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1.
Let x ≥ 0 and n ≥ 2 be integers. Suppose there exists an orthogonal array A(n, q, μ1) of strength 2 in n symbols with q rows and n2μ1 columns where q = q1 ? d, q1 = n2x + n + 1, μ1 = (n ? 1)x + 1 and d is a positive integer. Then d is called the deficiency of the orthogonal array. The question of embedding such an array into a complete array A(n, q1, μ1) is considered for the case d ≥ 3. It is shown that for d = 3 such an embedding is always possible if n ≥ 2(d ? 1)2(2d2 ? 2d + 1). Partial results are indicated if d ≥ 4 for the embedding of a related design in a corresponding balanced incomplete block design.  相似文献   

2.
Arnold and Stahlecker considered estimation of the regression coefficients in the linear model with a relative squared error and deterministic disturbances. They found an explicit form for a minimax linear affine solution d1 of that problem. In the paper we generalize the result of Arnold and Stahlecker proving that the decision rule d1 is also minimax when the class D of possible estimators of the regression coefficients is unrestricted. Then we show that d1 remains minimax in D when the disturbances are random with the mean vector zero and the identity covariance matrix.  相似文献   

3.
Let X11?X12???X1n be the order statistics of a random sample from a distribution on [0, 1]. Let Ak, the kth match, be the event that X1k?((k?1)nkn], and let Sn be the total number of matches. The consistency of Sn for testing uniform df, U, against df GU is investigated, and it is shown that Sn is consistent if the intersection of G with U has Lebesgue measure zero. It is also consistent against a sequence of alternatives approaching U at a rate less faster than n-12.  相似文献   

4.
Consider the p-dimensional unit cube [0,1]p, p≥1. Partition [0, 1]p into n regions, R1,n,…,Rn,n such that the volume Δ(Rj,n) is of order n?1,j=1,…,n. Select and fix a point in each of these regions so that we have x(n)1,…,x(n)n. Suppose that associated with the j-th predictor vector x(n)j there is an observable variable Y(n)j, j=1,…,n, satisfying the multiple regression model Y(n)j=g(x(n)j)+e(n)j, where g is an unknown function defined on [0, 1]pand {e(n)j} are independent identically distributed random variables with Ee(n)1=0 and Var e(n)12<∞. This paper proposes gn(x)=a-pnΣnj=1Y(n)jRj,nk[(x?u)?an]du as an estimator of g(x), where k(u) is a known p-dimensional bounded density and {an} is a sequence of reals converging to 0 asn→∞. Weak and strong consistency of gn(x) and rates of convergence are obtained. Asymptoticnormality of the estimator is established. Also proposed is σ2n=n?1Σnj=1(Y(n)j?gn(x(n)j))2 as a consistent estimate of σ2.  相似文献   

5.
Asymptotic expansions for the percentiles and c.d.f., up to terms of order 1n2 of the statistic T =mTrS1S-12, where mS1 and nS2 independently distributed W(m, p, Σ1) and W(n, p, Σ2) respectively, are obtained using methods similar to those of Ito [4], Chattopadhyay and Pillai [2]. These expansions hold when Σ1Σ-12 = I + F and|Chi(F)| < 1. Tables of powers of T for p = 3 and p = 4 for m = 4 and various values of n are given and comparison made with the exact powers for p = 3. These powers are useful for the study of (i) the test of equality of covariance matrices in two p-variate normal populations and (ii) robustness of test of equality of mean vectors of l normal populations against the violation of the assumption of equality of covariance matrices.  相似文献   

6.
Saha and Mohanty (1970) presented a main effect fold-over design consisting of 14 treatment combinations of the 24×33 factorial, which had the nice property of being even balanced. Calling this design DSM, this paper establishes the following specific results: (i) DSM is not d-optimal in the subclass Δe of all 14 point even balanced main effect fold-over designs of the 24×33 factorial; (ii) DSM is not d-optimal in the subclass Δ1e of all 14 point even and odd balanced main effect fold-over designs of the 24×33 factorial; (iii) DSM is even optimal in Δ1 and Δe. In addition to these results two 14 point designs in Δ1 are presented which are d-optimal and via a counter example it is shown that these designs are not odd optimal. Finally, several general matrix algebra results are given which should be useful in resolving d-optimality problems of fold-over designs of the kn11×kn22 factorial.  相似文献   

7.
We consider the signed linear rank statistics of the form
SΔN= i=1N cNiø(RΔNi(N+1))sgn YΔNi
where the cNi's are known real numbers, Δ∈[0,1] is an unknown real parameter,RΔNi is the rank of |YΔNi| among |YΔNj|, 1≤jN, ø is a score generating function, sgn y=1 or -1 according as y≥0 or <0, and YΔNj, 1≤jN, are independent random variables with continuous cumulative distribution functions F(y?ΔdNj), 1≤ jN, respectively where the dfNi's are known real numbers. Under suitable assumptions on the c's, d's, φ and F, it is proved that the random process {SΔN?S0N?ESΔN, 0≤Δ≤1}, properly normalized, converges weakly to a Gaussian process, and this result is also true if ESΔN is replaced by ΔbN, where
bN=4 i=1N cNidNi0 ø′(2F(x)?1)?2(x)dx and ?=F′
. As an application, we derive the asymptotic distribution of the properly normalized length of a confidence interval for Δ.  相似文献   

8.
The set of distinct blocks of a block design is known as its support. We construct complete designs with parameters v(?7), k=3, λ=v ? 2 which contain a block of maximal multiplicity and with support size b1 = (v3) ? 4(v ? 2). Any complete design which contains such a block, and has parameters v, k, λ as above, must be supported on at most (v3) ? 4(v ? 2) blocks. Attention is given to complete designs because of their direct relationship to simple random sampling.  相似文献   

9.
The convergence rates of empirical Bayes estimation in the exponential family are studied in this paper. We first develop an approach for obtaining the lower bound of empirical Bayes estimators. As an application of the approach, we demonstrate that O(n−1) is the lower bound rate for priors with bounded compact support. Second, we construct an empirical Bayes estimator using kernel sequence method and show that it has a rate of convergence of O(n−1(lnn)8). This upper bound rate is much faster compared to the earlier results published in the literature under the same assumption.  相似文献   

10.
We investigate the efficiences of Tiku's (1967) modified maximum likelihood estimators μc and σc (based on symmetrically censored normal samples) for estimating the location and scale parameters μ and σ of symmetric non-normal distributions. We show that μc and σc are jointly more efficient than x? and s for long-tailed distributions (kurtosis β21 = μ4μ22>4.2, β21 = 4.2 for the Logistic), and always more efficient than the trimmed mean μT and the matching sample estimate σT of σ. We also show that μc and σc are jointly at least as efficient as some of the more prominent “robust” estimators (Gross, 1976). We show that the statistic tc = μcmσc, m = n ?2r + 2rβ (r is the number of observations censored on each side of the sample and β is a constant), is robust and powerful for testing an assumed value of μ. We define a statistic Tc (based on μc andσc) for testing that two symmetric distributions are identical and show that Tc is robust and generally more poweerful than the well-known nonparametric statistics (Wilcoxon, normal-score, Kolmogorov-Smirnov), against the important location-shift alternatives. We generalize the statistic Tc to test that k symmetric distibutions are identical. The asymptotic distributions of tc and Tc are normal, under some very general regularity conditions. For small samples, the upper (lower) percentage points of tc and Tc are shown to be closely approximated by Student's t-distributions. Besides, the statistics μc and σc (and hence tc and Tc) are explicit and simple functions of sample observations and are easy to compute.  相似文献   

11.
The estimation of γ=∫?∞?2(x)dx, important in nonparametric inference, is discussed. Methods, based on either density estimators or on the lengths of nonparametric confidence intervals, are compared. We conclude that density type estimators have advantages over the estimators based on confidence intervals.  相似文献   

12.
This paper deals with a sequence-compound estimation. The component problem is the squared error loss estimation of θ?[a,b] based on an observation X whose p.d.f. is of the form u(x)c(θ)exp(?xθ). For each 0<t<12 a class of sequence-compound estimators ψ?=ψ?1,ψ?2,…) is exhibited whose compound risk (average of risks) up to stage n differs from the Bayes envelope (in the component problem) w.r.t. the empiric distribution Gn of the parameters involved up to stage n by a quantity of order O(n?δt) for a δ>0. It is also shown that at any stage i the difference of the risk of ψ?i and the risk of the Bayes response w.r.t. Gi?1 is O(i?δt). Examples of the above type of families are given where δ is min{1,2ab} and t is arbitrarily close to 12. Here it may be worthwhile to mention that a rate O(n?12) or better has not yet been obtained even in a very special family of densities.  相似文献   

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15.
In the usual two-way layout of ANOVA (interactions are admitted) let nij ? 1 be the number of observations for the factor-level combination(i, j). For testing the hypothesis that all main effects of the first factor vanish numbers n1ij are given such that the power function of the F-test is uniformly maximized (U-optimality), if one considers only designs (nij) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.  相似文献   

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17.
An estimating equation for a parameter θ, based on an observation ?, is an equation g(x,θ)=0 which can be solved for θ in terms of x. An estimating equation is unbiased if the funaction g has 0 mean for every θ. For the case when the form of the frequency function p(x,θ) is completely specified up to the unknown real parameter θ, the optimality of the m.1 equation ?logp=0 in the class of all unbiased estimating equations was established by Godambe (1960). In this paper we allow the form of the frequency function p to vary assuming that x=(x1,…,xn)?Rn and that under p, E(xi)=θ. x1,…, xn are independent observations on a variate x, it is shown that among all the unbiased estimating equations for θ, x??θ=0 is uniquely optimum up to a constant multiple.  相似文献   

18.
Let ρ1,…,ρp be the population canonical correlation coefficients from a normal distribution. This paper considers the estimation of δ1,…,δp, where δii2/(1−ρi2),i=1,…,p, in a decision theoretic way. Since the distribution of δi's is complicated, two-staged estimation has been a usual method so far; i.e., first find a good estimator of a matrix whose eigenvalues are the δi's, then use its eigenvalues as the estimators of δi's. In this paper we directly estimate δi's and evaluate the estimators with respect to a quadratic loss function. We propose a new class of estimators and prove its dominance over the usual estimator.  相似文献   

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