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Dr. Qiqing Yu 《Metrika》1990,37(1):245-252
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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
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K. F. Cheng 《Metrika》1982,29(1):215-225
For a specified distribution functionG with densityg, and unknown distribution functionF with densityf, the generalized failure rate function (x)=f(x)/gG
–1
F(x) may be estimated by replacingf andF byf
n and
, wheref
n is an empirical density function based on a sample of sizen from the distribution functionF, and
. Under regularity conditions we show
and, under additional restrictions
whereC is a subset ofR and n. Moreover, asymptotic normality is derived and the Berry-Esséen type bound is shown to be related to a theorem which concerns the sum of i.i.d. random variables. The order boundO(n–1/2+c
n
1/2
) is established under mild conditions, wherec
n is a sequence of positive constants related tof
n and tending to 0 asn.Research was supported in part by the Army, Navy and Air Force under Office of Naval Research contract No. N00014-76-C-0608. AMS 1970 subject classifications. Primary 62G05. Secondary 60F15. 相似文献
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Summary Minimizing
is discussed under the unbiasedness condition:
and the condition (A):f
i
(x) (i=1, ..., p) are linearly independent
, and
. 相似文献
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In the present paper families of truncated distributions with a Lebesgue density
forx=(x
1,...,x
n
) ε ℝ
n
are considered, wheref
0:ℝ → (0, ∞) is a known continuous function andC
n
(ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator
form a saddle point when the parameter interval is sufficiently small. 相似文献
12.
LetP be a probability measure on ℝ andI
x be the set of alln-dimensional rectangles containingx. If for allx ∈ ℝn and θ ∈ ℝ the inequality
holds,P is a normal distributioin with mean 0 or the unit mass at 0. The result generalizes Teicher’s (1961) maximum likelihood characterization
of the normal density to a characterization ofN(0, σ2) amongall distributions (including those without density). The m.l. principle used is that of Scholz (1980). 相似文献
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This paper deals with the estimation of survivor function
using optimally selected order statistics when the sample sizen is large. We use the estimates (μ*,σ*) based on the optimum set of order statistics
for largen and fixedk (≤n) such that the estimate
has optimum variance property. The asymptotic relative efficiency of such an estimator is compared with the one based on
the complete sample. The general theory of the problem and specific details with respect to a two-parameter Normal, Logistic,
Exponential and Pareto distributions is considered as an example. 相似文献
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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
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Summary Completeness of a family of probability distributions implies its bounded completeness but not conversely. An example of a
family which is boundedly complete but not complete was presented by Lehmann and Scheffe [5]. This appears to be the only
such example quoted in the statistical literature. The purpose of this note is to provide further examples of this type. It
is shown that any given family of power series distributions can be used to construct a class containing infinitely many boundedly
complete, but not complete, families. Furthermore, it is shown that the family of continuous distributions
, is boundedly complete, but not complete, whereU denotes the uniform distribution on [a, b] and {P
ϑ,ϑ ∈ IR}, is a translation family generated by a distributionP
0 with mean value zero, which is continuous with respect to the Lebesgue measure. 相似文献
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Dr. Arne Sandström 《Metrika》1987,34(1):129-142
Let T(
) be a linear function of concomitants of order statistics, whereT (·) denotes a statistical functional depending on some distribution function (df)F and
is an estimator ofF. Under an auxiliary model approach we consider statistics of the form
, where
denotes a weighted empirical df and
a finite population df (t denotes a triangular array). The results can be used to estimate income inequality in finite populations and especially when
the survey is based on some design.
The paper was written when the author was working at the Statistical Research Unit, Statistics Sweden, Stockholm, Sweden
The research was supported by the Joint Committé of the Nordic Social Research Council. 相似文献
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Prof. Dr. J. Pfanzagl 《Metrika》1970,15(1):141-148
Summary Let (X,A) be a measurable space andP
ϑη |A (ϑη) ∈ Θ x H, ∥A, (θ, η) ∈ Θ×H, a parametrized family of probability measures (for short:p-measures). This paper is concerned with the problem of consistently estimatingθ from realizations governed by
, where ηu ∈ H, v ∈ ℕ, are unknown. 相似文献