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1.
Let X
1, X
2, ..., X
n
be independent exponential random variables such that X
i
has failure rate λ for i = 1, ..., p and X
j
has failure rate λ* for j = p + 1, ..., n, where p ≥ 1 and q = n − p ≥ 1. Denote by D
i:n
(p,q) = X
i:n
−X
i-1:n
the ith spacing of the order statistics X
1:n
≤ X
2:n
≤ ... ≤ X
n:n
, i = 1, ..., n, where X
0:n
≡ 0. The purpose of this paper is to investigate multivariate likelihood ratio orderings between spacings D
i:n
(p,q), generalizing univariate comparison results in Wen et al.(J Multivariate Anal 98:743–756, 2007). We also point out that
such multivariate likelihood ratio orderings do not hold for order statistics instead of spacings.
Supported by National Natural Science Foundation of China, the Program for New Century Excellent Talents in University (No.:
NCET-04-0569), and by the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KJCX3-SYW-S02). 相似文献
2.
Let X
1,X
2,…,X
n be a random sample from a continuous distribution with the corresponding order statistics X
1:n≤X
2:n≤…≤X
n:n. All the distributions for which E(X
k+r: n|X
k:n)=a
X
k:n+b are identified, which solves the problem stated in Ferguson (1967).
Received February 1998 相似文献
3.
In this paper we consider the case of the scale-contaminated normal (mixture of two normals with equal mean components but
different component variances: (1−p)N(μ,σ2)+pN(μ,τ2) with σ and τ being non-negative and 0≤p≤1). Here is the scale error and p denotes the amount with which this error occurs. It's maximum deviation to the best normal distribution is studied and shown
to be montone increasing with increasing scale error. A closed-form expression is derived for the proportion which maximizes
the maximum deviation of the mixture of normals to the best normal distribution. Implications to power studies of tests for
normality are pointed out.
Received May 2001 相似文献
4.
Min-Hsiao Tsai 《Metrika》2009,70(3):355-367
Consider the problem of discriminating between two rival response surface models and estimating parameters in the identified
model. To construct designs serving for both model discrimination and parameter estimation, the M
γ-optimality criterion, which puts weight γ (0≤γ≤1) for model discrimination and 1 − γ for parameter estimation, is adopted.
The corresponding M
γ-optimal product design is explicitly derived in terms of canonical moments. With the application of the maximin principle
on the M
γ-efficiency of any M
γ'-optimal product design, a criterion-robust optimal product design is proposed. 相似文献
5.
Arturo J. Fernández 《Metrika》2000,50(3):211-220
In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t
k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t
r, …, t
s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t
k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also
provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications,
the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included.
Received: August 1999 相似文献
6.
Hagen Scherb 《Metrika》2001,53(1):71-84
Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H
0 and the alternative hypothesis H
1 are given by
(i) H
0 : θ≤θ0, H
1 : θ>θ0, and
(ii) H
0 : θ≤θ1 or θ≥θ2, H
1 : θ1<θ<θ2, where θ1<θ2.
Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H
0 and H
1 take the form
(iii) H
0 : θ1≤θ≤θ2, H
1 : θ<θ1 or θ>θ2, where θ1<θ2, and
(iv) H
0 : θ=θ0, H
1:θ≠θ0.
To determine tests in case (i), only one critical value c and one randomization constant γ have to be computed. In cases (ii) through (iv) tests are determined by two critical values
c
1, c
2 and two randomization constants γ1, γ2. Unlike determination of tests in case (i), computation of critical values and randomization constants in the remaining cases
is rather difficult, unless distributions are symmetric. No straightforward method to determine two-sided UMP tests in discrete
sample spaces seems to be known. The purpose of this note is to disclose a distribution independent principle for the determination
of UMP tests in cases (ii) through (iv).
Received: March 1999 相似文献
7.
Prof. Dr. W. Stute 《Metrika》1992,39(1):257-267
LetX
1, ...,X
n
be an i.i.d. sample from some parametric family {θ :θ (Θ} of densities. In the random censorship model one observesZ
i
=min (X
i
,Y
i
) andδ
i
=1{
x
i
≤Y
i}, whereY
i
is a censoring variable being independent ofX
i
. In this paper we investigate the strong consistency ofθ
n
maximizing the modified likelihood function based on (Z
i
,δ
i
, 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data.
Work partially supported by the “Deutsche Forschungsgemeinschaft”. 相似文献
8.
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
9.
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
10.
Paul J. Campbell 《Metrika》2007,66(3):305-313
We consider games of chance between two players: Player M can win only by amassing point totals in several categories before
player N scores a prescribed total number n of points. Let M have k objectives, with m
i
points required in category i and probability q
i
of scoring a point in that category. We resolve certain special cases:
(a) For all m
i
equal, the probabilities of M winning are ordered by majorization of the vectors (q
1,...,q
k
).
(b) For all q
i
equal, the probabilities of M winning are ordered by majorization of the vectors (m
1,...,m
k
).
(c) For all m
i
equal and all q
i
equal, the probability of M winning approaches 0 as n → ∞ or as k → ∞. The results, which follow from inequalities of majorization and Schur convexity, are in accord with intuition.
相似文献
11.
Andrej Pázman 《Metrika》2002,56(2):113-130
The nonlinear regression model with N observations y
i=η(x
i,θ) +εi, and with the parameter θ subject to q nonlinear constraints C
j (θ)=0; j=1, …,q, is considered. As an example, the spline regression with unknown nodes is taken. Expressions for the variances (variance
matrices) of the LSE are discussed. Because of the complexity of these expressions, and the singularity of the variance matrix
of the LSE for θ, the optimality criteria and their properties, in particular the convexity and the equivalence theorem are
considered from different aspects. Also the possibility of restriction to designs with limited values of measures of nonlinearity
is mentioned.
Research supported by the VEGA-grant of the Slovak grant agency No. 1/7295/20. 相似文献
12.
Shanbhag (J Appl Probab 9:580–587, 1972; Theory Probab Appl 24:430–433, 1979) showed that the diagonality of the Bhattacharyya
matrix characterizes the set of Normal, Poisson, Binomial, negative Binomial, Gamma or Meixner hypergeometric distributions.
In this note, using Shanbhag (J Appl Probab 9:580–587, 1972; Theory Probab Appl 24:430–433, 1979) and Pommeret (J Multivar
Anal 63:105–118, 1997) techniques, we evaluated the general form of the 5 × 5 Bhattacharyya matrix in the natural exponential
family satisfying
f(x|q)=\fracexp{xg(q)}b(g(q))y(x){f(x|\theta)=\frac{\exp\{xg(\theta)\}}{\beta(g(\theta))}\psi(x)} with cubic variance function (NEF-CVF) of
θ. We see that the matrix is not diagonal like distribution with quadratic variance function and has off-diagonal elements.
In addition, we calculate the 5 × 5 Bhattacharyya matrix for inverse Gaussian distribution and evaluated different Bhattacharyya
bounds for the variance of estimator of the failure rate, coefficient of variation, mode and moment generating function due
to inverse Gaussian distribution. 相似文献
13.
LetX
1,X
2, ...,X
n
(n≥3) be a random sample on a random variableX with distribution functionF having a unique continuous inverseF
−1 over (a,b), −∞≤a<b≤∞ the support ofF. LetX
1:n
<X
2:n
<...<X
n:n
be the corresponding order statistics. Letg be a nonconstant continuous function over (a,b). Then for some functionG over (a, b) and for some positive integersr ands, 1<r+1<s≤n
相似文献
14.
In literature, Williams Square change-over designs balanced for first residual effects of treatments have been proved to
be universally optimal. Here, an attempt has been made to show that these designs are robust against missing of last α [≤v−1; v being the number of periods in the design for v treatments] observations from an experimental unit.
Received May 2000 相似文献
15.
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
16.
Hirotoshi Yoshioka 《Quality and Quantity》2006,40(6):1061-1077
Q-analysis, introduced by a mathematician Ron Atkin, is a useful tool to explore social structures. I introduce essential concepts
and techniques of q-analysis to show q-analysis’s potential ability to analyze and extract information from census data – the work that most researchers have done
mainly with statistical methods. Using Mexican census and q-analysis, I examine whether children of female-headed household aged 15–19 were more likely to attend school than male-headed
household children in Chiapas, Mexico in 2000. My findings are consistent with the large body of previous research, many of
which were conducted with statistical methods: women’s control of income tends to results in a better welfare for their children,
defined as children’s school attendance by the sex of heads of household in my study. I evaluate the advantages and disadvantages
of a-analysis of census data. I conclude that while it has some weaknesses, q-analysis is a complimentary method to statistical
methods for analysis of census data that may overcome some limitations that statistical methods often face such as an incapability
of handling a small sample. 相似文献
17.
Nigm et al. (2003, statistics 37: 527–536) proposed Bayesian method to obtain predictive interval of future ordered observation Y
(j) (r < j≤ n ) based on the right type II censored samples Y
(1) < Y
(2) < ... < Y
(r) from the Pareto distribution. If some of Y
(1) < ... < Y
(r-1) are missing or false due to artificial negligence of typist or recorder, then Nigm et al.’s method may not be an appropriate
choice. Moreover, the conditional probability density function (p.d.f.) of the ordered observation Y
(j) (r < j ≤ n ) given Y
(1) <Y
(2) < ... < Y
(r) is equivalent to the conditional p.d.f. of Y
(j) (r < j ≤ n ) given Y
(r). Therefore, we propose another Bayesian method to obtain predictive interval of future ordered observations based on the
only ordered observation Y
(r), then compares the length of the predictive intervals when using the method of Nigm et al. (2003, statistics 37: 527–536) and our proposed method. Numerical examples are provided to illustrate these results. 相似文献
18.
We consider the codifference and the normalized codifference function as dependence measures for stationary processes. Based on the empirical characteristic function, we propose estimators
of the codifference and the normalized codifference function. We show consistency of the proposed estimators, where the underlying model is the ARMA with symmetric α-stable innovations, 0 < α ≤ 2. In addition, we derive their limiting distribution. We present a simulation study showing the dependence of the estimator
on certain design parameters. Finally, we provide an empirical example using some stocks from Indonesia Stock Exchange. 相似文献
19.
Professor O. D. Anderson 《Metrika》1988,35(1):349-376
Summary We derive the detailed correlation structure for the simple “staircase model”: a process where white noise is superimposed
on a deterministic step function that has equal rises and equal treads. It turns out that this structure is an immediate generalisation
of that for a linear trend (which, for discrete data, can be alternatively considered as a step function with equal rises
and unit treads). We compare the structure obtained with that for a random walk, and those for a subset of other ARIMA(p, 1,q) models, and those of general ARIMA(p, d, q) processes withd>1. 相似文献
20.
A mixture experiment is an experiment in which the k ingredients are nonnegative and subject to the simplex restriction on the (k − 1)-dimensional probability simplex S
k-1. In this work, an essentially complete class of designs under the Kiefer ordering for a linear log contrast model with a
mixture experiment is presented. Based on the completeness result, -optimal designs for all p,−∞ ≤ p ≤ 1 including D- and A-optimal are obtained, where the eigenvalues of the design moment matrix are used. By using the approach presented here, we
gain insight on how these -optimal designs behave.
Mong-Na Lo Huang was supported in part by the National Science Council of Taiwan, ROC under grant NSC 93-2118-M-110-001. 相似文献
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