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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.
Let Z
1, Z
2, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z
t
= 1) and q = Pr(Z
t
= 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E1{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E2{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E3{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by Nn,k(i){ N_{n,k}^{(i)}} (respectively Mn,k(i){M_{n,k}^{(i)}}) the number of occurrences of the pattern Ei{\mathcal{E}_{i}} , i = 1, 2, 3, in Z
1, Z
2, . . . , Z
n
when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let Tr,k(i){T_{r,k}^{(i)}} (resp. Wr,k(i)){W_{r,k}^{(i)})} be the waiting time for the r − th occurrence of the pattern Ei{\mathcal{E}_{i}}, i = 1, 2, 3, in Z
1, Z
2, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study
of Nn,k(i){N_{n,k}^{(i)}}, Mn,k(i){M_{n,k}^{(i)}}, Tr,k(i){T_{r,k}^{(i)}} and Wr,k(i){W_{r,k}^{(i)}} (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass
functions and moments. An application is given. 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.
Suppose [^(q)]{\widehat{\theta}} is an estimator of θ in
\mathbbR{\mathbb{R}} that satisfies the central limit theorem. In general, inferences on θ are based on the central limit approximation. These have error O(n
−1/2), where n is the sample size. Many unsuccessful attempts have been made at finding transformations which reduce this error to O(n
−1). The variance stabilizing transformation fails to achieve this. We give alternative transformations that have bias O(n
−2), and skewness O(n
−3). Examples include the binomial, Poisson, chi-square and hypergeometric distributions. 相似文献
5.
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
6.
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
7.
The mixture of Rayleigh random variables X
1and X
2 are identified in terms of relations between the conditional expectation of ( X2:22 -X1:22)r{\left( {X_{2:2}^2 -X_{1:2}^2}\right)^{r}} given X
1:2 (or X2:22k{X_{2:2}^{2k}} given X1:2,"k £ r){X_{1:2},\forall k\leq r)} and hazard rate function of the distribution, where X
1:2 and X
2:2 denote the corresponding order statistics, r is a positive integer. In addition, we also mention some related theorems to characterize the mixtures of Rayleigh distributions.
Finally, we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems) and a simulated example is used
to illustrate our results. 相似文献
8.
9.
Let {Tn, n ≥ 1} be an arbitrary sequence of nonlattice random variables and let {Sn, n ≥ 1} be another sequence of positive random variables. Assume that the sequences are independent. In this paper we obtain asymptotic expression for the density function of the ratio statistic Rn = Tn/Sn based on simple conditions on the moment generating functions of Tn and Sn. When Sn = re, our main result reduces to that of Chaganty and Sethura-man[Ann. Probab. 13(1985):97-114]. We also obtain analogous results when Tn and Sn are both lattice random variables. We call our theorems large deviation local limit theorems for Rn, since the conditions of our theorems imply that Rn → c in probability for some constant c. We present some examples to illustrate our theorems. 相似文献
10.
Summary A standard improper prior for the parameters of a MANOVA model is shown to yield an inference that is incoherent in the sense
of Heath and Sudderth. The proof of incoherence is based on the fact that the formal Bayes estimate, sayδ
0
, of the covariance matrix based on the improper prior and a certain bounded loss function is uniformly inadmissible in that
there is another estimatorδ
l
and an ɛ>0 such that the risk functions satisfyR(δ
l
,Σ)⩽R δ
0
,Σ)−ε for all values of the covariance matrix Σ. The estimatorδ
I
is formal Bayes for an alternative improper prior which leads to a coherent inference.
Research supported by National Science Foundation grants DMS-89-22607 (for Eaton) and DMS-9123358 (for Sudderth). 相似文献
11.
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
12.
Summary We consider a lotL formed byN apparently similar unitsW
1,…,W
N, where each of theW
i may come from one of two different populationsP
1 andP
2;T
1,…,T
N denote the corresponding lifetimes. The units fromP
i
undergo a failure of kindi and their survival function isS
i
(t).
We assume that the failure rate function
are known and that the units fromP
1 are ?substandard?: λ
1
(t)≥λ
2
(t), ∀t≥0.
We want to putW
1,…,W
N under a pre-operational test (burn-in test) in order to eliminate at least a great part of the substandard units and we face
the problem of obtaining a rule for stopping the test under the assumption that, with the failure of a unit, it is possible
to recognize the population from which the unit comes.
Such a problem will be formalized as an optimal stopping problem for a suitably defined Markov process. Our study shall evidentiate
some fundamental aspects of the problem and the role of the prior distribution of the (random) numberM
0 of those units inL coming fromP
1 (substandard). The latter distribution has a great influence on the form of the solution.
This research was supported by the C.N.R. Project ?Statistica Bayesiana e Simulazione in Affidalità e Modellistica Biologica?. 相似文献
13.
Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕj} be a complete orthonormal sequence of functions onL
2(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that
over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by
setting all but the firstm of the ϑj’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series.
Research partially supported by: CNR grant, n. 93. 00837. CT10. 相似文献
14.
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
15.
Maciej Wilczyński 《Statistical Papers》2012,53(1):151-164
Arnold and Stahlecker (Stat Pap 44:107–115, 2003) considered the prediction of future values of the dependent variable in
the linear regression model with a relative squared error and deterministic disturbances. They found an explicit form for
a minimax linear affine solution d* of that problem. In the paper we generalize this result proving that the decision rule d* is also minimax when the class D{\mathcal{D}} of possible predictors of the dependent variable is unrestricted. Then we show that d* remains minimax in D{\mathcal{D}} when the disturbances are random with the mean vector zero and the known positive definite covariance matrix. 相似文献
16.
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
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.
We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy
k
. This may correspond to the case of a regression model, where one observesy
k
=f(θ,x
k
)+ε
k
, with ε
k
some random error, or to the Bernoulli case wherey
k
∈{0, 1}, with Pr[y
k
=1|θ,x
k
|=f(θ,x
k
). Special attention is given to sequences given by
, with
an estimated value of θ obtained from (x1, y1),...,(x
k
,y
k
) andd
k
(x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon,
where one wants to maximize ∑
i=1
N
w
i
f(θ, x
i
) with {w
i
} a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method
for a binary response problem. 相似文献
19.
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
20.
Summary Let {X
n
} be a sequence of random variables conditionally independent and identically distributed given the random variable Θ. The
aim of this paper is to show that in many interesting situations the conditional distribution of Θ, given (X
1,…,X
n
), can be approximated by means of the bootstrap procedure proposed by Efron and applied to a statisticT
n
(X
1,…,X
n
) sufficient for predictive purposes. It will also be shown that, from the predictive point of view, this is consistent with
the results obtained following a common Bayesian approach. 相似文献