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1.
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
2.
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
3.
There are many situations where the usual random sample from a population of interest is not available, due to the data having
unequal probabilities of entering the sample. The method of weighted distributions models this ascertainment bias by adjusting
the probabilities of actual occurrence of events to arrive at a specification of the probabilities of the events as observed
and recorded. We consider two different classes of contaminated or mixture of weight functions, Γ
a
={w(x):w(x)=(1−ε)w
0(x)+εq(x),q∈Q} and Γ
g
={w(x):w(x)=w
0
1−ε
(x)q
ε(x),q∈Q} wherew
0(x) is the elicited weighted function,Q is a class of positive functions and 0≤ε≤1 is a small number. Also, we study the local variation of ϕ-divergence over classes
Γ
a
and Γ
g
. We devote on measuring robustness using divergence measures which is based on the Bayesian approach. Two examples will be
studied. 相似文献
4.
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
5.
A. Stepanov 《Statistical Papers》2007,48(1):63-79
LetX
1,X
2, … be a sequence of i.i.d. random variables with some continuous distribution functionF. LetX(n) be then-th record value associated with this sequence and μ
n
−
, μ
n
+
be the variables that count the number of record values belonging to the random intervals(f−(X(n)), X(n)), (X(n), f+(X(n))), wheref−, f+ are two continuous functions satisfyingf−(x)<x, f+(x)>x. Properties of μ
n
−
, μ
n
+
are studied in the present paper. Some statistical applications connected with these variables are also provided. 相似文献
6.
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
7.
In this paper, we study the asymptotic distributions of MLE and UMVUE of a parametric functionh(θ1, θ2) when sampling from a biparametric uniform distributionU(θ1, θ2). We obtain both limiting distributions as a convolution of exponential distributions, and we observe that the limiting distribution
of UMVUE is a shift of the limiting distribution of MLE. 相似文献
8.
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
9.
David D. Hanagal 《Statistical Papers》1999,40(1):99-106
In this paper, we estimate the reliability of a system with k components. The system functions when at least s (1≤s≤k) components survive a common random stress. We assume that the strengths of these k components are subjected to a common
stress which is independent of the strengths of these k components. If (X
1,X
2,…,X
k
) are strengths of k components subjected to a common stress (Y), then the reliability of the system or system reliability
is given byR=P[Y<X
(k−s+1)] whereX
(k−s+1) is (k−s+1)-th order statistic of (X
1,…,X
k
). We estimate R when (X
1,…,X
k
) follow an absolutely continuous multivariate exponential (ACMVE) distribution of Hanagal (1993) which is the submodel of
Block (1975) and Y follows an independent exponential distribution. We also obtain the asymptotic normal (AN) distribution
of the proposed estimator. 相似文献
10.
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
11.
In this article, we study the joint distribution of X and two linear combinations of order statistics, a
T
Y
(2) and b
T
Y
(2), where a = (a
1, a
2)
T
and b = (b
1, b
2)
T
are arbitrary vectors in R
2 and Y
(2) = (Y
(1), Y
(2))
T
is a vector of ordered statistics obtained from (Y
1, Y
2)
T
when (X, Y
1, Y
2)
T
follows a trivariate normal distribution with a positive definite covariance matrix. We show that this distribution belongs
to the skew-normal family and hence our work is a generalization of Olkin and Viana (J Am Stat Assoc 90:1373–1379, 1995) and
Loperfido (Test 17:370–380, 2008). 相似文献
12.
Two variance components model for which each invariant quadratic admissible estimator of a linear function of variance components
(under quadratic loss function) is a linear combination of two quadratic forms,Z
1,Z
2, say, is considered. A setD={(d
1,d
2)′:d
1
Z
1+d
2
Z
2 is admissible} is described by giving formulae on the boundary ofD. Different forms of the setD are presented on figures. 相似文献
13.
A basic concept for comparing spread among probability distributions is that of dispersive ordering. Let X and Y be two random variables with distribution functions F and G, respectively. Let F
−1 and G
−1 be their right continuous inverses (quantile functions). We say that Y is less dispersed than X (Y≤
disp
X) if G
−1(β)−G
−1(α)≤F
−1(β)−F
−1(α), for all 0<α≤β<1. This means that the difference between any two quantiles of G is smaller than the difference between the corresponding quantiles of F. A consequence of Y≤
disp
X is that |Y
1−Y
2| is stochastically smaller than |X
1−X
2| and this in turn implies var(Y)≤var(X) as well as E[|Y
1−Y
2|]≤E[|X
1−X
2|], where X
1, X
2 (Y
1, Y
2) are two independent copies of X(Y). In this review paper, we give several examples and applications of dispersive ordering in statistics. Examples include those
related to order statistics, spacings, convolution of non-identically distributed random variables and epoch times of non-homogeneous
Poisson processes.
This work was supported in part by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.
Subhash Kochar is thankful to Dr. B. Khaledi for many helpful discussions. 相似文献
14.
We introduce a new family of skew-normal distributions that contains the skew-normal distributions introduced by Azzalini
(Scand J Stat 12:171–178, 1985), Arellano-Valle et al. (Commun Stat Theory Methods 33(7):1465–1480, 2004), Gupta and Gupta
(Test 13(2):501–524, 2008) and Sharafi and Behboodian (Stat Papers, 49:769–778, 2008). We denote this distribution by GBSN
n
(λ1, λ2). We present some properties of GBSN
n
(λ1, λ2) and derive the moment generating function. Finally, we use two numerical examples to illustrate the practical usefulness
of this distribution. 相似文献
15.
L.Y. Chan J.H. Meng Y.C. Jiang & Y.N. Guan 《Australian & New Zealand Journal of Statistics》1998,40(3):359-372
The paper discusses D-optimal axial designs for the additive quadratic and cubic mixture models σ1≤i≤q(βixi + βiix2i) and σ1≤i≤q(βixi + βiix2i + βiiix3i), where xi≥ 0, x1 + . . . + xq = 1. For the quadratic model, a saturated symmetric axial design is used, in which support points are of the form (x1, . . . , xq) = [1 ? (q?1)δi, δi, . . . , δi], where i = 1, 2 and 0 ≤δ2 <δ1 ≤ 1/(q ?1). It is proved that when 3 ≤q≤ 6, the above design is D-optimal if δ2 = 0 and δ1 = 1/(q?1), and when q≥ 7 it is D-optimal if δ2 = 0 and δ1 = [5q?1 ? (9q2?10q + 1)1/2]/(4q2). Similar results exist for the cubic model, with support points of the form (x1, . . . , xq) = [1 ? (q?1)δi, δi, . . . , δi], where i = 1, 2, 3 and 0 = δ3 <δ2 < δ1 ≤1/(q?1). The saturated D-optimal axial design and D-optimal design for the quadratic model are compared in terms of their efficiency and uniformity. 相似文献
16.
The consequences of substituting the denominator Q
3(p) − Q
1(p) by Q
2 − Q
1(p) in Groeneveld’s class of quantile measures of kurtosis (γ
2(p)) for symmetric distributions, are explored using the symmetric influence function. The relationship between the measure
γ
2(p) and the alternative class of kurtosis measures κ2(p) is derived together with the relationship between their influence functions. The Laplace, Logistic, symmetric Two-sided
Power, Tukey and Beta distributions are considered in the examples in order to discuss the results obtained pertaining to
unimodal, heavy tailed, bounded domain and U-shaped distributions.
The authors thank the referee for the careful review. 相似文献
17.
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
18.
Consider the problem of obtaining a confidence interval for some function g(θ) of an unknown parameter θ, for which a (1-α)-confidence
interval is given. If g(θ) is one-to-one the solution is immediate. However, if g is not one-to-one the problem is more complex
and depends on the structure of g. In this note the situation where g is a nonmonotone convex function is considered. Based
on some inequality, a confidence interval for g(θ) with confidence level at least 1-α is obtained from the given (1-α) confidence
interval on θ. Such a result is then applied to the n(μ, σ
2) distribution with σ known. It is shown that the coverage probability of the resulting confidence interval, while being greater
than 1-α, has in addition an upper bound which does not exceed Θ(3z1−α/2)-α/2. 相似文献
19.
Andreas Kyriakoussis 《统计学通讯:理论与方法》2017,46(8):4088-4102
In this study, we introduce the Heine process, {Xq(t), t > 0}, 0 < q < 1, where the random variable Xq(t), for every t > 0, represents the number of events (occurrences or arrivals) during a time interval (0, t]. The Heine process is introduced as a q-analog of the basic Poisson process. Also, in this study, we prove that the distribution of the waiting time Wν, q, ν ? 1, up to the νth arrival, is a q-Erlang distribution and the interarrival times Tk, q = Wk, q ? Wk ? 1, q,?k = 1, 2, …, ν with W0, q = 0 are independent and equidistributed with a q-Exponential distribution. 相似文献
20.