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1.
For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  θ P θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.  相似文献   

2.
ABSTRACT

This article considers the problem of choosing between two possible treatments which are each modeled with a Poisson distribution. Win-probabilities are defined as the probabilities that a single potential future observation from one of the treatments will be better than, or at least as good as, a potential future observation from the other treatment. Using historical data from the two treatments, it is shown how estimates and confidence intervals can be constructed for the win-probabilities. Extensions to situations with three or more treatments are also discussed. Some examples and illustrations are provided, and the relationship between this methodology and standard inference procedures on the Poisson parameters is discussed.  相似文献   

3.
In this article, we establish some new results on stochastic comparisons of the maxima of two heterogenous gamma variables with different shape and scale parameters. Let X1 and X2 [X*1 and X*2] be two independent gamma variables with Xi?[X*i] having shape parameter ri?[r*i] and scale parameter λi?[λ*i], i = 1, 2. It is shown that the likelihood ratio order holds between the maxima, X2: 2 and X*2: 2 when λ1 = λ*1 ? λ2 = λ*2 and r1 ? r*1 ? r2 = r*2. We also prove that, if ri, r*i ∈ (0, 1], (r1, r2) majorizes (r*1, r2*), and (λ1, λ2) is p-larger than (λ*1, λ2*), then X2: 2 is larger than X*2: 2 in the sense of the hazard rate order [dispersive order]. Some numerical examples are provided to illustrate the main results. The new results established here strengthen and generalize some of the results known in the literature.  相似文献   

4.
In this article we study the effect of truncation on the performance of an open vector-at-a-time sequential sampling procedure (P* B) proposed by Bechhofer, Kiefer and Sobel , for selecting the multinomial event which has the largest probability. The performance of the truncated version (P* B T) is compared to that of the original basic procedure (P* B). The performance characteristics studied include the probability of a correct selection, the expected number of vector-observations (n) to terminate sampling, and the variance of n. Both procedures guarantee the specified probability of a correct selection. Exact results and Monte Carlo sampling results are obtained. It is shown that P* B Tis far superior to P* B in terms of E{n} and Var{n}, particularly when the event probabilities are equal.The performance of P* B T is also compared to that of a closed vector-at-a-time sequential sampling procedure proposed for the same problem by Ramey and Alam; this procedure has here to fore been claimed to be the best one for this problem. It is shown that p* B T is superior to the Ramey-Alam procedure for most of the specifications of practical interest.  相似文献   

5.
Let (X 1, X 2) be a bivariate L p -norm generalized symmetrized Dirichlet (LpGSD) random vector with parameters α12. If p12=2, then (X 1, X 2) is a spherical random vector. The estimation of the conditional distribution of Z u *:=X 2 | X 1>u for u large is of some interest in statistical applications. When (X 1, X 2) is a spherical random vector with associated random radius in the Gumbel max-domain of attraction, the distribution of Z u * can be approximated by a Gaussian distribution. Surprisingly, the same Gaussian approximation holds also for Z u :=X 2| X 1=u. In this paper, we are interested in conditional limit results in terms of convergence of the density functions considering a d-dimensional LpGSD random vector. Stating our results for the bivariate setup, we show that the density function of Z u * and Z u can be approximated by the density function of a Kotz type I LpGSD distribution, provided that the associated random radius has distribution function in the Gumbel max-domain of attraction. Further, we present two applications concerning the asymptotic behaviour of concomitants of order statistics of bivariate Dirichlet samples and the estimation of the conditional quantile function.  相似文献   

6.
7.
Well-known nonparametric confidence intervals for quantiles are of the form (X i : n , X j : n ) with suitably chosen order statistics X i : n and X j : n , but typically their coverage levels differ from those prescribed. It appears that the coverage level of the confidence interval of the form (X i : n , X j : n ) with random indices I and J can be rendered equal, exactly to any predetermined level γ?∈?(0, 1). Best in the sense of minimum E(J???I), i.e., ‘the shortest’, two-sided confidence intervals are constructed. If no two-sided confidence interval exists for a given γ, the most accurate one-sided confidence intervals are constructed.  相似文献   

8.
Suppose that we have a nonparametric regression model Y = m(X) + ε with XRp, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x0) for fixed x0Rp are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x0). We also construct an empirical likelihood (EL) statistic for m(x0) with limiting distribution of χ21, which is used to construct an EL confidence interval for m(x0).  相似文献   

9.
In survival data analysis it is frequent the occurrence of a significant amount of censoring to the right indicating that there may be a proportion of individuals in the study for which the event of interest will never happen. This fact is not considered by the ordinary survival theory. Consequently, the survival models with a cure fraction have been receiving a lot of attention in the recent years. In this article, we consider the standard mixture cure rate model where a fraction p 0 of the population is of individuals cured or immune and the remaining 1 ? p 0 are not cured. We assume an exponential distribution for the survival time and an uniform-exponential for the censoring time. In a simulation study, the impact caused by the informative uniform-exponential censoring on the coverage probabilities and lengths of asymptotic confidence intervals is analyzed by using the Fisher information and observed information matrices.  相似文献   

10.
Let X = (Xj : j = 1,…, n) be n row vectors of dimension p independently and identically distributed multinomial. For each j, Xj is partitioned as Xj = (Xj1, Xj2, Xj3), where pi is the dimension of Xji with p1 = 1,p1+p2+p3 = p. In addition, consider vectors Yji, i = 1,2j = 1,…,ni that are independent and distributed as X1i. We treat here the problem of testing independence between X11 and X13 knowing that X11 and X12 are uncorrected. A locally best invariant test is proposed for this problem.  相似文献   

11.
A random vector X = (X 1,…,X n ) is negatively associated if and only if for every pair of partitions X 1 = (X π(1),…,X π(k)), X 2 = (X π(k+1),…,X π(n)) of X , P( X 1 ? A, X 2 ? B) ≤ P( X 1 ? A)P( X 2 ? B) whenever A and B are open upper sets and π is any permutation of {1,…,n}. In this paper, we develop some of concepts of negative dependence, which are weaker than negative association but stronger than negative orthant dependence by requiring the above inequality to hold only for some upper sets A and B and applying the arguments in Shaked.  相似文献   

12.
Let X1,X2,…,Xm be distributed normally with mean μ and variance σ2 X; Let Y1,Y2,…,Yn be distributed normally with mean μ and variance σ2 Y; let X1,X2,…,Xm,Y1,Y2,…,Yn be jointly independent. There have been several papers written concerning point estimation of μ for this problem, but very little is available in the literature concerning confidence intervals on the common mean μ. In this paper a method is proposed that results in a confidence interval with confidence coefficient essentially equal to a prescribed value 1 - α. The method is evaluated and compnred with other methods through the expected length of the confidence interval.  相似文献   

13.
Viewing the future order statistics as latent variables at each Gibbs sampling iteration, several Bayesian approaches to predict future order statistics based on type-II censored order statistics, X(1), X(2), …, X(r), of a size n( > r) random sample from a four-parameter generalized modified Weibull (GMW) distribution, are studied. Four parameters of the GMW distribution are first estimated via simulation study. Then various Bayesian approaches, which include the plug-in method, the Monte Carlo method, the Gibbs sampling scheme, and the MCMC procedure, are proposed to develop the prediction intervals of unobserved order statistics. Finally, four type-II censored samples are utilized to investigate the predictions.  相似文献   

14.
The problem of calculating orthant probabilities for sets of variables (X1,., Xn) is considered in the case where they are jointly normally distributed with zero means and a correlation matrix such that the correlation between Xi and Xi is zero if |i-j|> 1. An effective method is given which works for quite large n when the correlations between Xi and Xi+1 have the values 1/2, 2/5, 3/10, 4/17, 5/26,. and more approximate methods are given for other values. The accuracy is investigated numerically.  相似文献   

15.
In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X1, X2, …, Xn ? p are independent with density function f(x) while the remaining, p observations (outliers) Xn ? p + 1, …, Xn are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.  相似文献   

16.
Distributions of a response y (height, for example) differ with values of a factor t (such as age). Given a response y* for a subject of unknown t*, the objective of inverse prediction is to infer the value of t* and to provide a defensible confidence set for it. Training data provide values of y observed on subjects at known values of t. Models relating the mean and variance of y to t can be formulated as mixed (fixed and random) models in terms of sets of functions of t, such as polynomial spline functions. A confidence set on t* can then be had as those hypothetical values of t for which y* is not detected as an outlier when compared to the model fit to the training data. With nonconstant variance, the p-values for these tests are approximate. This article describes how versatile models for this problem can be formulated in such a way that the computations can be accomplished with widely available software for mixed models, such as SAS PROC MIXED. Coverage probabilities of confidence sets on t* are illustrated in an example.  相似文献   

17.
Let X1, …, Xn be independent random variables with XiEWG(α, β, λi, pi), i = 1, …, n, and Y1, …, Yn be another set of independent random variables with YiEWG(α, β, γi, qi), i = 1, …, n. The results established here are developed in two directions. First, under conditions p1 = ??? = pn = q1 = ??? = qn = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ1 = ??? = λn = γ1 = ??? = γn = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p1, …, pn) and (q1, …, qn). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.  相似文献   

18.
Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X 1, X (2)', X '(3))', where X1 is one-dimension, X(2) is p2- dimensional, and so 1 + p1 + p2 = p. Let ρ1 and ρ be the multiple correlation coefficients of X1 with X(2) and with ( X '(2), X '(3))', respectively. Write ρ2/2 = ρ2 - ρ2/1. We shall cosider the following two problems  相似文献   

19.
Let X1X2,.be i.i.d. random variables and let Un= (n r)-1S?(n,r) h (Xi1,., Xir,) be a U-statistic with EUn= v, v unknown. Assume that g(X1) =E[h(X1,.,Xr) - v |X1]has a strictly positive variance s?2. Further, let a be such that φ(a) - φ(-a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S2n be the Jackknife estimator of n Var Un. Consider the stopping times N(d)= min {n: S2n: + n-12a-2},d > 0, and a confidence interval for v of length 2d,of the form In,d= [Un,-d, Un + d]. We assume that Var Un is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let IN(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. limd → 0P(v ∈ IN(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ IN(d),d) converges to α is investigated. We obtain that |P(v ∈ IN(d),d) - α| = 0 (d1/2-(1+k)/2(1+m)), d → 0, where K = max {0,4 - m}, under the condition that E|h(X1, Xr)|m < ∞m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).  相似文献   

20.
This article considers the construction of level 1?α fixed width 2d confidence intervals for a Bernoulli success probability p, assuming no prior knowledge about p and so p can be anywhere in the interval [0, 1]. It is shown that some fixed width 2d confidence intervals that combine sequential sampling of Hall [Asymptotic theory of triple sampling for sequential estimation of a mean, Ann. Stat. 9 (1981), pp. 1229–1238] and fixed-sample-size confidence intervals of Agresti and Coull [Approximate is better than ‘exact’ for interval estimation of binomial proportions, Am. Stat. 52 (1998), pp. 119–126], Wilson [Probable inference, the law of succession, and statistical inference, J. Am. Stat. Assoc. 22 (1927), pp. 209–212] and Brown et al. [Interval estimation for binomial proportion (with discussion), Stat. Sci. 16 (2001), pp. 101–133] have close to 1?α confidence level. These sequential confidence intervals require a much smaller sample size than a fixed-sample-size confidence interval. For the coin jamming example considered, a fixed-sample-size confidence interval requires a sample size of 9457, while a sequential confidence interval requires a sample size that rarely exceeds 2042.  相似文献   

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