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1.
i
, i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ
i
, i = 1, 2, ..., k and common known scale parameter σ. Let Y
i
denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π
i
is selected iff Y
i
≥Y
(1)−d, where Y
(1) is the largest of the Y
i
's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered
for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that
the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained.
The improved estimators are consistent and their risks are shown to be O(kn
−2). As a special case, we obtain the coresponding results for the estimation of θ(1), the parameter associated with Y
(1).
Received: January 6, 1998; revised version: July 11, 2000 相似文献
2.
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
3.
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
4.
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
5.
This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y
|
X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X
(t+1)|X
(t),θ), where X
(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X
(t+1)|X
(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected
dynamically over time.
We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics
(though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration
rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume
that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas
the distribution of the true allele frequencies is only indirectly specified through a transition model.
As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually
done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC
is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models.
We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting
is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases. 相似文献
6.
Christopher S. Withers 《Statistics》2013,47(5):1092-1105
A great deal of inference in statistics is based on making the approximation that a statistic is normally distributed. The error in doing so is generally O(n?1/2), where n is the sample size and can be considered when the distribution of the statistic is heavily biased or skewed. This note shows how one may reduce the error to O(n?(j+1)/2), where j is a given integer. The case considered is when the statistic is the mean of the sample values of a continuous distribution with a scale or location change after the sample has undergone an initial transformation, which may depend on an unknown parameter. The transformation corresponding to Fisher's score function yields an asymptotically efficient procedure. 相似文献
7.
Note on the bias in the estimation of the serial correlation coefficient of AR(1) processes 总被引:1,自引:0,他引:1
Manfred Mudelsee 《Statistical Papers》2001,42(4):517-527
We derive approximating formulas for the mean and the variance of an autocorrelation estimator which are of practical use
over the entire range of the autocorrelation coefficient ρ. The least-squares estimator ∑
n
−1
i
=1ε
i
ε
i
+1 / ∑
n
−1
i
=1ε2
i
is studied for a stationary AR(1) process with known mean. We use the second order Taylor expansion of a ratio, and employ
the arithmetic-geometric series instead of replacing partial Cesàro sums. In case of the mean we derive Marriott and Pope's
(1954) formula, with (n− 1)−1 instead of (n)−1, and an additional term α (n− 1)−2. This new formula produces the expected decline to zero negative bias as ρ approaches unity. In case of the variance Bartlett's
(1946) formula results, with (n− 1)−1 instead of (n)−1. The theoretical expressions are corroborated with a simulation experiment. A comparison shows that our formula for the mean
is more accurate than the higher-order approximation of White (1961), for |ρ| > 0.88 and n≥ 20. In principal, the presented method can be used to derive approximating formulas for other estimators and processes.
Received: November 30, 1999; revised version: July 3, 2000 相似文献
8.
S. Gurler 《Statistical Papers》2012,53(1):23-31
Sequential order statistics is an extension of ordinary order statistics. They model the successive failure times in sequential
k-out-of-n systems, where the failures of components possibly affect the residual lifetimes of the remaining ones. In this paper, we
consider the residual lifetime of the components after the kth failure in the sequential (n − k + 1)-out-of-n system. We extend some results on the joint distribution of the residual lifetimes of the remaining components in an ordinary
(n − k + 1)-out-of-n system presented in Bairamov and Arnold (Stat Probab Lett 78(8):945–952, 2008) to the case of the sequential (n − k + 1)-out-of-n system. 相似文献
9.
Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions 总被引:1,自引:0,他引:1
Hsiuying Wang 《Statistics and Computing》2009,19(2):139-148
For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf
θ
P
θ
(θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence
coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average
coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of
the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing
the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter
discrete distributions are proposed. With these methodologies, both exact values can be derived. 相似文献
10.
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
11.
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
12.
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
13.
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
14.
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
15.
16.
Tachen Liang 《统计学通讯:理论与方法》2013,42(8):1543-1553
17.
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
18.
In this paper, by relaxing the mixing coefficients to α(n) = O(n ?β), β > 3, we investigate the Bahadur representation of sample quantiles under α-mixing sequence and obtain the rate as ${O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})}$ . Meanwhile, for any δ > 0, by strengthening the mixing coefficients to α(n) = O(n ?β ), ${\beta > \max\{3+\frac{5}{1+\delta},1+\frac{2}{\delta}\}}$ , we have the rate as ${O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$ . Specifically, if ${\delta=\frac{\sqrt{41}-5}{4}}$ and ${\beta > \frac{\sqrt{41}+7}{2}}$ , then the rate is presented as ${O(n^{-\frac{\sqrt{41}+5}{16}}(\log\log n\cdot \log n)^{\frac{1}{2}})}$ . 相似文献
19.
Classical saddlepoint methods, which assume that the cumulant generating function is known, result in an approximation to the distribution that achieves an error of order O(n?1). The authors give a general theorem to address the accuracy of saddlepoint approximations in which the cumulant generating function has been estimated or approximated. In practice, the resulting saddlepoint approximations are typically of the order O(n?1/2). The authors give simulation results for small sample examples to compare estimated saddlepoint approximations. 相似文献
20.
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