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1.
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
2.
Rasul A. Khan 《Journal of statistical planning and inference》1981,5(3):243-251
Let X1,X2, … be iid random variables with the pdf f(x,θ)=exp(θx?b(θ)) relative to a σ-finite measure μ, and consider the problem of deciding among three simple hypotheses Hi:θ=θi (1?i?3) subject to P(acceptHi|θi)=1?α (1?i?3). A procedure similar to Sobel–Wald procedure is discussed and its asymptotic efficiency as compared with the best nonsequential test is obtained by finding the limit lima→0(EiN(a)/n(a)), where N (a) is the stopping time of the proposed procedure and n(a) is the sample size of the best non-sequential test. It is shown that the same asymptotic limit holds for the original Sobel–Wald procedure. Specializing to N(θ,1) distribution it is found that lima→0 (i=1,2) and lima→0 (E3N(α)n(α))=δ21/4δ, where δi=(θi+1?θi) with 0<δ1?δ2. Also, the asymptotic efficiency evaluated when the X's have an exponential distribution. 相似文献
3.
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
4.
Let {W(s); 8 ≥ 0} be a standard Wiener process, and let βN = (2aN (log (N/aN) + log log N)-1/2, 0 < αN ≤ N < ∞, where αN↑ and αN/N is a non-increasing function of N, and define γN(t) = βN[W(nN + taN) ? W(nN)), 0 ≥ t ≥ 1, with nN = N – aN. Let K = {x ? C[0,1]: x is absolutely continuous, x(0) = 0 and }. We prove that, with probability one, the sequence of functions {γN(t), t ? [0,1]} is relatively compact in C[0,1] with respect to the sup norm ||·||, and its set of limit points is K. With aN = N, our result reduces to Strassen's well-known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered. 相似文献
5.
Christian Genest 《Revue canadienne de statistique》1984,12(2):153-163
In this paper, we consider the problem of combining a number of opinions which have been expressed as probability measures P1, …, Pn, over some space. It is shown that a pooling formula which has the marginalization property of McConway (1981) must be of the form T = Σni=1Wi Pi + (1 - Σni =1Wi)Q, where Q is an arbitrary measure and W1, …, Wn ϵ [—1,1] are weights such that| ΣJ Σ j wj | ≤ 1 for every subset J of {1, …, n}. If, in addition, T is required to preserve the independence of arbitrary events A and B whenever these events are independent under each Pi, then either T = Pi for some 1 ≤ i ≤ n or T = Q, in which case Q takes values in {0, l}. 相似文献
6.
Rasul A. Khan 《Revue canadienne de statistique》1978,6(1):113-119
Let X1, X2, …, Xn be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss. 相似文献
7.
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
8.
Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (qij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., qij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability pij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p0N(t) (p(m)(N0)(t)), 0 ≤ m ≤N, where f(m)(t) denotes the mth derivative of a function f(t) with f(0)(t) =f(t). If X(t) is a birth-death process, then pij(t) is represented as a linear combination of p0N(m)(t), 0 ≤m≤N - |i-j|. 相似文献
9.
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
10.
Summary Let {X
n
} be a sequence of random variables conditionally independent and identically distributed given the random variable Θ. The
aim of this paper is to show that in many interesting situations the conditional distribution of Θ, given (X
1,…,X
n
), can be approximated by means of the bootstrap procedure proposed by Efron and applied to a statisticT
n
(X
1,…,X
n
) sufficient for predictive purposes. It will also be shown that, from the predictive point of view, this is consistent with
the results obtained following a common Bayesian approach. 相似文献
11.
In this paper, we consider the simple linear errors-in-variables (EV) regression models: ηi=θ+βxi+εi,ξi=xi+δi,1≤i≤n, where θ,β,x1,x2,… are unknown constants (parameters), (ε1,δ1),(ε2,δ2),… are errors and ξi,ηi,i=1,2,… are observable. The asymptotic normality for the least square (LS) estimators of the unknown parameters β and θ in the model are established under the assumptions that the errors are m-dependent, martingale differences, ?-mixing, ρ-mixing and α-mixing. 相似文献
12.
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
13.
Sumanta Adhya Tathagata Banerjee Gaurangadeb Chattopadhyay 《AStA Advances in Statistical Analysis》2012,96(1):69-98
Suppose that a finite population consists of N distinct units. Associated with the ith unit is a polychotomous response vector, d
i
, and a vector of auxiliary variable x
i
. The values x
i
’s are known for the entire population but d
i
’s are known only for the units selected in the sample. The problem is to estimate the finite population proportion vector
P. One of the fundamental questions in finite population sampling is how to make use of the complete auxiliary information
effectively at the estimation stage. In this article a predictive estimator is proposed which incorporates the auxiliary information
at the estimation stage by invoking a superpopulation model. However, the use of such estimators is often criticized since
the working superpopulation model may not be correct. To protect the predictive estimator from the possible model failure,
a nonparametric regression model is considered in the superpopulation. The asymptotic properties of the proposed estimator
are derived and also a bootstrap-based hybrid re-sampling method for estimating the variance of the proposed estimator is
developed. Results of a simulation study are reported on the performances of the predictive estimator and its re-sampling-based
variance estimator from the model-based viewpoint. Finally, a data survey related to the opinions of 686 individuals on the
cause of addiction is used for an empirical study to investigate the performance of the nonparametric predictive estimator
from the design-based viewpoint. 相似文献
14.
Ricardo Aníbal Leiva 《Statistical Methods and Applications》1994,3(2):271-289
Summary The problem of predicting the number of change points in a piecewise linear model is studied from a Bayesian viewpoint. For
a given a priori joint probability functionf
R,C=fRf
C/R, whereR is the number of change points andC=C′(R)=(C1,…,CR) is the change-point epoch vector, the marginal posterior probability functionf
R.C/Y
is obtained, and then used to find predictors forR andC(R). 相似文献
15.
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
16.
ABSTRACT Suppose independent random samples are available from k(k ≥ 2) exponential populations ∏1,…,∏ k with a common location θ and scale parameters σ1,…,σ k , respectively. Let X i and Y i denote the minimum and the mean, respectively, of the ith sample, and further let X = min{X 1,…, X k } and T i = Y i ? X; i = 1,…, k. For selecting a nonempty subset of {∏1,…,∏ k } containing the best population (the one associated with max{σ1,…,σ k }), we use the decision rule which selects ∏ i if T i ≥ c max{T 1,…,T k }, i = 1,…, k. Here 0 < c ≤ 1 is chosen so that the probability of including the best population in the selected subset is at least P* (1/k ≤ P* < 1), a pre-assigned level. The problem is to estimate the average worth W of the selected subset, the arithmetic average of means of selected populations. In this article, we derive the uniformly minimum variance unbiased estimator (UMVUE) of W. The bias and risk function of the UMVUE are compared numerically with those of analogs of the best affine equivariant estimator (BAEE) and the maximum likelihood estimator (MLE). 相似文献
17.
In this paper we describe active set type algorithms for minimization of a smooth function under general order constraints,
an important case being functions on the set of bimonotone r×s matrices. These algorithms can be used, for instance, to estimate a bimonotone regression function via least squares or (a
smooth approximation of) least absolute deviations. Another application is shrinkage estimation in image denoising or, more
generally, regression problems with two ordinal factors after representing the data in a suitable basis which is indexed by
pairs (i,j)∈{1,…,r}×{1,…,s}. Various numerical examples illustrate our methods. 相似文献
18.
Konanur G. Janardan 《Statistical Papers》2005,46(4):587-597
A discrete distribution associated with a pure birth process starting with no individuals, with birth rates λ
n
=λ forn=0, 2, …,m−1 and λ
n
forn≥m is considered in this paper. The probability mass function is expressed in terms of an integral that is very convenient for
computing probabilities, moments, generating functions and others. Using this representation, the mean and the k-th factorial
moments of the distribution are obtained. Some nice characterizations of this distribution are also given. 相似文献
19.
Let X1,…,Xr?1,Xr,Xr+1,…,Xn be independent, continuous random variables such that Xi, i = 1,…,r, has distribution function F(x), and Xi, i = r+1,…,n, has distribution function F(x?Δ), with -∞ <Δ< ∞. When the integer r is unknown, this is refered to as a change point problem with at most one change. The unknown parameter Δ represents the magnitude of the change and r is called the changepoint. In this paper we present a general review discussion of several nonparametric approaches for making inferences about r and Δ. 相似文献
20.
We consider a nonparametric regression model where m noise-perturbed functions f
1,…,f
m
are randomly observed. For a fixed ν∈{1,…,m}, we want to estimate f
ν from the observations. To reach this goal, we develop an adaptive wavelet estimator based on a hard thresholding rule. Adopting
the mean integrated squared error over Besov balls, we prove that it attains a sharp rate of convergence. Simulation results
are reported to support our theoretical findings. 相似文献