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1.
Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and ci(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[11100121], c1=[10], c2=[02].) Let Ti (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the sn?r solutions of the equations At=ci, wheret′=(t1,…,tn) is a vector of variables.(Thus, EG(4, 3) consists of the 34=81 points of the form (t1,t2,t3,t4), where t's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=ci is sn?r, where r=Rank(A), and the set of such solutions is said to form an (n?r)-flat, i.e. a flat of (n?r) dimensions. In our example, both T1 and T2 are 2-flats consisting of 34?2=9 points each. The flats T1,T2,…,Tf are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T1 are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T2 consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sn symmetric factorial experiment obtained by taking T1,T2,…,Tf together. (Thus, in the example, 34=81 treatments of the 34 factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T1 and T2 enumerated above.)In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X (n×v), occuring in the corresponding linear model, into components X.j(j=0,1,…,g), such that the information matrix X is the direct sum of the X′.jX.j. Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying (s?1) degrees of freedom as usual). For j≠0, we show that the spectrum of X′.jX.j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X.j into the Xij (i=1,…,f). Furthermore, the Xij are in turn partitioned into smaller matrices (which we shall here call the) Xijh. We show that each Xijh can be factored into a product of 3 matrices J, ζ (not depending on i,j, and h) and Q(j,h,i)where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF(s), and show that the Q(j,h,i) belong to a ring isomorphic to R. When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X.j and X′.jX.j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R.  相似文献   

2.
ABSTRACT

Let (Xi, Yi), i = 1, …, n be a pair where the first coordinate Xi represents the lifetime of a component, and the second coordinate Yi denotes the utility of the component during its lifetime. Then the random variable Y[r: n] which is known to be the concomitant of the rth order statistic defines the utility of the component which has the rth smallest lifetime. In this paper, we present a dynamic analysis for an n component system under the above-mentioned concomitant setup.  相似文献   

3.
This paper develops a conditional approach to testing hypotheses set up after viewing the data. For example, suppose Xi are estimates of location parameters θi, i = 1,…n. We show how to compute p-values for testing whether θ1 is one of the three largest θi after observing that X1 is one of the three largest Xi, or for testing whether θ1 > θ2 > … > θn after observing X1 >X2> … >Xn.  相似文献   

4.
This paper introduces a new class of bivariate lifetime distributions. Let {Xi}i ? 1 and {Yi}i ? 1 be two independent sequences of independent and identically distributed positive valued random variables. Define T1 = min?(X1, …, XM) and T2 = min?(Y1, …, YN), where (M, N) has a discrete bivariate phase-type distribution, independent of {Xi}i ? 1 and {Yi}i ? 1. The joint survival function of (T1, T2) is studied.  相似文献   

5.
Let X1,…, Xn be mutually independent non-negative integer-valued random variables with probability mass functions fi(x) > 0 for z= 0,1,…. Let E denote the event that {X1X2≥…≥Xn}. This note shows that, conditional on the event E, Xi-Xi+ 1 and Xi+ 1 are independent for all t = 1,…, k if and only if Xi (i= 1,…, k) are geometric random variables, where 1 ≤kn-1. The k geometric distributions can have different parameters θi, i= 1,…, k.  相似文献   

6.
Let X1,…,Xn be exchangeable normal variables with a common correlation p, and let X(1) > … > X(n) denote their order statistics. The random variable σni=nk+1xi, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.  相似文献   

7.
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   

8.
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   

9.
ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (XT1, X2T, …, XTk, ZT)T = (X11, …, X1n, …, Xk1, …, Xkn, Z1, …, Zn)T and derive the distribution of concomitant of multivariate order statistics arising from X1, X2, …, Xk. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.  相似文献   

10.
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   

11.
Abstract. We consider N independent stochastic processes (X i (t), t ∈ [0,T i ]), i=1,…, N, defined by a stochastic differential equation with drift term depending on a random variable φ i . The distribution of the random effect φ i depends on unknown parameters which are to be estimated from the continuous observation of the processes Xi. We give the expression of the exact likelihood. When the drift term depends linearly on the random effect φ i and φ i has Gaussian distribution, an explicit formula for the likelihood is obtained. We prove that the maximum likelihood estimator is consistent and asymptotically Gaussian, when T i =T for all i and N tends to infinity. We discuss the case of discrete observations. Estimators are computed on simulated data for several models and show good performances even when the length time interval of observations is not very large.  相似文献   

12.
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   

13.
Let (X1,X2, …,Xn) be jointly distributed random variables. Define Xn:n = max(X1,X2, …,Xn).Bounds on E(Xn:n), obtained by putting constraints on the distributions and/or dependence structure of the Xi's, are surveyed.  相似文献   

14.
Let X1 X2 … XN be independent normal p-vectors with common mean vector $$ = ($$) and common nonsingular covariance matrix $$ = Diag ($sGi) [(1–p) I + pE] Diag ($sGi), $sGi> 0, i = 1… p, 1>p>=1/p–1. Write rij = sample correlation between the i th and the j th variable i j = 1,… p. It has been proved that for testing the hypothesis H0 : p = 0 against the alternative H1 : p>0 where $$ and $sG1,…, $sGp are unknown, the test which rejects H0 for large value of $$ rij is locally best invariant for every $aL: 0 > $aL > 1 and locally minimax as p $$ 0 in the sense of Giri and Kiefer, 1964, for every $aL: 0 > $aL $$ $aL0 > 1 where$aL0 = Pp=0 $$.  相似文献   

15.
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   

16.
Simulating a stationary AR(p), Xt = ∑pi=1αiXti + Zt, when the innovations {Zt} are assumed to be i.i.d. is straightforward. Starting the process in the stationary state, however, requires generation of (X1,X2,…,Xp) from the stationary p-dimensional distribution. When Zt is normal this may be achieved by generating Xi as a linear function of X1,X2,…,Xi−1 and an independent normal variate for i = 2,3,…, p. It is shown that the ability to initialize a stationary AR(p) in this way characterizes the normal distribution.  相似文献   

17.
18.
《随机性模型》2013,29(1):25-37
For a shot-noise process X(t) with Poisson arrival times and exponentially diminishing shocks of i.i.d. sizes, we consider the first time T b at which a given level b > 0 is exceeded. An integral equation for the joint density of T b and X(T b ) is derived and, for the case of exponential jumps, solved explicitly in terms of Laplace transforms (LTs). In the general case we determine the ordinary LT of the function ? P(T b > t) in terms of certain LTs derived from the distribution function H(x; t) = P(X(t) ≤ x), considered as a function of both variables x and t. Moreover, for G(t, u) = P(T b > t, X(t) < u), that is the joint distribution function of sup0 ≤ st X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u).  相似文献   

19.
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   

20.
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   

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