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1.
Summary In case of absolute error loss we investigate for an arbitrary class of probability distributions, if or if not a two point
prior can be least favourable and a corresponding Bayes estimator can be minimax when the parameter is restricted to a closed
and bounded interval of ℝ. The general results are applied to several examples, for instance location and scale parameter
families are considered. We give examples for which, independent of the length of the parameter interval, no two point priors
exist. On the other hand examples are given having a least favourable two point prior when the parameter interval is sufficiently
small. 相似文献
2.
Sequences of integers defined by a non-linear recursive congruential pseudo random number generator with prime modulusp and maximal period length are divided into vectors ofd consecutive numbers. The lattice spanned by these vectors is studied. i.e. the minimal lattice which contains all these vectors.
It is shown that this lattice coincides with the full integer lattice, at least ford=2 andd=3, i.e. non-linear generators pass Marsaglia’s lattice test at least ford ⩽ 3. For a special class of non-linear generators introduced in Eichenauer and Lehn (1986) it is proved that these generators
pass the test even for dimensionsd ⩽ (p − 1)/2.
Supported by Deutsche Forschungsgemeinschaft. 相似文献
3.
Alicja Jokiel-Rokita 《Metrika》2011,74(2):211-219
The problem of sequentially estimating an unknown distribution parameter of a particular exponential family of distributions
is considered under LINEX loss function for estimation error and a cost c > 0 for each of an i.i.d. sequence of potential observations X
1, X
2, . . . A Bayesian approach is adopted and conjugate prior distributions are assumed. Asymptotically pointwise optimal and
asymptotically optimal procedures are derived. 相似文献
4.
In the present paper families of truncated distributions with a Lebesgue density
forx=(x
1,...,x
n
) ε ℝ
n
are considered, wheref
0:ℝ → (0, ∞) is a known continuous function andC
n
(ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator
form a saddle point when the parameter interval is sufficiently small. 相似文献
5.
The problem of estimating a linear combination,μ, of means ofp-independent, first-order autoregressive models is considered. Sequential procedures are derived (i) to estimateμ pointwise using the linear combination of sample means, subject to a loss function (squared error plus cost per observation),
and (ii) to arrive at a fixed-width confidence interval forμ. It is observed that in the case of point estimation we do not require a sampling scheme, where as in the case of interval
estimation we do require a sampling scheme and a scheme similar to the one given in Mukhopadhyay and Liberman (1989) is proposed.
All the first order efficiency properties of the sequential procedures involved here are derived. This paper is an extension
of results of Sriram (1987) involving one time series to multiple time series.
Research supported by AFOSR Grant number 89-0225. 相似文献
6.
We consider the sequential point estimation problem of the powers of a normal scale parameter σr with r≠ 0 when the loss function is squared error plus linear cost. It is shown that the regret due to using our fully sequential
procedure in ignorance of σ is asymptotically minimized for estimating σ−2. We also propose a bias-corrected procedure to reduce the risk and show that the larger the distance between r and −2 is, the more effective our bias-corrected procedure is.
Received August 2000 相似文献
7.
In this paper we consider the case of the scale-contaminated normal (mixture of two normals with equal mean components but
different component variances: (1−p)N(μ,σ2)+pN(μ,τ2) with σ and τ being non-negative and 0≤p≤1). Here is the scale error and p denotes the amount with which this error occurs. It's maximum deviation to the best normal distribution is studied and shown
to be montone increasing with increasing scale error. A closed-form expression is derived for the proportion which maximizes
the maximum deviation of the mixture of normals to the best normal distribution. Implications to power studies of tests for
normality are pointed out.
Received May 2001 相似文献
8.
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
9.
Dr. J. Eichenauer-Herrmann 《Metrika》1992,39(1):199-208
Summary Admissibility of estimators under vague prior information on the distribution of the unknown parameter is studied which leads
to the notion of gamma-admissibility. A sufficient condition for an estimator of the formδ(x)=(ax+b)/(cx+d) to be gamma-admissible in the one-parameter exponential family under squared error loss is established. As an application
of this result two equalizer rules are shown to be unique gamma-minimax estimators by proving their gamma-admissibility. 相似文献
10.
We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known
apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth
estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized
Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are
compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995),
and the restricted maximum likelihood estimators.
Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation. 相似文献
11.
W. Molenaar 《Statistica Neerlandica》1969,23(1):19-40
Summary In these days some sportsmen are given dangerous drugs in order to stimulate them into still higher efforts. The present paper wants to add some poison to the classical approximation of a binomial by a Poisson distribution. The relatively harmless drug of a little extra calculation shall be seen to result in a much better accuracy. This paper reports both on theoretical considerations, following from series expansions of Poisson-type parameters, and on extensive numerical investigations of accuracy by means of an Electrologica X8 computer. The main conclusion is that the probability of at most k successes in n Bernoulli trials with success probability p <.5 can be very closely approximated by the probability of at most k events in a Poisson distribution with expectation not np but (12–2p)n-7kζ= (12–8p)n-k+k/nnp. In the case p > 5 one should make p < .5 by a reversal, i.e. the interchanging of successes and failures, before the application of a Poisson-type approximation (with one exception mentioned in section 4). For the probability of at least j successes one should approximate the complementary probability of at most j—1 successes. After an introductory section 1 and a discussion of measures of accuracy in section 2, the existing Poisson-type approximations are treated in section 3. The fourth section introduces a new parameter and discusses the advantages of reversal in other situations than p > 5. In section 5 a series expansion for the exact Poisson parameter is derived. Two new parameters, among which is ζ, are introduced in section 6, where the series expansions for parameters given in table 4 lead to conclusions about the various approximations, summarized in table 6. The final section 7 gives some numerical results, which confirm to a large extent the conclusions from the asymptotic expansions. 相似文献
12.
For the invariant decision problem of estimating a continuous distribution function F with two entropy loss functions, it is proved that the best invariant estimators d
0 exist and are the same as the best invariant estimator of a continuous distribution function under the squared error loss
function L (F, d)=∫|F (t) −d (t) |2
dF (t). They are minimax for any sample size n≥1. 相似文献
13.
A bivariate normal distribution is considered whose mean lies in an equilateral triangle. We show by a convexity argument that the three point prior having mass 1/3 at each of the edges is least favourable if the length of a side of the equilateral triangle is less than or equal to
. Thus the corresponding Bayes estimator is minimax in that case. Numerical studies are given as well. 相似文献
14.
Summary The distribution of the square of the distance between a random point and a fixed point on ap-dimensional unit sphere when (i) the two points lie on the whole sphere and (ii) the two points lie in the positive quadrant, has been derived, assuming that the random point is distributed proportionally to exp (ky
1), wherek is a concentration parameter. Then-th order moment in both cases is also obtained. 相似文献
15.
Summary For a linear modelY =ϑ + Z,ϑ ∈V,V ⊂ ℝ
n
a linear space, the following theorem is proved under simple conditions on the subspaceV: The projection onV (i.e. the least squares estimate forϑ) is a sufficient statistic iffZ is normally distributed. Further, this result is extended to the case of a multivariate linear model. 相似文献
16.
Draper and Guttman (1997) shows that for basic 2k−p designs, p≥0, k − p replicates of blocks designs of size two are needed to estimate all the usual (estimable) effects. In this work, we provide
an algebraic formal proof for the two-level blocks designs results and present results applicable to the general case; that
is, for the case of s
k factorial (p=0) or s
k−p fractional factorial (p >0) designs in s
b blocks, where 0<b<k− p, at least replicates are needed to clear up all possible effects. Through the theoretical development presented in this work, it can
provide a clearer view on why those results would hold. We will also discuss the estimation equations given in Draper and
Guttman (1997).
Research supported in part by the National Science Council of Taiwan, R.O.C., Grant No. NSC 89-2118-M110-010.
Acknowledgement. The authors would like to thank the referee for very helpful comments. 相似文献
17.
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
18.
Prof. Dr. J. Pfanzagl 《Metrika》1970,15(1):30-39
Summary
Lehmann [p. 83] has shown that some families of probability measures with monotone likelihood ratios (m.l.r.) admit median unbiased
estimates which are optimum in the sense that among all median unbiased estimates they minimize the expected loss for any
loss function which assumes its minimal value zero for the “true” parameter value and is nondecreasing as the parameter moves
away from the true value in either direction. This very strong optimum property was proved under the assumption that all probability
measures of the m.l.r.-family have continuous distribution functions, that they are mutually absolutely continuous and that
each element of the support is the median of somep-measure of the family. This result does therefore not cover important cases such as the binomial families or thePoisson family.
The purpose of the present paper is to show the existence ofrandomized median unbiased estimates with the same optimum property for m.l.r.-families which are closed and connected with respect
to the strong topology. Such families are always dominated. We do, however, neither assume that thep-measures are mutually absolutely continuous nor that the distribution functions are continuous. We remark that the use of
randomized estimates is indispensable here because nonrandomized median unbiased estimates do not always exist in the general
case. 相似文献
19.
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
20.
Alicja Jokiel-Rokita 《Metrika》2006,64(3):259-269
Characterizations of gamma-minimax predictors for the linear combinations of the unknown parameter and the random variable having the multinomial distribution under arbitrary squared error loss are established in two situations – when the sample size is fixed and when the sample size is a realization of a random variable. It is always assumed that the available vague prior information about the unknown parameter can be described by a class of priors whose vector of first moments belongs to a suitable convex and compact set. Several known gamma-minimax and minimax results can be obtained from the characterizations derived in the present paper. 相似文献