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1.
Let X1 be a strictly stationary multiple time series with values in Rd and with a common density f. Let X1,.,.,Xn, be n consecutive observations of X1. Let k = kn, be a sequence of positive integers, and let Hni be the distance from Xi to its kth nearest neighbour among Xj, j i. The multivariate variable-kernel estimate fn, of f is defined by where K is a given density. The complete convergence of fn, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.  相似文献   

2.
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   

3.
Consider a sequence of independent random variables X 1, X 2,…,X n observed at n equally spaced time points where X i has a probability distribution which is known apart from the values of a parameter θ i R which may change from observation to observation. We consider the problem of estimating θ = (θ1, θ2,…,θ n ) given the observed values of X 1, X 2,…,X n . The paper proposes a prior distribution for the parameters θ for which sets of parameter values exhibiting no change, or no change apart from a few sudden large changes, or lots of small changes, all have positive prior probability. Markov chain sampling may be used to calculate Bayes estimates of the parameters. We report the results of a Monte Carlo study based on Poisson distributed data which compares the Bayes estimator with estimators obtained using cubic splines and with estimators derived from the Schwarz criterion. We conclude that the Bayes method is preferable in a minimax sense since it never produces the disastrously large errors of the other methods and pays only a modest price for this degree of safety. All three methods are used to smooth mortality rates for oesophageal cancer in Irish males aged 65–69 over the period 1955 through 1994.  相似文献   

4.
The probability density function (pdf) of a two parameter exponential distribution is given by f(x; p, s?) =s?-1 exp {-(x - ρ)/s?} for x≥ρ and 0 elsewhere, where 0 < ρ < ∞ and 0 < s?∞. Suppose we have k independent random samples where the ith sample is drawn from the ith population having the pdf f(x; ρi, s?i), 0 < ρi < ∞, 0 < s?i < s?i < and f(x; ρ, s?) is as given above. Let Xi1 < Xi2 <… < Xiri denote the first ri order statistics in a random sample of size ni, drawn from the ith population with pdf f(x; ρi, s?i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρi, s?i, i = 1, 2,…, k based on the above observations are asymptotically optimal in the sense of Bahadur efficiency. Our results are similar to those for normal distributions.  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 i.i.d. random variables with distribution function F, and let Y1,.,.,Yn be i.i.d. with distribution function G. For i = 1, 2,.,., n set δi, = 1 if Xi ≤ Yi, and 0 otherwise, and Xi, = min{Xi, Ki}. A kernel-type density estimate of f, the density function of F w.r.t. Lebesgue measure on the Borel o-field, based on the censored data (δi, Xi), i = 1,.,.,n, is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher-order differentiability assumption on f. A procedure for relaxing such assumptions is also proposed.  相似文献   

7.
Let FN(.) be the density function of X2N. Values of C1/N, i= 1, 2, satisfying the twin conditions Pr (C1≤X2N≤C2)=1-α and the conditional expectation of X2N given C1≤X2N≤C2 is N are tabulated for α=.2, .1, .05, .01, .005, .001, N=1(1)20(2)50(5)150(10)350. The second condition may be replaced by the condition fN+2(C1)=fN+2V(C2). The author has with him a bigger table giving C1 and C2 for α=.2, .1, .05, .01, .005, .001, N=1(1)350 to three decimals (to three significant digits, if some decimals are not significant). Several applications are mentioned. A practical application that is perhaps not obvious is to test whether two or more counts are distributed as independent Poisson variables. The new simple formulae used in the construction of the table are given and should prove useful in obtaining accurate values for omitted entries and in increasing the accuracy of entries.  相似文献   

8.
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   

9.
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   

10.
Suppose that Xi are independent random variables, and that Xi has cdf Fi (x), 1 ≤ ik. Many statistical problems involve the probability Pr{X 1 < X 2 < ··· < Xk }. In this note a numerical method is proposed for computing this probability.  相似文献   

11.
Let (X, Y) be a bivariate random vector with joint distribution function FX, Y(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (Xi, Yi), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (Xi, Yi) for which Xi ? Yi. We denote such pairs as (X*i, Yi*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function FX, Y(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*i, Yi*). The copula of bivariate random vector (X*, Y*) is also derived.  相似文献   

12.
Let X1,X2,…,Xn be n normal variates with zero means, unit variances and correlation matrix {pij). The orthant probability is the probability that all of the X1's are simultaneously positive. This paper presents a general reduction method by extending the method of Childs (1967), and shows that the probability can be represented by a linear combination of some multivariate integrals of order([n/2]?1). As illustrations, we apply the proposed method to the quadrivariate and six–variate cases. Some numerical results are also given.  相似文献   

13.
Consider the process with, cf. (1.2) on page 265 in B1, X1, …, XN a sample from a distribution F and, for i = 1, …, N, R |x 1 , - q 1 ø| , the rank of |X1 - q1ø| among |X1 - q1ø|, …, |XN - qNø|. It is shown that, under certain regularity conditions on F and on the constants pi and qi, TøN(t) is asymptotically approximately a linear function of ø uniformly in t and in ø for |ø| ≤ C. The special case where the pi and the qi, are independent of i is considered.  相似文献   

14.
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   

15.
We consider n pairs of random variables (X11,X21),(X12,X22),… (X1n,X2n) having a bivariate elliptically contoured density of the form where θ1 θ2 are location parameters and Δ = ((λik)) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product-moment correlation coefficient between X1 and X2 is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.  相似文献   

16.
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   

17.
Let (X i , Y i ), i = 1, 2,…, n be independent and identically distributed random variables from some continuous bivariate distribution. If X (r) denotes the rth-order statistic, then the Y's associated with X (r) denoted by Y [r] is called the concomitant of the rth-order statistic. In this article, we derive an analytical expression of Shannon entropy for concomitants of order statistics in FGM family. Applying this expression for some well-known distributions of this family, we obtain the exact form of Shannon entropy, some of the information properties, and entropy bounds for concomitants of order statistics. Some comparisons are also made between the entropy of order statistics X (r) and the entropy of its concomitants Y [r]. In this family, we show that the mutual information between X (r) and Y [r], and Kullback–Leibler distance among the concomitants of order statistics are all distribution-free. Also, we compare the Pearson correlation coefficient between X (r) and Y [r] with the mutual information of (X (r), Y [r]) for the copula model of FGM family.  相似文献   

18.
For X1, …, XN a random sample from a distribution F, let the process SδN(t) be defined as where K2N = σNi=1(ci ? c?)2 and R xi, + Δd, is the rank of Xi + Δdi, among X1 + Δd1, …, XN + ΔdN. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants ci and di, SΔN (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.  相似文献   

19.
20.
Let Sn = X1 + … + Xn, where X1,…, Xn are independent Bernoulli random variables. In this paper, we evaluate probability metrics of the Wasserstein type between the distribution of Sn and a Poisson distribution. Our results show that, if E(Sn) = O(1) and if the individual probabilities of success of the Xi's tend uniformly to zero, then the general rate of convergence of the above mentioned metrics to zero is O(∑ni = 1P2i). We also show that this rate is sharp and discuss applications of these results.  相似文献   

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