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1.
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
2.
Let {v
n(θ)} be a sequence of statistics such that whenθ =θ
0,v
n(θ
0)
N
p(0,Σ), whereΣ is of rankp andθ εR
d. Suppose that underθ =θ
0, {Σ
n} is a sequence of consistent estimators ofΣ. Wald (1943) shows thatv
n
T
(θ
0)Σ
n
−1
v
n(θ
0)
x
2(p). It often happens thatv
n(θ
0)
N
p(0,Σ) holds butΣ is singular. Moore (1977) states that under certain assumptionsv
n
T
(θ
0)Σ
n
−
v
n(θ
0)
x
2(k), wherek = rank (Σ) andΣ
n
−
is a generalized inverse ofΣ
n. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (Σ
n) =k forn sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier
to verify, assumptions.
Research partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University. 相似文献
3.
Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl
p (θ, d)=|θ-d|p
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
4.
Hagen Scherb 《Metrika》2001,53(1):71-84
Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H
0 and the alternative hypothesis H
1 are given by
(i) H
0 : θ≤θ0, H
1 : θ>θ0, and
(ii) H
0 : θ≤θ1 or θ≥θ2, H
1 : θ1<θ<θ2, where θ1<θ2.
Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H
0 and H
1 take the form
(iii) H
0 : θ1≤θ≤θ2, H
1 : θ<θ1 or θ>θ2, where θ1<θ2, and
(iv) H
0 : θ=θ0, H
1:θ≠θ0.
To determine tests in case (i), only one critical value c and one randomization constant γ have to be computed. In cases (ii) through (iv) tests are determined by two critical values
c
1, c
2 and two randomization constants γ1, γ2. Unlike determination of tests in case (i), computation of critical values and randomization constants in the remaining cases
is rather difficult, unless distributions are symmetric. No straightforward method to determine two-sided UMP tests in discrete
sample spaces seems to be known. The purpose of this note is to disclose a distribution independent principle for the determination
of UMP tests in cases (ii) through (iv).
Received: March 1999 相似文献
5.
Yongge Tian 《Metrika》2010,72(3):313-330
Estimations of parametric functions under a general linear model and its restricted models involve some complicated operations
of matrices and their generalized inverses. In the past several years, a powerful tool—the matrix rank method was utilized
to manipulate various complicated matrix expressions that involve generalized inverses of matrices. In this paper, we use
this method to derive necessary and sufficient conditions for six equalities of the ordinary least-squares estimators and
the best linear unbiased estimators of parametric functions to equal under a general linear model and its corresponding restricted
model. 相似文献
6.
Summary Dynamic exponential family regression provides a framework for nonlinear regression analysis with time dependent parametersβ
0,β
1, …,β
t, …, dimβ
t=p. In addition to the familiar conditionally Gaussian model, it covers e.g. models for categorical or counted responses. Parameters
can be estimated by extended Kalman filtering and smoothing. In this paper, further algorithms are presented. They are derived
from posterior mode estimation of the whole parameter vector (β′0, …,β′t) by Gauss-Newton resp. Fisher scoring iterations. Factorizing the information matrix into block-bidiagonal matrices, algorithms
can be given in a forward-backward recursive form where only inverses of “small”p×p-matrices occur. Approximate error covariance matrices are obtained by an inversion formula for the information matrix, which
is explicit up top×p-matrices.
Heinz Leo Kaufmann, my friend and coauthor for many years, died in a tragical rock climbing accident in August 1989. This
paper is dedicated to his memory. 相似文献
7.
The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, ${\beta\in\Re^r}The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted
model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, b ? ?r{\beta\in\Re^r} may hold. In this area we show that the Bayes and empirical Bayes estimators dominate the unrestricted estimator (when nothing
is known about the mean vector θ). 相似文献
8.
In two recent papers by Balakrishnan et al. (J Qual Technol 39:35–47, 2007; Ann Inst Stat Math 61:251–274, 2009), the maximum
likelihood estimators [^(q)]1{\hat{\theta}_{1}} and [^(q)]2{\hat{\theta}_{2}} of the parameters θ
1 and θ
2 have been derived in the framework of exponential simple step-stress models under Type-II and Type-I censoring, respectively.
Here, we prove that these estimators are stochastically monotone with respect to θ
1 and θ
2, respectively, which has been conjectured in these papers and then utilized to develop exact conditional inference for the
parameters θ
1 and θ
2. For proving these results, we have established a multivariate stochastic ordering of a particular family of trinomial distributions
under truncation, which is also of independent interest. 相似文献
9.
Satya R. Chakravarty Bhargav Maharaj 《Journal of Economic Interaction and Coordination》2012,7(1):99-123
An ethnic polarization index is a summary statistic of ethnic diversity in a population. Reynal-Querol (J Confl Resolut 46:29–54,
2002) suggested an index of ethnic polarization, the RQ index, and discussed its properties. In this paper we develop two ethnic polarization orderings that can rank ethnic distributions
in terms of all ethnic polarization indices satisfying certain intuitively reasonable postulates. Some of these postulates
and some additional ones taken from the earlier literature are employed to develop some axiomatic characterizations of the
RQ index. In the process, a generalized form of the RQ index is also characterized. 相似文献
10.
Berthold Heiligers 《Metrika》2002,54(3):191-213
E-optimality of approximate designs in linear regression models is paired with a dual problem of nonlinear Chebyshev approximation.
When the regression functions form a totally positive system, then the information matrices of designs for subparameters turn
out to be “almost” totally positive, a property which allows to solve the nonlinear Chebyshev problem. Thereby we obtain explicit
formulae for E-optimal designs in terms of equi-oscillating generalized polynomials. The considerations unify and generalize known results
on E-optimality for particular regression setups. 相似文献
11.
Let (T
n
)
n≥1 be a sequence random variables (rv) of interest distributed as T. In censorship models the rv T is subject to random censoring by another rv C. Let θ be the mode of T. In this paper we define a new smooth kernel estimator [^(q)]n{\hat{\theta}_n} of θ and establish its almost sure convergence under an α-mixing condition. 相似文献
12.
Various optimal criteria have been proposed to rank asymmetrical fractional factorial designs. Among them, the generalized
minimum aberration and the minimum moment aberration criteria are the most popular ones and have received much attention.
Recently, Liu et al. (Stat Sin 16:1285–1297, 2006) proposed the minimum
χ
2 criterion in terms of level-combinations. In this paper, the equivalency of the generalized minimum aberration and the minimum
χ
2 criteria is reported, which not only provides another justification for each other but also develops some theoretical results
for designs with generalized minimum aberration and some lower bounds for the generalized wordlength pattern. Besides, an
analytic relationship between generalized minimum aberration and minimum moment aberration is obtained for asymmetrical fractional
factorial designs. 相似文献
13.
The paper considers n-dimensional VAR models for variables exhibiting cointegration and common cyclical features. Two specific reduced rank vector error correction models are discussed. In one, named the “strong form” and denoted by SF, the collection of all coefficient matrices of a VECM has rank less than n, in the other, named the “weak form” and denoted by WF, the collection of all coefficient matrices except the matrix of coefficient of error correction terms has rank less than n. The paper explores the theoretical connections between these two forms, suggests asymptotic tests for each form and examines the small sample properties of these tests by Monte Carlo simulations. 相似文献
14.
Shanbhag (J Appl Probab 9:580–587, 1972; Theory Probab Appl 24:430–433, 1979) showed that the diagonality of the Bhattacharyya
matrix characterizes the set of Normal, Poisson, Binomial, negative Binomial, Gamma or Meixner hypergeometric distributions.
In this note, using Shanbhag (J Appl Probab 9:580–587, 1972; Theory Probab Appl 24:430–433, 1979) and Pommeret (J Multivar
Anal 63:105–118, 1997) techniques, we evaluated the general form of the 5 × 5 Bhattacharyya matrix in the natural exponential
family satisfying
f(x|q)=\fracexp{xg(q)}b(g(q))y(x){f(x|\theta)=\frac{\exp\{xg(\theta)\}}{\beta(g(\theta))}\psi(x)} with cubic variance function (NEF-CVF) of
θ. We see that the matrix is not diagonal like distribution with quadratic variance function and has off-diagonal elements.
In addition, we calculate the 5 × 5 Bhattacharyya matrix for inverse Gaussian distribution and evaluated different Bhattacharyya
bounds for the variance of estimator of the failure rate, coefficient of variation, mode and moment generating function due
to inverse Gaussian distribution. 相似文献
15.
Arturo J. Fernández 《Metrika》2000,50(3):211-220
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≤r≤s<k≤n), 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
16.
17.
Consider a family of distribution functions ${\{F(x, \theta),\,\theta \in \Theta\}}Consider a family of distribution functions {F(x, q), q ? Q}{\{F(x, \theta),\,\theta \in \Theta\}} . Suppose that there exists an estimator of the unknown parameter vector θ based on given data set. Then it is readily to obtain an estimator of any quantity given as an explicit function g(θ). Particularly, it is the case when the maximum likelihood estimator of θ is available. However, often some quantities of interest can not be expressed as an explicit function, rather it is determined
as an implicit function of θ. The present article studies this problem. Sufficient conditions are given for deriving estimators of these quantities. The
results are then applied to estimate change point of failure rate function, and change point of mean residual life function. 相似文献
18.
Prof. Dr. J. Pfanzagl 《Metrika》1970,15(1):141-148
Summary Let (X,A) be a measurable space andP
ϑη |A (ϑη) ∈ Θ x H, ∥A, (θ, η) ∈ Θ×H, a parametrized family of probability measures (for short:p-measures). This paper is concerned with the problem of consistently estimatingθ from realizations governed by
, where ηu ∈ H, v ∈ ℕ, are unknown. 相似文献
19.
Prof. Dr. W. Stute 《Metrika》1992,39(1):257-267
LetX
1, ...,X
n
be an i.i.d. sample from some parametric family {θ :θ (Θ} of densities. In the random censorship model one observesZ
i
=min (X
i
,Y
i
) andδ
i
=1{
x
i
≤Y
i}, whereY
i
is a censoring variable being independent ofX
i
. In this paper we investigate the strong consistency ofθ
n
maximizing the modified likelihood function based on (Z
i
,δ
i
, 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data.
Work partially supported by the “Deutsche Forschungsgemeinschaft”. 相似文献
20.
We consider the problem of estimating the scale parameter θ of the shifted exponential distribution with unknown shift based on a set of observed records drawn from a sequential sample
of independent and identically distributed random variables. Under a large class of bowl-shaped loss functions, the best affine
equivariant estimator (BAEE) of θ is shown to be inadmissible. Two dominating procedures are proposed. A numerical study is performed to show the extent of
risk reduction that the improved estimators provide over the BAEE. 相似文献