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1.
Summary LetN=[n ij ] (i=1, …,r;j=1, …,c) be the matrix of observed frequencies in anr×c contingency table fromr possibly different multinomial populations with respective probabilitiesp i =(p i1, …,p ic ).Freeman andHalton have proposed an exact conditional test for the hypothesisH 0 :p i =(p 1, …p c ) of the exact test is derived. Numerical values forβ(p) were previously computed for the special case:r=3,c=2 [Bennett andNakamura, 1964].  相似文献   

2.
Holger Dette 《Metrika》1993,40(1):37-50
The optimal design problem for the estimation of several linear combinationsc′ l ϑ (l=1, …,m) is considered in the usual linear regression modely=f′(x)ϑ (f(x) ∈ ℝ k ,ϑ ∈ ℝ k ). An optimal design minimizes a (weighted)p-norm of the variances of the least squares estimates for the different linear combinationsc′ l ϑ. A generalized Elfving theorem is used to derive the relation of the new optimality criterion to theE-optimal design problem. It is shown that theE-optimal design for the parameterϑ minimizes such a (weighted)p-norm whenever the vectorc=(c′ 1, …, c′k)′ is an inball vector of a symmetric convex and compact “Elfving set” in.  相似文献   

3.
Andrej Pázman 《Metrika》2002,56(2):113-130
The nonlinear regression model with N observations y i=η(x i,θ) +εi, and with the parameter θ subject to q nonlinear constraints C j (θ)=0; j=1, …,q, is considered. As an example, the spline regression with unknown nodes is taken. Expressions for the variances (variance matrices) of the LSE are discussed. Because of the complexity of these expressions, and the singularity of the variance matrix of the LSE for θ, the optimality criteria and their properties, in particular the convexity and the equivalence theorem are considered from different aspects. Also the possibility of restriction to designs with limited values of measures of nonlinearity is mentioned. Research supported by the VEGA-grant of the Slovak grant agency No. 1/7295/20.  相似文献   

4.
In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t r, …, t s (1≤rs<kn), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999  相似文献   

5.
F. Brodeau 《Metrika》1999,49(2):85-105
This paper is devoted to the study of the least squares estimator of f for the classical, fixed design, nonlinear model X (t i)=f(t i)+ε(t i), i=1,2,…,n, where the (ε(t i))i=1,…,n are independent second order r.v.. The estimation of f is based upon a given parametric form. In Brodeau (1993) this subject has been studied in the homoscedastic case. This time we assume that the ε(t i) have non constant and unknown variances σ2(t i). Our main goal is to develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another for comparing two such classes. The fundamental tool is an approximation of the elements of these classes by more regular functions, which leads to asymptotic properties of estimators based on the least squares estimator of the unknown parameters. We point out that Neubauer and Zwanzig (1995) have obtained interesting results for connected subjects by using the same technique of approximation. Received: February 1996  相似文献   

6.
LetX 1,X 2,… be i.i.d. with finite meanμ>0,S n =X 1+…+X n . Forf(n)=n β ,c>0 we consider the stopping timesT c =inf{n:S n >c+f(n)} with overshootR c =S T c −(c+f(T c )). For 0<β<1 we give a bound for sup c≥0 ER c in the spirit of Lorden’s well-known inequality forf=0.  相似文献   

7.
The center of a univariate data set {x 1,…,x n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x 1−μ|,…,|x n−μ|). As the median and the mean are the minimizers of respectively the L 1- and the L 2-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x 1,…,x n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L 1-distances y1=(∥x 1- μ1,…,∥x n- μ1) and the vector of L 2-distances y2=(∥x 1- μ2,…,∥x n-μ2). We define the L 1-median and the L 1-mean as the minimizers of respectively the L 1- and the L 2-norm of y 1; and then the L 2-median and the L 2-mean as the minimizers of respectively the L 1- and the L 2-norm of y 2. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L 1-mean appears to be a new concept. Received January 1999  相似文献   

8.
Summary LetX andY be two random vectors with values in ℝ k and ℝ∝, respectively. IfZ=(X T,Y T) T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X 1 T , …,X t T ) T is splitted intot≥3 partsX 1, …,X t.  相似文献   

9.
W. Bischoff  W. Fieger 《Metrika》1992,39(1):185-197
Summary Let the random variableX be normal distributed with known varianceσ 2>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl p (θ, d)=|θ-d| p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.  相似文献   

10.
Consider the standard linear model Y=X θ + ε. If the parameter of interest is a full rank subsystem K′θ of mean parameters, the associated information matrix can be defined via an extremal representation. For rank deficient subsystems, Pukelsheim (1993) introduced the notion of generalized information matrices that inherit many properties of the information matrices. However, this notion is not a direct extension of the full rank case in the sense that the definition of the generalized information matrix applied to full rank subsystems does not lead to the usual information matrix. In this paper, we propose a definition of the information matrix via an extremal representation that encompasses the full rank and the non-full rank cases. We also study its properties and show its links with the generalized information matrices.  相似文献   

11.
Tang Qingguo 《Metrika》2009,69(1):55-67
Suppose that the longitudinal observations (Y ij , X ij , t ij ) for i = 1, . . . ,n; j = 1, . . . ,m i are modeled by the semiparamtric model where β 0 is a k × 1 vector of unknown parameters, g(·) is an unknown estimated function and e ij are unobserved disturbances. This article consider M-type regressions which include mean, median and quantile regressions. The M-estimator of the slope parameter β 0 is obtained through piecewise local polynomial approximation of the nonparametric component. The local M-estimator of g(·) is also obtained by replacing β 0 in model with its M-estimator and using local linear approximation. The asymptotic distribution of the estimator of β 0 is derived. The asymptotic distributions of the local M-estimators of g(·) at both interior and boundary points are also established. Various applications of our main results are given. The research is supported in part by National Natural Science Foundation of China (Grant No. 10671089).  相似文献   

12.
N. Giri  M. Behara  P. Banerjee 《Metrika》1992,39(1):75-84
Summary LetX=(X ij )=(X 1, ...,X n )’,X i =(X i1, ...,X ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ22 -2 be the squared multiple correlation coefficient between the first and the remainingp 2+p 3=p−1 components of eachX i . We have considered here the problem of testingH 02=0 against the alternativesH 11 -2 =0, ϱ 2 -2 >0 on the basis ofX andn 1 additional observationsY 1 (n 1×1) on the first component,n 2 observationsY 2(n 2×p 2) on the followingp 2 components andn 3 additional observationsY 3(n 3×p 3) on the lastp 3 components and we have derived here the locally minimax test ofH 0 againstH 1 when ϱ 2 -2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.  相似文献   

13.
Consider the heteroscedastic regression model Y (j)(x in , t in ) = t in βg(x in ) + σ in e (j)(x in ), 1 ≤ j ≤ m, 1 ≤ i ≤ n, where sin2=f(uin){\sigma_{in}^{2}=f(u_{in})}, (x in , t in , u in ) are fixed design points, β is an unknown parameter, g(·) and f(·) are unknown functions, and the errors {e (j)(x in )} are mean zero NA random variables. The moment consistency for least-squares estimators and weighted least-squares estimators of β is studied. In addition, the moment consistency for estimators of g(·) and f(·) is investigated.  相似文献   

14.
Dr. P. N. Rathie 《Metrika》1972,18(1):216-219
Equivalence of the generalized entropyH β (P, Φ t ) defined in this paper andKapur’s entropy of orderα and typeβ, ie.H α β (P), is established. The results given recently byCampbell follow as special cases. International Conference on System Sciences, Honolulu, January 1968.  相似文献   

15.
We propose a class of nonparametric tests for testing non-stochasticity of the regression parameterβ in the regression modely i =βx i +ɛ i ,i=1, ...,n. We prove that the test statistics are asymptotically normally distributed both underH 0 and under contiguous alternatives. The asymptotic relative efficiencies (in the Pitman sense) with respect to the best parametric test have also been computed and they are quite high. Some simulation studies are carried out to illustrate the results. Research was supported by the University Grants Commission, India.  相似文献   

16.
Let X 1,X 2,…,X n be a random sample from a continuous distribution with the corresponding order statistics X 1:nX 2:n≤…≤X n:n. All the distributions for which E(X k+r: n|X k:n)=a X k:n+b are identified, which solves the problem stated in Ferguson (1967). Received February 1998  相似文献   

17.
LetX 1,X 2, …,X n be independent identically distributed random vectors in IR d ,d ⩾ 1, with sample mean and sample covariance matrixS n. We present a practicable and consistent test for the composite hypothesisH d: the law ofX 1 is a non-degenerate normal distribution, based on a weighted integral of the squared modulus of the difference between the empirical characteristic function of the residualsS n −1/2 (X j − ) and its pointwise limit exp (−1/2|t|2) underH d. The limiting null distribution of the test statistic is obtained, and a table with critical values for various choices ofn andd based on extensive simulations is supplied.  相似文献   

18.
In the linear model Y i = x i + e i, i=1,,n, with unknown (, ), {\open R}p, >0, and with i.i.d. errors e 1,,e n having a continuous distribution F, we test for the goodness-of-fit hypothesis H 0:F(e)F 0(e/), for a specified symmetric distribution F 0, not necessarily normal. Even the finite sample null distribution of the proposed test criterion is independent of unknown (,), and the asymptotic null distribution is normal, as well as the distribution under local (contiguous) alternatives. The proposed tests are consistent against a general class of (nonparametric) alternatives, including the case of F having heavier (or lighter) tails than F 0. A simulation study illustrates a good performance of the tests. Received July 2001  相似文献   

19.
Herbert Vogt 《Metrika》1996,44(1):207-221
Let ζ t be the number of events which will be observed in the time interval [0;t] and define as the average number of events per time unit if this limit exists. In the case of i.i.d. waiting-times between the events,E t ] is the renewal function and it follows from well-known results of renewal theory thatA exists and is equal to 1/τ, if τ>0 is the expectation of the waiting-times. This holds true also when τ = ∞.A may be estimate by ζ t /t or where is the mean of the firstn waiting-timesX 1,X 2, ...,X n . Both estimators converage with probability 1 to 1/τ if theX i are i.i.d.; but the expectation of may be infinite for alln and also if it is finite, is in general a positively biased estimator ofA. For a stationary renewal process, ζ t /t is unbiased for eacht; if theX i are i.i.d. with densityf(x), then ζ t /t has this property only iff(x) is of the exponential type and only for this type the numbers of events in consecutive time intervals [0,t], [t, 2t], ... are i.i.d. random variables for arbitraryt > 0.  相似文献   

20.
Suppose the observations (X i,Y i), i=1,…, n, are ϕ-mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied by serveral authors, e.g. G. Collomb (1984), L. Gy?rfi et al. (1989). But the optimal convergence rates are not reached unless the Y i are bounded or the E exp (a|Y i|) are bounded for some a>0. Compared with the i.i.d. case the convergence of the Nadaraya-Watson estimator under ϕ-mixing variables needs strong moment conditions. In this paper we study the strong uniform convergence and convergence rate for the improved kernel estimator of the regression function which has been suggested by Cheng P. (1983). Compared with Theorem A in Y. P. Mack and B. Silverman (1982) or Theorem 3.3.1 in L. Gy?rfi et al. (1989), we prove the convergence for this kind of estimators under weaker moment conditions. The optimal convergence rate for the improved kernel estimator is attained under almost the same conditions of Theorem 3.3.2 in L. Gy?rfi et al. (1989). Received: September 1999  相似文献   

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