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1.
This paper is devoted to the problem of estimating the square of population mean (μ2) in normal distribution when a prior estimate or guessed value σ0
2 of the population variance σ2 is available. We have suggested a family of shrinkage estimators , say, for μ2 with its mean squared error formula. A condition is obtained in which the suggested estimator is more efficient than Srivastava et al’s (1980) estimator Tmin. Numerical illustrations have been carried out to demonstrate the merits of the constructed estimator over Tmin. It is observed that some of these estimators offer improvements over Tmin particularly when the population is heterogeneous and σ2 is in the vicinity of σ0
2. 相似文献
2.
Estimation of a normal mean relative to balanced loss functions 总被引:3,自引:0,他引:3
LetX
1,…,X
nbe a random sample from a normal distribution with mean θ and variance σ2. The problem is to estimate θ with Zellner's (1994) balanced loss function,
% MathType!End!2!1!, where 0<ω<1. It is shown that the sample mean
% MathType!End!2!1!, is admissible. More generally, we investigate the admissibility of estimators of the form
% MathType!End!2!1! under
% MathType!End!2!1!. We also consider the weighted balanced loss function,
% MathType!End!2!1!, whereq(θ) is any positive function of θ, and the class of admissible linear estimators is obtained under such loss withq(θ) =e
θ
. 相似文献
3.
Improvement of the Liu estimator in linear regression model 总被引:2,自引:0,他引:2
In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed
Estimator
for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ2
I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993).
In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator
for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator
to exceed the mean squared error matrix of the mixed estimator
will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we
show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator
and to the biased estimator introduced by Liu (1993). 相似文献
4.
Milan Merkle 《Statistical Methods and Applications》1996,5(3):323-334
Summary Let
, whereX
i are i.i.d. random variables with a finite variance σ2 and
is the usual estimate of the mean ofX
i. We consider the problem of finding optimal α with respect to the minimization of the expected value of |S
2(σ)−σ2|k for variousk and with respect to Pitman's nearness criterion. For the Gaussian case analytical results are obtained and for some non-Gaussian
cases we present Monte Carlo results regarding Pitman's criteron.
This research was supported by Science Fund of Serbia, grant number 04M03, through Mathematical Institute, Belgrade. 相似文献
5.
Essential graphs and largest chain graphs are well-established graphical representations of equivalence classes of directed acyclic graphs and chain graphs respectively,
especially useful in the context of model selection. Recently, the notion of a labelled block ordering of vertices
was introduced as a flexible tool for specifying subfamilies of chain graphs. In particular, both the family of directed
acyclic graphs and the family of “unconstrained” chain graphs can be specified in this way, for the appropriate choice of
. The family of chain graphs identified by a labelled block ordering of vertices is partitioned into equivalence classes each represented by means of a -essential graph. In this paper, we introduce a topological ordering of meta-arrows and use this concept to devise an efficient procedure for the construction of -essential graphs. In this way we also provide an efficient procedure for the construction of both largest chain graphs and
essential graphs. The key feature of the proposed procedure is that every meta-arrow needs to be processed only once. 相似文献
6.
Janusz Wywiał 《Statistical Papers》2004,45(3):413-431
LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF
x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X
1, Y1)…(Xn, Yn)]=[X Y]. LetX=[X
1…X
n]andY=[Y
1…Y
n].This sample is drawn from a distribution determined by the functionF(x,y). LetX
(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples:
% MathType!End!2!1! and
% MathType!End!2!1!.Let
% MathType!End!2!1! and
% MathType!End!2!1! be the sample means from the sub-samplesU
k,1 andU
k,2, respectively. The linear combination
% MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx
(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived.
The variance of the statistic
% MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation
of the mean is considered, too. 相似文献
7.
In this paper we introduce the distribution of , with c > 0, where X
i
, i = 1, 2, are independent generalized beta-prime-distributed random variables, and establish a closed form expression of its
density. This distribution has as its limiting case the generalized beta type I distribution recently introduced by Nadarajah
and Kotz (2004). Due to the presence of several parameters the density can take a wide variety of shapes.
相似文献
8.
In this paper we consider the inferential aspect of the nonparametric estimation of a conditional function , where X
t,m
represents the vector containing the m conditioning lagged values of the series. Here is an arbitrary measurable function. The local polynomial estimator of order p is used for the estimation of the function g, and of its partial derivatives up to a total order p. We consider α-mixing processes, and we propose the use of a particular resampling method, the local polynomial bootstrap, for the approximation
of the sampling distribution of the estimator. After analyzing the consistency of the proposed method, we present a simulation
study which gives evidence of its finite sample behaviour. 相似文献
9.
Suppose there are k
1 (k
1 ≥ 1) test treatments that we wish to compare with k
2 (k
2 ≥ 1) control treatments. Assume that the observations from the ith test treatment and the jth control treatment follow a two-parameter exponential distribution and , where θ is a common scale parameter and and are the location parameters of the ith test and the jth control treatment, respectively, i = 1, . . . ,k
1; j = 1, . . . ,k
2. In this paper, simultaneous one-sided and two-sided confidence intervals are proposed for all k
1
k
2 differences between the test treatment location and control treatment location parameters, namely , and the required critical points are provided. Discussions of multiple comparisons of all test treatments with the best
control treatment and an optimal sample size allocation are given. Finally, it is shown that the critical points obtained
can be used to construct simultaneous confidence intervals for Pareto distribution location parameters. 相似文献
10.
The distributions of the product and ratio of independent random variables arise in many applied problems. These have been
extensively studied by many researchers. In this paper, the distributions of the product | XY | and ratio have been derived, when X and Y are Maxwell and Rayleigh random variables and are distributed independently of each other. The associated cdfs, pdfs, kth moments, entropies, etc., have been given. To describe the possible shapes of the associated pdfs and entropies, the respective
plots are provided. The percentage points associated with the cdfs of the product and ratio have been tabulated. 相似文献
11.
Martin Bachmaier 《Statistical Papers》2000,41(1):53-64
Using Fisher's information fort-distributions, the absolute asymptotic efficiency of some M-estimates for scale with known location parameter is calculated
and graphically illustrated. The compared estimators are the standard deviationS
*, the mean absolute deviation, called mean deviationD
*, the median absolute deviation, called MAD*, and some M-estimates for scale, one, which is very robust, and another one with high asymptotic efficiency fort-distributions close to the normal. The last one is considered with monotone (in the positive field) and with very late redescending
χ-function too. Also the
, an alternative and generalized excess measure defined as the double relative asymptotic variance of the underlying scale
estimator
in the previous paper, is calculated fort-distributions and graphically illustrated, because there is the relation that the higher the asymptotic efficiency of
is, the lower is the corresponding
. 相似文献
12.
The objective of this paper is to construct an unbiased estimator (up to order 0(1/n)) of the population mean
of the study variatey which is more efficient than the sample mean
of the ‘n’ obsrvedy-values. In particular, the unbiased estimators are discussed for the cases of positive and negative correlations of the study
variatey and the auxiliary variatex. 相似文献
13.
14.
The unbiased estimator of a population variance σ2, S
2 has traditionally been overemphasized, regardless of sample size. In this paper, alternative estimators of population variance
are developed. These estimators are biased and have the minimum possible mean-squared error [and we define them as the “minimum mean-squared error biased estimators” (MBBE)]. The comparative merit of these estimators over the unbiased estimator is explored using relative efficiency (RE)
(a ratio of mean-squared error values). It is found that, across all population distributions investigated, the RE of the
MBBE is much higher for small samples and progressively diminishes to 1 with increasing sample size. The paper gives two applications
involving the normal and exponential distributions. 相似文献
15.
The value for which the mean square error of a biased estimatoraT for the mean μ is less than the variance of an unbiased estimatorT is derived by minimizingMSE(aT). The resulting optimal value is 1/[1+c(n)v
2], wherev=σ/μ, is the coefficient of variation. WhenT is the UMVUE
, thenc(n)=1/n, and the optimal value becomes 1/(n+v
2) (Searls, 1964). Whenever prior information about the size ofv is available the shrinkage procedure is useful. In fact for some members of the one-parameter exponential families it is
known that the variance is at most a quadratic function of the mean. If we identify the pertinent coefficients in the quadratic
function, it becomes easy to determinev. 相似文献
16.
Helmut Rieder Matthias Kohl Peter Ruckdeschel 《Statistical Methods and Applications》2008,17(1):13-40
Robust Statistics considers the quality of statistical decisions in the presence of deviations from the ideal model, where
deviations are modelled by neighborhoods of a certain size about the ideal model. We introduce a new concept of optimality
(radius-minimaxity) if this size or radius is not precisely known: for this notion, we determine the increase of the maximum
risk over the minimax risk in the case that the optimally robust estimator for the false neighborhood radius is used. The
maximum increase of the relative risk is minimized in the case that the radius is known only to belong to some interval [r
l
,r
u
]. We pursue this minmax approach for a number of ideal models and a variety of neighborhoods. Also, the effect of increasing
parameter dimension is studied for these models. The minimax increase of relative risk in case the radius is completely unknown,
compared with that of the most robust procedure, is 18.1% versus 57.1% and 50.5% versus 172.1% for one-dimensional location
and scale, respectively, and less than 1/3 in other typical contamination models. In most models considered so far, the radius
needs to be specified only up to a factor , in order to keep the increase of relative risk below 12.5%, provided that the radius–minimax robust estimator is employed.
The least favorable radii leading to the radius–minimax estimators turn out small: 5–6% contamination, at sample size 100.
相似文献
17.
Mariusz Grządziel 《Statistical Papers》2008,49(3):399-419
Gnot et al. (J Statist Plann Inference 30(1):223–236, 1992) have presented the formulae for computing Bayes invariant quadratic
estimators of variance components in normal mixed linear models of the form
where the matrices V
i
, 1 ≤ i ≤ k − 1, are symmetric and nonnegative definite and V
k
is an identity matrix. These formulae involve a basis of a quadratic subspace containing MV
1
M,...,MV
k-1
M,M, where M is an orthogonal projector on the null space of X′. In the paper we discuss methods of construction of such a basis. We survey Malley’s algorithms for finding the smallest
quadratic subspace including a given set of symmetric matrices of the same order and propose some modifications of these algorithms.
We also consider a class of matrices sharing some of the symmetries common to MV
1
M,...,MV
k-1
M,M. We show that the matrices from this class constitute a quadratic subspace and describe its explicit basis, which can be
directly used for computing Bayes invariant quadratic estimators of variance components. This basis can be also used for improving
the efficiency of Malley’s algorithms when applied to finding a basis of the smallest quadratic subspace containing the matrices
MV
1
M,...,MV
k-1
M,M. Finally, we present the results of a numerical experiment which confirm the potential usefulness of the proposed methods.
Dedicated to the memory of Professor Stanisław Gnot. 相似文献
18.
When constructing uniform random numbers in [0, 1] from the output of a physical device, usually n independent and unbiased bits B
j
are extracted and combined into the machine number
. In order to reduce the number of data used to build one real number, we observe that for independent and exponentially distributed random variables X
n
(which arise for example as waiting times between two consecutive impulses of a Geiger counter) the variable U
n : = X
2n – 1/(X
2n – 1 + X
2n
) is uniform in [0, 1]. In the practical application X
n
can only be measured up to a given precision (in terms of the expectation of the X
n
); it is shown that the distribution function obtained by calculating U
n
from these measurements differs from the uniform by less than /2.We compare this deviation with the error resulting from the use of biased bits B
j
with P
{B
j
= 1{ =
(where ] –
[) in the construction of Y above. The influence of a bias is given by the estimate that in the p-total variation norm Q
TV
p
= (
|Q()|
p
)1/p
(p 1) we have P
Y
– P
0
Y
TV
p
(c
n
· )1/p
with c
n
p
for n . For the distribution function F
Y
– F
0
Y
2(1 – 2–n
)|| holds. 相似文献
19.
20.
Summary In this paper likelihood is characterized as an index which measures how much a model fits a sample. Some properties required
to an index of fit are introduced and discussed, while stressing how they describe aspects inner to idea of fit. Finally we
prove that, if an index of fit is maximal when the model reaches the distribution of the sample, then such an index is an
increasing continuous transform of
, where thep
i's are the theoretical relative frequencies provided by the model and theq
i's are the actual relative frequencies of the sample. 相似文献