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1.
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   

2.
Given the regression model Yi = m(xi) +εi (xi ε C, i = l,…,n, C a compact set in R) where m is unknown and the random errors {εi} present an ARMA structure, we design a bootstrap method for testing the hypothesis that the regression function follows a general linear model: Ho : m ε {mθ(.) = At(.)θ : θ ε ? ? Rq} with A a functional from R to Rq. The criterion of the test derives from a Cramer-von-Mises type functional distance D = d2([mcirc]n, At(.)θn), between [mcirc]n, a Gasser-Miiller non-parametric estimator of m, and the member of the class defined in Ho that is closest to mn in terms of this distance. The consistency of the bootstrap distribution of D and θn is obtained under general conditions. Finally, simulations show the good behavior of the bootstrap approximation with respect to the asymptotic distribution of D = d2.  相似文献   

3.
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   

4.
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   

5.
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   

6.
The use of a range estimator of the population standard deviation, sigma (σ), for determining sample sizes is discussed in this study. Standardized mean ranges (dn's), when divided into the ranges of sampling frames, provide estimates of the standard deviation of the population. These estimates can be used for determining sample sizes. The dn's are provided for seven different distributions for sampling frame sizes that range from 2 to 2000, For each of the seven distributions, functional relationships are developed such that dn = f(nSF) where nSF is the size of the sample frame. From these functions, dn's can be estimated for sampling frame sizes which are not presented in the study.  相似文献   

7.
Let (θ1,x1),…,(θn,xn) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ2. Let ti be the linear Bayes estimator of θi and θ~i be the linear empirical Bayes estimator of θi as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~i instead of ti because of ignorance of the mean and the variance is ri = E(θi ? θi)2 ?E(tii)2. Under appropriate conditions cumulative regret Rn = r1+…rn is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x.  相似文献   

8.
Let T2 i=z′iS?1zi, i==,…k be correlated Hotelling's T2 statistics under normality. where z=(z′i,…,z′k)′ and nS are independently distributed as Nkp((O,ρ?∑) and Wishart distribution Wp(∑, n), respectively. The purpose of this paper is to study the distribution function F(x1,…,xk) of (T2 i,…,T2 k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T2 i,…,T2 k) up to the order n?2. Next we give asymptotic expansions for (T2 i,…,T2 k) in two cases (i)ρ=Ik and (ii) k=2 by inverting the expanded characteristic function up to the orders n?2 and n?1, respectively. Our results can be applied to the distribution function of max (T2 i,…,T2 k) as a special case.  相似文献   

9.
Fix r ≥ 1, and let {Mnr} be the rth largest of {X1,X2,…Xn}, where X1,X2,… is a sequence of i.i.d. random variables with distribution function F. It is proved that P[Mnr ≤ un i.o.] = 0 or 1 according as the series Σn=3Fn(un)(log log n)r/n converges or diverges, for any real sequence {un} such that n{1 -F(un)} is nondecreasing and divergent. This generalizes a result of Bamdorff-Nielsen (1961) in the case r = 1.  相似文献   

10.
Let X1, X2, …, Xn be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss.  相似文献   

11.
Consider n independent random variables Zi,…, Zn on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = Fθ(t),t ≥ x0, where θ ∈ ? ? R d. A necessary and sufficient condition for the family Fθ, θ ∈ ?, is established such that the k-th largest order statistic Zn?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Zn?i+1:n)1 i=kof the k largest order statistics. This is achieved for k = k(n)→n→∞∞ with k/n→n→∞ 0.

In the case of vectors of central order statistics ( Zr:n, Zr+1:n,…, Zs:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Zm:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only  相似文献   

12.
The supremum of random variables representing a sequence of rewards is of interest in establishing the existence of optimal stopping rules. Necessary and sufficient conditions are given for existence of moments of supn(Xn ? cn) and supn(Sn ? cn) where X1, X2, … are i.i.d. random variables, Sn = X1 + … + Xn, and cn = (nL(n))1/r, 0 < r < 2, L = 1, L = log, and L = log log. Following Cohn (1974), “rates of convergence” results are used in the proof.  相似文献   

13.
Let X ∈ R be a random vector with a distribution which is invariant under rotations within the subspaces Vj (dim Vj. = qj) whose direct sum is R. The large sample distributions of the eigenvalues and vectors of Mn= n-1Σnl xixi are studied. In particular it is shown that several eigenvalue results of Anderson & Stephens (1972) for uniformly distributed unit vectors hold more generally.  相似文献   

14.
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   

15.
Suppose (X, Y) has a Downton's bivariate exponential distribution with correlation ρ. For a random sample of size n from (X, Y), let X r:n be the rth X-order statistic and Y [r:n] be its concomitant. We investigate estimators of ρ when all the parameters are unknown and the available data is an incomplete bivariate sample made up of (i) all the Y-values and the ranks of associated X-values, i.e. (i, Y [i:n]), 1≤in, and (ii) a Type II right-censored bivariate sample consisting of (X i:n , Y [i:n]), 1≤ir<n. In both setups, we use simulation to examine the bias and mean square errors of several estimators of ρ and obtain their estimated relative efficiencies. The preferred estimator under (i) is a function of the sample correlation of (Y i:n , Y [i:n]) values, and under (ii), a method of moments estimator involving the regression function is preferred.  相似文献   

16.
Data which is grouped and truncated is considered. We are given numbers n1<…<nk=n and we observe Xni ),i=1,…k, and the tottal number of observations available (N> nk is unknown. If the underlying distribution has one unknown parameter θ which enters as a scale parameter, we examine the form of the equations for both conditional, unconditional and modified maximum likelihood estimators of θ and N and examine when these estimators will be finite, and unique. We also develop expressions for asymptotic bias and search for modified estimators which minimize the maximum asymptotic bias. These results are specialized tG the zxponential distribution. Methods of computing the solutions to the likelihood equatims are also discussed.  相似文献   

17.
Consider a family of square-integrable Rd-valued statistics Sk = Sk(X1,k1; X2,k2;…; Xm,km), where the independent samples Xi,kj respectively have ki i.i.d. components valued in some separable metric space Xi. We prove a strong law of large numbers, a central limit theorem and a law of the iterated logarithm for the sequence {Sk}, including both the situations where the sample sizes tend to infinity while m is fixed and those where the sample sizes remain small while m tends to infinity. We also obtain two almost sure convergence results in both these contexts, under the additional assumption that Sk is symmetric in the coordinates of each sample Xi,kj. Some extensions to row-exchangeable and conditionally independent observations are provided. Applications to an estimator of the dimension of a data set and to the Henze-Schilling test statistic for equality of two densities are also presented.  相似文献   

18.
Consider the semiparametric regression model Yi = x′iβ +g(ti)+ei for i=1,2, …,n. Here the design points (xi,ti) are known and nonrandom and the ei are iid random errors with Ee1 = 0 and Ee2 1 = α2<∞. Based on g(.) approximated by a B-spline function, we consider using atest statistic for testing H0 : β = 0. Meanwhile, an adaptive parametric test statistic is constructed and a large sample study for this adaptive parametric test statistic is presented.  相似文献   

19.
Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a1<a2<a3<…. In particular, ai could be equal to i for all i. Let X1nX2n≦?≦Xnn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦kn), independence of the event {Xkn=X1n} and X1n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{Xkn = X1n | X1n = ai} is equivalent to X having the IFR (DFR) property. Let ai = i and G(i) = P(X≧i), i = 1, 2, …. We prove that the independence of {X2n ? X1nB} and X1n for all i is equivalent to X being geometric, where B = {m} (B = {m,m+1,…}), provided G(i) = qi?1, 1≦im+2 (1≦im+1), where 0<q<1.  相似文献   

20.
Morteza Amini 《Statistics》2013,47(5):393-405
In a sequence of bivariate random variables {(X i , Y i ), i≥1} from a continuous distribution with a real parameter θ, general comparison results between the amount of Fisher information about θ contained in the sequence of the first n records and their concomitants, and the desired information in an i.i.d. sample of size n from the parent distribution are established. Some relationships between reliability properties and the proposed criteria are obtained in situations in which the univariate counterpart of the underlying bivariate family belongs to location, scale or shape families. It is also shown that in some classes of bivariate families, the concerned information property is equivalent to that of its univariate counterpart. The proposed procedure is illustrated by considering several examples.  相似文献   

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