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1.
Let X1:n ≤ X2:n ≤···≤ Xn:n denote the order statistics of a sample of n independent random variables X1, X2,…, Xn, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (Xk1:n, Xk2:n ? Xk1:n,…, Xkr:n ? Xkr?1:n) is MTP2 [S-MRR2] whenever 1 ≤ k1 < k2 <···< kr ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.  相似文献   

2.
3.
Let X(1,n,m1,k),X(2,n,m2,k),…,X(n,n,m,k) be n generalized order statistics from a continuous distribution F which is strictly increasing over (a,b),−a<b, the support of F. Let g be an absolutely continuous and monotonically increasing function in (a,b) with finite g(a+),g(b) and E(g(X)). Then for some positive integer s,1<sn, we give characterization of distributions by means of
  相似文献   

4.
We are considering the ABLUE’s – asymptotic best linear unbiased estimators – of the location parameter μ and the scale parameter σ of the population jointly based on a set of selected k sample quantiles, when the population distribution has the density of the form
where the standardized function f(u) being of a known functional form.A set of selected sample quantiles with a designated spacing
or in terms of u=(x−μ)/σ
where
λi=∫−∞uif(t) dt, i=1,2,…,k
are given by
x(n1)<x(n2)<<x(nk),
where
Asymptotic distribution of the k sample quantiles when n is very large is given by
h(x(n1),x(n2),…,x(nk);μ,σ)=(2πσ2)k/212−λ1)(λk−λk−1)(1−λk)]−1/2nk/2 exp(−nS/2σ2),
where
fi=f(ui), i=0,1,…,k,k+1,
f0=fk+1=0, λ0=0, λk+1=1.
The relative efficiency of the joint estimation is given by
where
and κ being independent of the spacing . The optimal spacing is the spacing which maximizes the relative efficiency η(μ,σ).We will prove the following rather remarkable theorem. Theorem. The optimal spacing for the joint estimation is symmetric, i.e.
λiki+1=1,
or
ui+uki+1=0, i=1,2,…,k,
if the standardized density f(u) of the population is differentiable infinitely many times and symmetric
f(−u)=f(u), f′(−u)=−f′(u).
  相似文献   

5.
Let X 1,X 2,…,X n be independent exponential random variables such that X i has hazard rate λ for i = 1,…,p and X j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D i:n (λ, λ*) = X i:n  ? X i?1:n the ith spacing of the order statistics X 1:n  ≤ X 2:n  ≤ ··· ≤ X n:n , i = 1,…,n, where X 0:n ≡ 0. It is shown that the spacings (D 1,n ,D 2,n ,…,D n:n ) are MTP2, strengthening one result of Khaledi and Kochar (2000), and that (D 1:n 2, λ*),…,D n:n 2, λ*)) ≤ lr (D 1:n 1, λ*),…,D n:n 1, λ*)) for λ1 ≤ λ* ≤ λ2, where ≤ lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ1 < λ2.  相似文献   

6.
In this paper, by considering a (3n+1) -dimensional random vector (X0, XT, YT, ZT)T having a multivariate elliptical distribution, we derive the exact joint distribution of (X0, aTX(n), bTY[n], cTZ[n])T, where a, b, c∈?n, X(n)=(X(1), …, X(n))T, X(1)<···<X(n), is the vector of order statistics arising from X, and Y[n]=(Y[1], …, Y[n])T and Z[n]=(Z[1], …, Z[n])T denote the vectors of concomitants corresponding to X(n) ((Y[r], Z[r])T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X0, X(r), Y[r], Z[r])T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X(r), say, based on the concomitants Y[r] and Z[r]. Finally, we illustrate the usefulness of our results by a real data.  相似文献   

7.
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   

8.
For non-negative integral valued interchangeable random variables v1, v2,…,vn, Takács (1967, 70) has derived the distributions of the statistics ?n' ?1n' ?(c)n and ?(-c)n concerning the partial sums Nr = v1 + v2 + ··· + vrr = 1,…,n. This paper deals with the joint distributions of some other statistics viz., (α(c)n, δ(c)n, Zn), (β(c)n, Zn) and (β(-c)n, Zn) concerning the partial sums Nr = ε1 + ··· + εrr = 1,2,…,n, of geometric random variables ε1, ε2,…,εn.  相似文献   

9.
Let X 1, X 2,…, X n be independent exponential random variables with X i having failure rate λ i for i = 1,…, n. Denote by D i:n  = X i:n  ? X i?1:n the ith spacing of the order statistics X 1:n  ≤ X 2:n  ≤ ··· ≤ X n:n , i = 1,…, n, where X 0:n ≡ 0. It is shown that if λ n+1 ≤ [≥] λ k for k = 1,…, n then D n:n  ≤ lr D n+1:n+1 and D 1:n  ≤ lr D 2:n+1 [D 2:n+1 ≤ lr D 2:n ], and that if λ i  + λ j  ≥ λ k for all distinct i,j, and k then D n?1:n  ≤ lr D n:n and D n:n+1 ≤ lr D n:n , where ≤ lr denotes the likelihood ratio order. We also prove that D 1:n  ≤ lr D 2:n for n ≥ 2 and D 2:3 ≤ lr D 3:3 for all λ i 's.  相似文献   

10.
ABSTRACT

Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 < α < 1 and n = 1,2,…; and (iii) 1 < α < 2 and n = 1,2,…. New expressions are provided for the more general case where β > 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.  相似文献   

11.
Consider the regression model Yi= g(xi) + ei, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x0 < x1 < … < xn≤ 1 are chosen so that max1≤i≤n(xi-xi- 1) = 0(n-1), and where {ei} are i.i.d. with Ee1= 0 and Var e1 - s?2. In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g1n of Cheng & Lin (1979), g2n of Clark (1977), and g3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &in, i = 1, 2, 3, and that of the isotonic estimator g**n are considered.  相似文献   

12.
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   

13.
Suppose that data {(x l,i,n , y l,i,n ): l?=?1, …, k; i?=?1, …, n} are observed from the regression models: Y l,i,n ?=?m l (x l,i,n )?+?? l,i,n , l?=?1, …, k, where the regression functions {m l } l=1 k are unknown and the random errors {? l,i,n } are dependent, following an MA(∞) structure. A new test is proposed for testing the hypothesis H 0: m 1?=?·?·?·?=?m k , without assuming that {m l } l=1 k are in a parametric family. The criterion of the test derives from a Crámer-von-Mises-type functional based on different distances between {[mcirc]} l and {[mcirc]} s , l?≠?s, l, s?=?1, …, k, where {[mcirc] l } l=1 k are nonparametric Gasser–Müller estimators of {m l } l=1 k . A generalization of the test to the case of unequal design points, with different sample sizes {n l } l=1 k and different design densities {f l } l=1 k , is also considered. The asymptotic normality of the test statistic is obtained under general conditions. Finally, a simulation study and an analysis with real data show a good behavior of the proposed test.  相似文献   

14.
15.
Asymptotic behavior of the number of independent identically distributed observations in a left or right neighborhood of k n th order statistic from the sample of size n, for k n /n → α ? [0, 1], is studied. It appears that the limiting laws are of the Poisson type.  相似文献   

16.
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   

17.
18.
Let {X, Xn; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (ank; 1 ≤ kn, n ≥ 1); ank, ? R and supn, k |an,k| < ∞}. Set Sn( A ) = ∑nk=1an, kXk for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{supn ≥ 1, |Sn( A )|/n1/r}p < ∞ or E{supn ≥ 2 |Sn( A )|/2n log n)1/2}p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).  相似文献   

19.
Let Z 1, Z 2, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z t  = 1) and q = Pr(Z t  = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E1{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E2{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E3{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by Nn,k(i){ N_{n,k}^{(i)}} (respectively Mn,k(i){M_{n,k}^{(i)}}) the number of occurrences of the pattern Ei{\mathcal{E}_{i}} , i = 1, 2, 3, in Z 1, Z 2, . . . , Z n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let Tr,k(i){T_{r,k}^{(i)}} (resp. Wr,k(i)){W_{r,k}^{(i)})} be the waiting time for the rth occurrence of the pattern Ei{\mathcal{E}_{i}}, i = 1, 2, 3, in Z 1, Z 2, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of Nn,k(i){N_{n,k}^{(i)}}, Mn,k(i){M_{n,k}^{(i)}}, Tr,k(i){T_{r,k}^{(i)}} and Wr,k(i){W_{r,k}^{(i)}} (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.  相似文献   

20.
We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy k . This may correspond to the case of a regression model, where one observesy k =f(θ,x k )+ε k , with ε k some random error, or to the Bernoulli case wherey k ∈{0, 1}, with Pr[y k =1|θ,x k |=f(θ,x k ). Special attention is given to sequences given by , with an estimated value of θ obtained from (x1, y1),...,(x k ,y k ) andd k (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑ i=1 N w i f(θ, x i ) with {w i } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.  相似文献   

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