共查询到20条相似文献,搜索用时 62 毫秒
1.
Pawel R. Pordzik 《Statistical Papers》2012,53(2):299-304
Let
[^(\varveck)]{\widehat{\varvec{\kappa}}} and
[^(\varveck)]r{\widehat{\varvec{\kappa}}_r} denote the best linear unbiased estimators of a given vector of parametric functions
\varveck = \varvecKb{\varvec{\kappa} = \varvec{K\beta}} in the general linear models
M = {\varvecy, \varvecX\varvecb, s2\varvecV}{{\mathcal M} = \{\varvec{y},\, \varvec{X\varvec{\beta}},\, \sigma^2\varvec{V}\}} and
Mr = {\varvecy, \varvecX\varvecb | \varvecR \varvecb = \varvecr, s2\varvecV}{{\mathcal M}_r = \{\varvec{y},\, \varvec{X}\varvec{\beta} \mid \varvec{R} \varvec{\beta} = \varvec{r},\, \sigma^2\varvec{V}\}}, respectively. A bound for the Euclidean distance between
[^(\varveck)]{\widehat{\varvec{\kappa}}} and
[^(\varveck)]r{\widehat{\varvec{\kappa}}_r} is expressed by the spectral distance between the dispersion matrices of the two estimators, and the difference between sums
of squared errors evaluated in the model M{{\mathcal M}} and sub-restricted model Mr*{{\mathcal M}_r^*} containing an essential part of the restrictions
\varvecR\varvecb = \varvecr{\varvec{R}\varvec{\beta} = \varvec{r}} with respect to estimating
\varveck{\varvec{\kappa}}. 相似文献
2.
The selection of copulas is an important aspect of dependence modeling issues. In many practical applications, only a limited
number of copulas is tested and the copula with the best result for a goodness-of-fit test is chosen, which, however, does
not always lead to the best possible fit. In this paper we develop a practical and logical method for improving the goodness-of-fit
of a particular Archimedean copula by means of transforms. In order to do this, we introduce concordance invariant transforms which can also be tail dependence preserving, based on an analysis on the λ-function,
l = \fracjj¢{\lambda=\frac{\varphi}{\varphi'}}, where j{\varphi} is the Archimedean generator. The methodology is applied to the data set studied in Cook and Johnson (J R Stat Soc B 43:210–218,
1981) and Genest and Rivest (J Am Stat Assoc 88:1043–1043, 1993), where we improve the fit of the Frank copula and obtain
statistically significant results. 相似文献
3.
For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form Lr, w, d0(q, d)=wr(d0, d)+ (1-w) r(q, d){L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}, as well as the weighted version q(q) Lr, w, d0(q, d){q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}, where ρ(θ, δ) is an arbitrary loss function, δ
0 is a chosen a priori “target” estimator of q, w ? [0,1){\theta, \omega \in[0,1)}, and q(·) is a positive weight function. we develop Bayesian estimators under Lr, w, d0{L_{\rho, \omega, \delta_0}} with ω > 0 by relating such estimators to Bayesian solutions under Lr, w, d0{L_{\rho, \omega, \delta_0}} with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria
including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections
between optimal actions derived under balanced and unbalanced losses. 相似文献
4.
Yongge Tian 《Statistical Methods and Applications》2017,26(4):583-608
Assume that a linear random-effects model \(\mathbf{y}= \mathbf{X}\varvec{\beta }+ \varvec{\varepsilon }= \mathbf{X}(\mathbf{A}\varvec{\alpha }+ \varvec{\gamma }) + \varvec{\varepsilon }\) is transformed as \(\mathbf{T}\mathbf{y}= \mathbf{T}\mathbf{X}\varvec{\beta }+ \mathbf{T}\varvec{\varepsilon }= \mathbf{T}\mathbf{X}(\mathbf{A}\varvec{\alpha }+ \varvec{\gamma }) + \mathbf{T}\varvec{\varepsilon }\) by pre-multiplying a given matrix \(\mathbf{T}\) of arbitrary rank. These two models are not necessarily equivalent unless \(\mathbf{T}\) is of full column rank, and we have to work with this derived model in many situations. Because predictors/estimators of the parameter spaces under the two models are not necessarily the same, it is primary work to compare predictors/estimators in the two models and to establish possible links between the inference results obtained from two models. This paper presents a general algebraic approach to the problem of comparing best linear unbiased predictors (BLUPs) of parameter spaces in an original linear random-effects model and its transformations, and provides a group of fundamental and comprehensive results on mathematical and statistical properties of the BLUPs. In particular, we construct many equalities for the BLUPs under an original linear random-effects model and its transformations, and obtain necessary and sufficient conditions for the equalities to hold. 相似文献
5.
Arjun K. Gupta Johanna Marcela Orozco-Castañeda Daya K. Nagar 《Statistical Papers》2011,52(1):139-152
Let U, V and W be independent random variables, U and V having a gamma distribution with respective shape parameters a and b, and W having a non-central gamma distribution with shape and non-centrality parameters c and δ, respectively. Define X = U/(U + W) and Y = V/(V + W). Clearly, X and Y are correlated each having a non-central beta type 1 distribution, X ~ NCB1 (a,c;d){X \sim {\rm NCB1} (a,c;\delta)} and Y ~ NCB1 (b,c;d){Y \sim {\rm NCB1} (b,c;\delta)} . In this article we derive the joint probability density function of X and Y and study its properties. 相似文献
6.
Krämer (Sankhy $\bar{\mathrm{a }}$ 42:130–131, 1980) posed the following problem: “Which are the $\mathbf{y}$ , given $\mathbf{X}$ and $\mathbf{V}$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $\mathbf{y}$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $\varvec{\beta }$ coincide under the general Gauss–Markov model $\mathbf{y} = \mathbf{X} \varvec{\beta } + \mathbf{u}$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $\varvec{\beta }$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $\varvec{\beta }$ , we consider the estimation of the systematic part $\mathbf{X} \varvec{\beta }$ , which is a natural consequence of relaxing the assumption that $\mathbf{X}$ and $\mathbf{V}$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $\mathbf{X} \varvec{\beta }$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors. 相似文献
7.
The paper introduces a new difference-based ridge regression estimator [^(b)](k){\hat{\beta}(k)} of the regression parameters β in the partial linear model. Its mean-squared error is compared analytically with the non-ridge version [^(b)](0){\hat{\beta}(0)} . Finally, the performance of the new estimator is evaluated for a real data set. 相似文献
8.
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
9.
We consider equalities between the ordinary least squares estimator ( $\mathrm {OLSE} $ ), the best linear unbiased estimator ( $\mathrm {BLUE} $ ) and the best linear unbiased predictor ( $\mathrm {BLUP} $ ) in the general linear model $\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$ extended with the new unobservable future value $ \mathbf y _{*}$ of the response whose expectation is $ \mathbf X _{*}\varvec{\beta }$ . Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the $\mathrm {BLUP} $ and provide new results giving upper bounds for the Euclidean norm of the difference between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$ and between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$ . A remark is made on the application to small area estimation. 相似文献
10.
A pair of polychotomous random variables \((Y_1,Y_2)^\top =:{\varvec{Y}}\), where each \(Y_j\) has a totally ordered support, is studied within a penalized generalized linear model framework. We deal with a triangular generating process for \({\varvec{Y}}\), a structure that has been employed in the literature to control for the presence of residual confounding. Differently from previous works, however, the proposed model allows for a semi-parametric estimation of the covariate-response relationships. In this way, the risk of model mis-specification stemming from the imposition of fixed-order polynomial functional forms is also reduced. The proposed estimation methods and related inferential results are finally applied to study the effect of education on alcohol consumption among young adults in the UK. 相似文献
11.
The general mixed linear model can be denoted by y = X β + Z u + e , where β is a vector of fixed effects, u is a vector of random effects, and e is a vector of random errors. In this article, the problem of admissibility of Q y and Q y + q for estimating linear functions, ? = L ′β + M ′ u , of the fixed and random effects is considered, and the necessary and sufficient conditions for Q y (resp. Q y + q ) to be admissible in the set of homogeneous (resp. potentially inhomogeneous) linear estimators with respect to the MSE and MSEM criteria are investigated. We provide a straightforward alternative proof to the method that was utilized by Wu (1988), Baksalary and Markiewicz (1990), and Groß and Markiewicz (1999). In addition, we derive the corresponding results on the admissibility problem under the generalized MSE criterion. 相似文献
12.
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
13.
Estimation of population parameters is considered by several statisticians when additional information such as coefficient
of variation, kurtosis or skewness is known. Recently Wencheko and Wijekoon (Stat Papers 46:101–115, 2005) have derived minimum
mean square error estimators for the population mean in one parameter exponential families when coefficient of variation is
known. In this paper the results presented by Gleser and Healy (J Am Stat Assoc 71:977–981, 1976) and Arnholt and Hebert (, 2001) were generalized by considering T (X) as a minimal sufficient estimator of the parametric function g(θ) when the ratio t2=[ g(q) ]-2Var[ T(X ) ]{\tau^{2}=[ {g(\theta )} ]^{-2}{\rm Var}[ {T(\boldsymbol{X} )} ]} is independent of θ. Using these results the minimum mean square error estimator in a certain class for both population mean and variance can
be obtained. When T (X) is complete and minimal sufficient, the ratio τ2 is called “WIJLA” ratio, and a uniformly minimum mean square error estimator can be derived for the population mean and variance.
Finally by applying these results, the improved estimators for the population mean and variance of some distributions are
obtained. 相似文献
14.
We consider the linear regression modely=Xβ+u with prior information on the unknown parameter vector β. The additional information on β is given by a fuzzy set. Using
the mean squared error criterion we derive linear estimators that optimally combine the data with the fuzzy prior information.
Our approach generalizes the classical minimax procedure firstly proposed by Kuks and Olman. 相似文献
15.
We extend the study of weak local conditional independence (WCLI) based on a measurability condition made by (Commenges and
Gégout-Petit J R Stat Soc B 71:1–18) to a larger class of processes that we call D¢{\bf {\mathcal{D}'}}. We also give a definition related to the same concept based on certain likelihood processes, using the Girsanov theorem.
Under certain conditions, the two definitions coincide on D¢{\bf {\mathcal{D}'}}. These results may be used in causal models in that we define what may be the largest class of processes in which influences
of one component of a stochastic process on another can be described without ambiguity. From WCLI we can construct a concept
of strong local conditional independence (SCLI). When WCLI does not hold, there is a direct influence while when SCLI does
not hold there is direct or indirect influence. We investigate whether WCLI and SCLI can be defined via conventional independence
conditions and find that this is the case for the latter but not for the former. Finally we recall that causal interpretation
does not follow from mere mathematical definitions, but requires working with a good system and with the true probability. 相似文献
16.
In this paper, by relaxing the mixing coefficients to α(n) = O(n ?β), β > 3, we investigate the Bahadur representation of sample quantiles under α-mixing sequence and obtain the rate as ${O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})}$ . Meanwhile, for any δ > 0, by strengthening the mixing coefficients to α(n) = O(n ?β ), ${\beta > \max\{3+\frac{5}{1+\delta},1+\frac{2}{\delta}\}}$ , we have the rate as ${O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$ . Specifically, if ${\delta=\frac{\sqrt{41}-5}{4}}$ and ${\beta > \frac{\sqrt{41}+7}{2}}$ , then the rate is presented as ${O(n^{-\frac{\sqrt{41}+5}{16}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$ . 相似文献
17.
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
18.
M. A. Alkhamisi 《Statistical Papers》2010,51(3):651-672
In this paper, we propose two SUR type estimators based on combining the SUR ridge regression and the restricted least squares
methods. In the sequel these estimators are designated as the restricted ridge Liu estimator and the restricted ridge HK estimator
(see Liu in Commun Statist Thoery Methods 22(2):393–402, 1993; Sarkar in Commun Statist A 21:1987–2000, 1992). The study has
been made using Monte Carlo techniques, (1,000 replications), under certain conditions where a number of factors that may
effect their performance have been varied. The performance of the proposed and some of the existing estimators are evaluated
by means of the TMSE and the PR criteria. Our results indicate that the proposed SUR restricted ridge estimators based on
K
SUR, K
Sratio, K
Mratio and [(K)\ddot]{\ddot{K}} produced smaller TMSE and/or PR values than the remaining estimators. In contrast with other ridge estimators, components
of [(K)\ddot]{\ddot{K}} are defined in terms of the eigenvalues of X*¢ X*{X^{{\ast^{\prime}}} X^{\rm \ast}} and all lie in the open interval (0, 1). 相似文献
19.
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
20.
We introduce some new mathematical tools in the analysis of dispersion matrices of the two well-known OLSEs and BLUEs under general linear models with parameter restrictions. We first establish some formulas for calculating the ranks and inertias of the differences of OLSEs’ and BLUEs’ dispersion matrices of parametric functions under the general linear model \({\mathscr {M}}= \{\mathbf{y}, \ \mathbf{X }\pmb {\beta }, \ \pmb {\Sigma }\}\) and the constrained model \({\mathscr {M}}_r = \{\mathbf{y}, \, \mathbf{X }\pmb {\beta }\, | \, \mathbf{A }\pmb {\beta }= \mathbf{b}, \ \pmb {\Sigma }\}\), where \(\mathbf{A }\pmb {\beta }= \mathbf{b}\) is a consistent linear matrix equation for the unknown parameter vector \(\pmb {\beta }\) to satisfy. As applications, we derive necessary and sufficient conditions for many equalities and inequalities of OLSEs’ and BLUEs’ dispersion matrices to hold under \({\mathscr {M}}\) and \({\mathscr {M}}_r\). 相似文献