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1.
Let {X
j
} be a strictly stationary sequence of negatively associated random variables with the marginal probability density function
f(x). The recursive kernel estimators of f(x) are defined by
and the Rosenblatt–Parzen’s kernel estimator of f(x) is defined by , where 0 < b
n
→ 0 are bandwidths and K is some kernel function. In this paper, we study the uniformly Berry–Esseen bounds for these estimators of f(x). In particular, by choice of the bandwidths, the Berry–Esseen bounds of the estimators attain . 相似文献
2.
Minimax estimation of a cumulative distribution function by converting to a parametric problem 总被引:2,自引:1,他引:2
Let X = (X
1,...,X
n
) be a sample from an unknown cumulative distribution function F defined on the real line
. The problem of estimating the cumulative distribution function F is considered using a decision theoretic approach. No assumptions are imposed on the unknown function F. A general method of finding a minimax estimator d(t;X) of F under the loss function of a general form is presented. The method of solution is based on converting the nonparametric problem
of searching for minimax estimators of a distribution function to the parametric problem of searching for minimax estimators
of the probability of success for a binomial distribution. The solution uses also the completeness property of the class of
monotone decision procedures in a monotone decision problem. Some special cases of the underlying problem are considered in
the situation when the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute
error. 相似文献
3.
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
4.
F. Brodeau 《Metrika》1999,49(2):85-105
This paper is devoted to the study of the least squares estimator of f for the classical, fixed design, nonlinear model X (t
i)=f(t
i)+ε(t
i), i=1,2,…,n, where the (ε(t
i))i=1,…,n are independent second order r.v.. The estimation of f is based upon a given parametric form. In Brodeau (1993) this subject has been studied in the homoscedastic case. This time
we assume that the ε(t
i) have non constant and unknown variances σ2(t
i). Our main goal is to develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another for comparing two such
classes. The fundamental tool is an approximation of the elements of these classes by more regular functions, which leads
to asymptotic properties of estimators based on the least squares estimator of the unknown parameters. We point out that Neubauer
and Zwanzig (1995) have obtained interesting results for connected subjects by using the same technique of approximation.
Received: February 1996 相似文献
5.
Tang Qingguo 《Metrika》2009,69(1):55-67
Suppose that the longitudinal observations (Y
ij
, X
ij
, t
ij
) for i = 1, . . . ,n; j = 1, . . . ,m
i
are modeled by the semiparamtric model where β
0 is a k × 1 vector of unknown parameters, g(·) is an unknown estimated function and e
ij
are unobserved disturbances. This article consider M-type regressions which include mean, median and quantile regressions.
The M-estimator of the slope parameter β
0 is obtained through piecewise local polynomial approximation of the nonparametric component. The local M-estimator of g(·) is also obtained by replacing β
0 in model with its M-estimator and using local linear approximation. The asymptotic distribution of the estimator of β
0 is derived. The asymptotic distributions of the local M-estimators of g(·) at both interior and boundary points are also established. Various applications of our main results are given.
The research is supported in part by National Natural Science Foundation of China (Grant No. 10671089). 相似文献
6.
Herbert Vogt 《Metrika》1996,44(1):207-221
Let ζ
t
be the number of events which will be observed in the time interval [0;t] and define
as the average number of events per time unit if this limit exists. In the case of i.i.d. waiting-times between the events,E[ζ
t
] is the renewal function and it follows from well-known results of renewal theory thatA exists and is equal to 1/τ, if τ>0 is the expectation of the waiting-times.
This holds true also when τ = ∞.A may be estimate by ζ
t
/t or
where
is the mean of the firstn waiting-timesX
1,X
2, ...,X
n
. Both estimators converage with probability 1 to 1/τ if theX
i are i.i.d.; but the expectation of
may be infinite for alln and also if it is finite,
is in general a positively biased estimator ofA. For a stationary renewal process, ζ
t
/t is unbiased for eacht; if theX
i
are i.i.d. with densityf(x), then ζ
t
/t has this property only iff(x) is of the exponential type and only for this type the numbers of events in consecutive time intervals [0,t], [t, 2t], ... are i.i.d. random variables for arbitraryt > 0. 相似文献
7.
Consider the heteroscedastic regression model Y
(j)(x
in
, t
in
) = t
in
β + g(x
in
) + σ
in
e
(j)(x
in
), 1 ≤ j ≤ m, 1 ≤ i ≤ n, where sin2=f(uin){\sigma_{in}^{2}=f(u_{in})}, (x
in
, t
in
, u
in
) are fixed design points, β is an unknown parameter, g(·) and f(·) are unknown functions, and the errors {e
(j)(x
in
)} are mean zero NA random variables. The moment consistency for least-squares estimators and weighted least-squares estimators
of β is studied. In addition, the moment consistency for estimators of g(·) and f(·) is investigated. 相似文献
8.
We consider the problem of constructing simultaneous fixed-width confidence intervals for all pairwise treatment differences
μ1−μ
J
, in the presence ofk(≥2) independent populationsN
p
(μ1,Σ), 1≤i≠j≤k. Appropriate purely sequential, accelerated sequential and three-stage sampling strategies have been developed and variousfirst-order asymptotic properties are then derived when Σ
pxp
is completely unknown, but positive definite (p.d.). In the two special cases when the largest component variance in Σ is
a known multiple of one of the variances or Σ=σ2
H where σ(>0) is unknown, butH
pxp is known and p.d., the original multistage sampling strategies are specialized. Under such special circumstances, associatedsecond-order characteristics are then developed. It is to be noted that our present formulation and the methodologies fill important voids
in the context of multivariate multiple comparisons which is a challenging area that has not yet been fully explored. Moderate
sample performances of the proposed techniques were very encouraging and detailed remarks on these were included in Mukhopadhyay
and Aoshima (1997). 相似文献
9.
In two recent papers by Balakrishnan et al. (J Qual Technol 39:35–47, 2007; Ann Inst Stat Math 61:251–274, 2009), the maximum
likelihood estimators [^(q)]1{\hat{\theta}_{1}} and [^(q)]2{\hat{\theta}_{2}} of the parameters θ
1 and θ
2 have been derived in the framework of exponential simple step-stress models under Type-II and Type-I censoring, respectively.
Here, we prove that these estimators are stochastically monotone with respect to θ
1 and θ
2, respectively, which has been conjectured in these papers and then utilized to develop exact conditional inference for the
parameters θ
1 and θ
2. For proving these results, we have established a multivariate stochastic ordering of a particular family of trinomial distributions
under truncation, which is also of independent interest. 相似文献
10.
A consistent test for multivariate normality based on the empirical characteristic function 总被引:2,自引:1,他引:1
LetX
1,X
2, …,X
n be independent identically distributed random vectors in IR
d
,d ⩾ 1, with sample mean
and sample covariance matrixS
n. We present a practicable and consistent test for the composite hypothesisH
d: the law ofX
1 is a non-degenerate normal distribution, based on a weighted integral of the squared modulus of the difference between the
empirical characteristic function of the residualsS
n
−1/2
(X
j −
) and its pointwise limit exp (−1/2|t|2) underH
d. The limiting null distribution of the test statistic is obtained, and a table with critical values for various choices ofn andd based on extensive simulations is supplied. 相似文献
11.
It is well-known that the naive bootstrap yields inconsistent inference in the context of data envelopment analysis (DEA)
or free disposal hull (FDH) estimators in nonparametric frontier models. For inference about efficiency of a single, fixed
point, drawing bootstrap pseudo-samples of size m < n provides consistent inference, although coverages are quite sensitive to the choice of subsample size m. We provide a probabilistic framework in which these methods are shown to valid for statistics comprised of functions of
DEA or FDH estimators. We examine a simple, data-based rule for selecting m suggested by Politis et al. (Stat Sin 11:1105–1124, 2001), and provide Monte Carlo evidence on the size and power of our tests. Our methods (i) allow for heterogeneity in the inefficiency
process, and unlike previous methods, (ii) do not require multivariate kernel smoothing, and (iii) avoid the need for solutions
of intermediate linear programs. 相似文献
12.
This paper presents a method for estimating the model Λ(Y)=min(β′X+U, C), where Y is a scalar, Λ is an unknown increasing function, X is a vector of explanatory variables, β is a vector of unknown parameters, U has unknown cumulative distribution function F, and C is a censoring threshold. It is not assumed that Λ and F belong to known parametric families; they are estimated nonparametrically. This model includes many widely used models as special cases, including the proportional hazards model with unobserved heterogeneity. The paper develops n1/2-consistent, asymptotically normal estimators of Λ and F. Estimators of β that are n1/2-consistent and asymptotically normal already exist. The results of Monte Carlo experiments illustrate the finite-sample behavior of the estimators. 相似文献
13.
In this paper we deal with the problem of classifying a p-dimensional random vector into one of two elliptically contoured populations with unknown and distinct mean vectors and a common, but unknown, scale matrix. The classification procedure is based on two-step monotone training samples, one from each population, with the same monotone pattern. Our aim is to extend the classification procedure, which proposed recently by Chung and Han (Ann Ins Stat Math 52:544–556, 2000). This procedure is a linear combination of two discriminant functions, one based on the complete samples and the other on the incomplete samples. The performance of the proposed classification rule is compared with the plug-in method, this means with the classification rule which arises if the unknown parameters are substituted, into the usual classification rule, by their estimators. In order to apply the plug-in method, the MLE of the location parameters and of the common scale matrix of g ≥ 2 elliptically contoured populations are analytically obtained on the basis of two-step monotone training samples. 相似文献
14.
Multi-stage point and interval estimation of the largest mean ofK normal populations and the associated second-order properties 总被引:1,自引:0,他引:1
Summary We havek independent normal populations with unknown meansμ
1, …,μ
k and a common unknown varianceσ
2. Both point and interval estimation procedures for the largest mean are proposed by means of sequential and three-stage procedures.
For the point estimation problem, we require that the maximal risk be at mostW, a preassigned positive number. For the other problem, we wish to construct a fixed-width confidence interval having the
confidence coefficient at least 1-α, a preassigned number between zero and one. Asymptotic second order expansions are provided for various characteristics,
such as average sample size, associated risks etc., for the suggested multi-stage estimation procedures. 相似文献
15.
LetX be ap-normal random vector with unknown mean and unknown covariance matrix and letX be partitioned asX=(X
(1)
,X
(2)
, ...,X
(r)
) whereX
(j) is a subvector of dimensionp
j such that
j=1
r
p
j
=p. We show that the tests, obtained by Dahel (1988), are locally minimax. These tests have been derived to confront Ho: =0 versusH
1: 0 on the basis of sample of sizeN, X
1, ..., XN, drawn fromX andr additional samples of sizeN
j, U
i
(j)
, i=1, ..., Nj, drawn fromX
(1), ...X
(r) respectively. We assume that the (r+1) samples are independent and thatN
j>p
j forj=0, 1, ..., r (N
oN andp
op). Whenr=2 andp=2, a Monte Carlo study is performed to compare these tests with the likelihood ratio test (LRT) given by Srivastava (1985). We also show that no locally most powerful invariant test exists for this problem. 相似文献
16.
A minimal characterization of the covariance matrix 总被引:1,自引:0,他引:1
Summary LetX be ak-dimensional random vector with mean vectorμ and non-singular covariance matrix Σ. We show that among all pairs (a, Δ),a ∈ IR
k
, Δ ∈ IR
k×k
positive definite and symmetric andE(X−a)′ Δ−1(X−a)=k, (μ, Σ) is the unique pair which minimizes det Δ. This motivates certain robust estimators of location and scale.
Research supported by the Nuffield Foundation. 相似文献
17.
Biao Zhang 《Metrika》1997,46(1):221-244
For estimating the distribution functionF of a population, the empirical or sample distribution functionF
n
has been studied extensively. Qin and Lawless (1994) have proposed an alternative estimator
for estimatingF in the presence of auxiliary information under a semiparametric model. They have also proved the point-wise asymptotic normality
of
. In this paper, we establish the weak convergence of
to a Gaussian process and show that the asymptotic variance function of
is uniformly smaller than that ofF
n
. As an application of
, we propose to employ the mean
and varianceŜ
n
2
of
to estimate the population mean and variance in the presence of auxiliary information. A simulation study is presented to
assess the finite sample performance of the proposed estimators
, andŜ
n
2
. 相似文献
18.
In this paper, we study a robust and efficient estimation procedure for the order of finite mixture models based on the minimizing
a penalized density power divergence estimator. For this task, we use the locally conic parametrization approach developed
by Dacunha-Castelle and Gassiate (ESAIM Probab Stat 285–317, 1997a; Ann Stat 27:1178–1209, 1999), and verify that the minimizing
a penalized density power divergence estimator is consistent. Simulation results are provided for illustration. 相似文献
19.
In this paper we generalize the quality and cost trade-off problem of Chang and Hung (Qual Quant 41: 291–301, 2007) under
the LINEX loss function. We consider the general input characteristic given by the random variable X with moment generating function m
X
(t) and output characteristic given by the deterministic transformation Y = g(X). The two cases we consider are when g(X) is an affine function of X and X follows (1) the gamma distribution, and (2) the double exponential distribution. 相似文献
20.
This paper is concerned with the comparison of seven estimators of the mean of the selected population from two normal populations
with unknown means and common known variance under an asymmetric loss namely the LINEX loss function. The proposed estimators
are invariant under location transformation. The bias and risks of the seven estimators are computed and compared. The conclusion
recommend the use of δP (σ) which is simple to use and it is minimax.
Received: January 1999 相似文献