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1.
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
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
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
10.
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
11.
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
12.
A distributionF is said to be “more IFR” than another distributionG ifG
−1
F is convex. WhenF(0) =G(0) = 0, the problem of testingH
0 :F(x) =G (θx) for someθ > 0 andx ⩾ 0, against the alternativeH
A:F is more IFR thanG, is considered in this paper. Both cases, whenG is completely specified (one-sample case) and when it is not specified but a random sample form it is available (two-sample
case) are considered. The proposed tests are based onU-statistics. The asymptotic relative efficiency of the tests are compared with several other tests and the test statistics
remain asymptotically normal under certain dependency assumptions.
Research supported in part by a grant from the US Air Force Office of Scientific Research. 相似文献
13.
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
14.
Subject of this paper is the analysis of consensus within small groups of respondents, based on a proportionally large number
of variables. The target group is researchers who are interested in Q-mode research. Measures of agreement are compared, and
an application from a recent project is presented. Cohen’s κ is the preferable measure, Krippendorff’s α is an alternative, which is based on a different concept of expected disagreement. At group level, along with κ and α for multiple raters, additional measures are r
wg, intraclass correlation, and κ
SC. Predictions about level differences between groups can be assessed by a t-test and θ 相似文献
15.
Klaus Ziegler 《Metrika》2001,53(2):141-170
In the nonparametric regression model with random design and based on i.i.d. pairs of observations (X
i, Y
i), where the regression function m is given by m(x)=?(Y
i|X
i=x), estimation of the location θ (mode) of a unique maximum of m by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve m is considered. In order to obtain asymptotic confidence intervals for θ, the suitably normalized distribution of is bootstrapped in two ways: we present a paired bootstrap (PB) where resampling is done from the empirical distribution
of the pairs of observations and a smoothed paired bootstrap (SPB) where the bootstrap variables are generated from a smooth
bivariate density based on the pairs of observations. While the PB requires only relatively small computational effort when
carried out in practice, it is shown to work only in the case of vanishing asymptotic bias, i.e. of “undersmoothing” when
compared to optimal smoothing for mode estimation. On the other hand, the SPB, although causing more intricate computations,
is able to capture the correct amount of bias if the pilot estimator for m oversmoothes.
Received: May 2000 相似文献
16.
Bootstrap based goodness-of-fit-tests 总被引:1,自引:1,他引:0
Summary Let ℱ={F
θ} be a parametric family of distribution functions, and denote withF
n
the empirical d.f. of an i.i.d. sample. Goodness-of-fit tests of a composite hypothesis (contained in ℱ) are usually based
on the so-called estimated empirical process. Typically, they are not distribution-free. In such a situation the bootstrap
offers a useful alternative. It is the purpose of this paper to show that this approximation holds with probability one. A
simulation study is included which demonstrates the validity of the bootstrap for several selected parametric families. 相似文献
17.
LetP be a probability measure on ℝ andI
x be the set of alln-dimensional rectangles containingx. If for allx ∈ ℝn and θ ∈ ℝ the inequality
holds,P is a normal distributioin with mean 0 or the unit mass at 0. The result generalizes Teicher’s (1961) maximum likelihood characterization
of the normal density to a characterization ofN(0, σ2) amongall distributions (including those without density). The m.l. principle used is that of Scholz (1980). 相似文献
18.
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
19.
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
20.
A method to obtain new copulas from a given one 总被引:1,自引:0,他引:1
Patricia Mariela Morillas 《Metrika》2005,61(2):169-184
Given a strictly increasing continuous function φ from [0, 1] to [0, 1] and its pseudo-inverse φ[−1], conditions that φ must satisfy for Cφ(x1, . . . ,xn)=φ[−1](C(φ(x1), . . . ,φ(xn))) to be a copula for any copula C are studied. Some basic properties of the copulas obtained in this way are analyzed and
several examples of generator functions φ that can be used to construct copulas Cφ are presented. In this manner, a method to obtain from a given copula C a variety of new copulas is provided. This method
generalizes that used to construct Archimedean copulas in which the original copula C is the product copula, and it is related
with mixtures 相似文献