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1.
ABSTRACT

Consider k(≥ 2) independent exponential populations Π1, Π2, …, Π k , having the common unknown location parameter μ ∈ (?∞, ∞) (also called the guarantee time) and unknown scale parameters σ1, σ2, …σ k , respectively (also called the remaining mean lifetimes after the completion of guarantee times), σ i  > 0, i = 1, 2, …, k. Assume that the correct ordering between σ1, σ2, …, σ k is not known apriori and let σ[i], i = 1, 2, …, k, denote the ith smallest of σ j s, so that σ[1] ≤ σ[2] ··· ≤ σ[k]. Then Θ i  = μ + σ i is the mean lifetime of Π i , i = 1, 2, …, k. Let Θ[1] ≤ Θ[2] ··· ≤ Θ[k] denote the ranked values of the Θ j s, so that Θ[i] = μ + σ[i], i = 1, 2, …, k, and let Π(i) denote the unknown population associated with the ith smallest mean lifetime Θ[i] = μ + σ[i], i = 1, 2, …, k. Based on independent random samples from the k populations, we propose a selection procedure for the goal of selecting the population having the longest mean lifetime Θ[k] (called the “best” population), under the subset selection formulation. Tables for the implementation of the proposed selection procedure are provided. It is established that the proposed subset selection procedure is monotone for a general k (≥ 2). For k = 2, we consider the loss measured by the size of the selected subset and establish that the proposed subset selection procedure is minimax among selection procedures that satisfy a certain probability requirement (called the P*-condition) for the inclusion of the best population in the selected subset.  相似文献   

2.
Abstract

Let the data from the ith treatment/population follow a distribution with cumulative distribution function (cdf) F i (x) = F[(x ? μ i )/θ i ], i = 1,…, k (k ≥ 2). Here μ i (?∞ < μ i  < ∞) is the location parameter, θ i i  > 0) is the scale parameter and F(?) is any absolutely continuous cdf, i.e., F i (?) is a member of location-scale family, i = 1,…, k. In this paper, we propose a class of tests to test the null hypothesis H 0 ? θ1 = · = θ k against the simple ordered alternative H A  ? θ1 ≤ · ≤ θ k with at least one strict inequality. In literature, use of sample quasi range as a measure of dispersion has been advocated for small sample size or sample contaminated by outliers [see David, H. A. (1981). Order Statistics. 2nd ed. New York: John Wiley, Sec. 7.4]. Let X i1,…, X in be a random sample of size n from the population π i and R ir  = X i:n?r  ? X i:r+1, r = 0, 1,…, [n/2] ? 1 be the sample quasi range corresponding to this random sample, where X i:j represents the jth order statistic in the ith sample, j = 1,…, n; i = 1,…, k and [x] is the greatest integer less than or equal to x. The proposed class of tests, for the general location scale setup, is based on the statistic W r  = max1≤i<jk (R jr /R ir ). The test is reject H 0 for large values of W r . The construction of a three-decision procedure and simultaneous one-sided lower confidence bounds for the ratios, θ j i , 1 ≤ i < j ≤ k, have also been discussed with the help of the critical constants of the test statistic W r . Applications of the proposed class of tests to two parameter exponential and uniform probability models have been discussed separately with necessary tables. Comparisons of some members of our class with the tests of Gill and Dhawan [Gill A. N., Dhawan A. K. (1999). A One-sided test for testing homogeneity of scale parameters against ordered alternative. Commun. Stat. – Theory and Methods 28(10):2417–2439] and Kochar and Gupta [Kochar, S. C., Gupta, R. P. (1985). A class of distribution-free tests for testing homogeneity of variances against ordered alternatives. In: Dykstra, R. et al., ed. Proceedings of the Conference on Advances in Order Restricted Statistical Inference at Iowa city. Springer Verlag, pp. 169–183], in terms of simulated power, are also presented.  相似文献   

3.
ABSTRACT

Suppose independent random samples are available from k(k ≥ 2) exponential populations ∏1,…,∏ k with a common location θ and scale parameters σ1,…,σ k , respectively. Let X i and Y i denote the minimum and the mean, respectively, of the ith sample, and further let X = min{X 1,…, X k } and T i  = Y i  ? X; i = 1,…, k. For selecting a nonempty subset of {∏1,…,∏ k } containing the best population (the one associated with max{σ1,…,σ k }), we use the decision rule which selects ∏ i if T i  ≥ c max{T 1,…,T k }, i = 1,…, k. Here 0 < c ≤ 1 is chosen so that the probability of including the best population in the selected subset is at least P* (1/k ≤ P* < 1), a pre-assigned level. The problem is to estimate the average worth W of the selected subset, the arithmetic average of means of selected populations. In this article, we derive the uniformly minimum variance unbiased estimator (UMVUE) of W. The bias and risk function of the UMVUE are compared numerically with those of analogs of the best affine equivariant estimator (BAEE) and the maximum likelihood estimator (MLE).  相似文献   

4.
For XN p (μ, Σ) testing H o:Σ = Σ 0, with Σ 0 known, relies at present on an approximation of the null-distribution of the likelihood ratio statistic.

We present here the exact null distribution and also its computation, hence providing a precise tool that can be used in small sample cases.  相似文献   

5.
Consider k independent random samples with different sample sizes such that the ith sample comes from the cumulative distribution function (cdf) F i  = 1 ? (1 ? F)α i , where α i is a known positive constant and F is an absolutely continuous cdf. Also, suppose that we have observed the maximum and minimum of the first k samples. This article shows how one can construct the nonparametric prediction intervals for the order statistics of the future samples on the basis of these information. Three schemes are studied and in each case exact expressions for the prediction coefficients of prediction intervals are derived. Numerical computations are given for illustrating the results. Also, a comparison study is done while the complete samples are available.  相似文献   

6.
7.
Progressively Type-II censored conditionally N-ordered statistics (PCCOS-N) arising from iid random vectors Xi = (X1i, X2i, …, Xip), i = 1, 2…, n, were investigated by Bairamov (2006 Bairamov, I. (2006). Progressive Type II censored order statistics for multivariate observations. J. Mult. Anal. 97:797809.[Crossref], [Web of Science ®] [Google Scholar]), with respect to the magnitudes of N(Xi), i = 1, 2, …, n, where N( · ) is a p-variate measurable function defined on the support set of X1 satisfying certain regularity conditions and N(Xi) denotes the lifetime of the random vector Xi, i = 1, …, n. Under the PCCOS-N sampling scheme, n independent units are placed on a life-test and after the ith failure, Ri (i = 1, …, m) of the surviving units are removed at random from the remaining observations. In this article, we consider PCCOS-N arising from a vector with identical as well as non identical dependent components, jointly distributed according to a unified elliptically contoured copula (PCCOSDUECC-N). Results established here contain the previous results as particular cases. Illustrative examples and simulation studies show that PCCOSDUECC-N enables us to analyze the lifetime of several systems, including repairable systems and systems with standby components, more efficiently than PCCOS-N.  相似文献   

8.
A new procedure for testing the H 0: μ1 = ··· = μ k against the alternative H u 1 ≥ ··· ≥μ r  ≤ ··· ≤ μ k with at least one strict inequality, where μ i is the location parameter of the ith two-parameter exponential distribution, i = 1,…, k, is proposed. Exact critical constants are computed using a recursive integration algorithm. Tables containing these critical constants are provided to facilitate the implementation of the proposed test procedure. Simultaneous confidence intervals for certain contrasts of the location parameters are derived by inverting the proposed test statistic. In comparison to existing tests, it is shown, by a simulation study, that the new test statistic is more powerful in detecting U-shaped alternatives when the samples are derived from exponential distributions. As an extension, the use of the critical constants for comparing Pareto distribution parameters is discussed.  相似文献   

9.
Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ2 ? μ1)/σ, where μ1 and μ2 are the component means and σ2 is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)2 × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.  相似文献   

10.
In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.  相似文献   

11.
12.
Suppose we have {(x i , y i )} i = 1, 2,…, n, a sequence of independent observations. We wish to find approximate 1 ? α simultaneous confidence bands for the regression curve. Many previous confidence bands in the literature have practical difficulties. In this article, the local linear smoother is used to estimate the regression curve. The bias of the estimator is considered. Different methods of constructing confidence bands are discussed. Finally, a possible method incorporating logistic regression in an innovative way is proposed to construct the bands for random designs. Simulations are used to study the performance or properties of the methods. The procedure for constructing confidence bands is entirely data-driven. The advantage of the proposed method is that it is simple to use and can be applied to random designs. It can be considered as a practically useful and efficient method.  相似文献   

13.
《随机性模型》2013,29(1):31-42
Abstract

We give a sufficient condition for the exponential decay of the tail of a discrete probability distribution π = (π n ) n≥0 in the sense that lim n→∞(1/n) log∑ i>n π i  = ?θ with 0 < θ < ∞. We focus on analytic properties of the probability generating function of a discrete probability distribution, especially, the radius of convergence and the number of poles on the circle of convergence. Furthermore, we give an example of an M/G/1 type Markov chain such that the tail of its stationary distribution does not decay exponentially.  相似文献   

14.
i , i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ i , i = 1, 2, ..., k and common known scale parameter σ. Let Y i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π i is selected iff Y i Y (1)d, where Y (1) is the largest of the Y i 's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn −2). As a special case, we obtain the coresponding results for the estimation of θ(1), the parameter associated with Y (1). Received: January 6, 1998; revised version: July 11, 2000  相似文献   

15.
ABSTRACT

This article deals with a distribution associated with a pure birth process starting with no individuals, with birth rates λ n  = λ for n = 0, 2,…, m ? 1 and λ n  = μ for n ≥ m. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions, and others. Using this representation, the kth factorial moments of the distribution is obtained. Some other forms of this distribution are also given.  相似文献   

16.
Suppose exponential populations πi with parameters (μii) (i = 1, 2, …, K) are given. The σi can be unknown and unequal. This article discusses how to select the k (≥1) best populations. Under the subset selection formulation, a one-stage procedure is proposed. Under the indifference zone formulation, a two-stage procedure is proposed. An appealing feature of these procedures is that no statistical tables are needed for their implementation.  相似文献   

17.
In this article, we consider a partially linear single-index model Y = g(Z τθ0) + X τβ0 + ? when the covariate X may be missing at random. We propose weighted estimators for the unknown parametric and nonparametric part by applying weighted estimating equations. We establish normality of the estimators of the parameters and asymptotic expansion for the estimator of the nonparametric part when the selection probabilities are unknown. Simulation studies are also conducted to illustrate the finite sample properties of these estimators.  相似文献   

18.
Suppose particles are randomly distributed in a certain medium, powder or liquid, which is conceptually divided into N cells. Let pi denote the probability that a particle falls in the ith cell and Yi denote the number of particles in the ith cell. Assume that the joint probability function of the Yi follows a multinomial distribution with cell probabilities pi respectively. Take n (≤N) cells at random without replacement and put each of the cells separately through a mixing mechanism of dilution and swirl. These n cells constitute the first stage samples and the number of particles in these cells are not observable. Now conceptually divide each of n cells into M subcells of equal size and let Xij denote the number of particles in the jth subcell of the ith cell selected in the first stage; i=1,2,…,N and j=1,2,…,M. Consequently assume that the conditional joint probability function of the Xij given Yi=yi follows a multinomial distribution with equal cell probabilities. Now take m (≤M) subcells at random from each of the cells selected in the first stage sample. Assume that the numbers of particles in M×N subcells are observable. The properties of the estimator of the particle density per sample unit are investigated under the modified two-stage cluster sampling method. A laboratory experiment for Xanthan Gum Products is analyzed in order to examine the appropriateness of the model assumed in this paper.  相似文献   

19.
For each n, k ∈ ?, let Y i  = (Y i1, Y i2,…, Y ik ), 1 ≤ i ≤ n be independent random vectors in ? k with finite third moments and Y ij are independent for all j = 1, 2,…, k. In this article, we use the Stein's technique to find constants in uniform bounds for multidimensional Berry-Esseen inequality on a closed sphere, a half plane and a rectangular set.  相似文献   

20.
Consider observations (representing lifelengths) taken on a random field indexed by lattice points. Estimating the distribution function F(x) = P(X i  ≤ x) is an important problem in survival analysis. We propose to estimate F(x) by kernel estimators, which take into account the smoothness of the distribution function. Under some general mixing conditions, our estimators are shown to be asymptotically unbiased and consistent. In addition, the proposed estimator is shown to be strongly consistent and sharp rates of convergence are obtained.  相似文献   

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