1988) is one of its competitors, This paper is concerned with the distribution properties of the run lengths of these two control charts. In context of continuous time, we first give the expansions of the higher moments of these run lengths. Then, we show that the asymptotic distributions of these run lengths are either some exponential distributions, or the distribution of the suprema of a standard Brownian motion, or some normal distributions, according to whether the μ<δ/2,μ =δ/2 and μ>δ/2. Here δ is the reference value of the above charts. Some similar results are also obtained in the context of discrete time.  相似文献   

12.
The Steffensen inequality is applied to derive quantile bounds for the expectations of generalized order statistics from a distribution belonging to a particular subclass of distributions. The subclass consists of F having the property that F?1(0+)=x0>0 and that x →[1? F(x)]xz is nonincreasing for all x > X0 and some z > 0.  相似文献   

13.
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15.
Consider a one-way layout of the analysis of variance assuming independence, normality, and homogeneity of variance. Test the null hypothesis Ho that the means, j., of each of Amp; columns, i = 1,…, k are equal versus the alternative that they follow an umbrella pattern. That is, the alternative is H1-H0 where H1: μ1> μ2>… > μk, and m is known. We derive a class of tests which are unbiased and lie in a nontrivial complete class. We recommend specific tests within the class. A simulation of the power functions of some tests is contrasted with the simulated power function of the likelihood ratio test.  相似文献   

16.
It is well known that a Bayesian credible interval for a parameter of interest is derived from a prior distribution that appropriately describes the prior information. However, it is less well known that there exists a frequentist approach developed by Pratt (1961 Pratt , J. W. ( 1961 ). Length of confidence intervals . J. Amer. Statist. Assoc. 56 : 549657 .[Taylor & Francis Online], [Web of Science ®] [Google Scholar]) that also utilizes prior information in the construction of frequentist confidence intervals. This frequentist approach produces confidence intervals that have minimum weighted average expected length, averaged according to some weight function that appropriately describes the prior information. We begin with a simple model as a starting point in comparing these two distinct procedures in interval estimation. Consider X 1,…, X n that are independent and identically N(μ, σ2) distributed random variables, where σ2 is known, and the parameter of interest is μ. Suppose also that previous experience with similar data sets and/or specific background and expert opinion suggest that μ = 0. Our aim is to: (a) develop two types of Bayesian 1 ? α credible intervals for μ, derived from an appropriate prior cumulative distribution function F(μ) more importantly; (b) compare these Bayesian 1 ? α credible intervals for μ to the frequentist 1 ? α confidence interval for μ derived from Pratt's frequentist approach, in which the weight function corresponds to the prior cumulative distribution function F(μ). We show that the endpoints of the Bayesian 1 ? α credible intervals for μ are very different to the endpoints of the frequentist 1 ? α confidence interval for μ, when the prior information strongly suggests that μ = 0 and the data supports the uncertain prior information about μ. In addition, we assess the performance of these intervals by analyzing their coverage probability properties and expected lengths.  相似文献   

17.
In this paper, Fisher information matrix about the five parameters ρ, μ:1, μ2, λ1and λ2of a mixture of two Inverse Gaussian density functions is obtained. The Leguerre-Gauss quadrature formula is used to evaluate the essential integral on which the twenty five elements of the information matrix are based. Results involving the computation of the information about p are compared with those involving both the power series expansion and Simpson's method of integration. Laguerre-Gauss quadra-ture was found to lead to good approximations as compared with other methods. It was therefore chosen for the computations of the elements of the information matrix.  相似文献   

18.
In this article, we consider the product-limit quantile estimator of an unknown quantile function under a censored dependent model. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate O[(log n)] for some λ > 0. The strong Gaussian approximation of the product-limit process is then applied to derive the laws of the iterated logarithm for product-limit process.  相似文献   

19.
20.
Let X= (X1,…, Xk)’ be a k-variate (k ≥ 2) normal random vector with unknown population mean vector μ = (μ1 ,…, μk)’ and covariance matrix Σ of order k and let μ[1] ≤ … ≤ μ[k] be the ordered values of the μ ’ s. No prior knowledge of the pairing of the μ[i] with the Xj. (or μ[i] with the σj 2) is assumed for any i and j (1 ≤ i, j ≤ k). Based on a random sample of N independent vector observations on X, this paper considers both upper and lower (one-sided) and two-sided 100γ% (0 < γ < 1) confidence intervals for μ[k] and μ[1], the largest and the smallest mean, respectively, when Σ is known and when Σ is equal to σ2R with common unknown variance σ2 > 0 and correlation matrix R known, respectively. An optimum two-sided confidence interval via finding the shortest length from this class is also considered. Necessary tables and computer program to actually apply these procedures are provided.  相似文献   

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