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1.
Consider a nonparametric nonseparable regression model Y = ?(Z, U), where ?(Z, U) is strictly increasing in U and UU[0, 1]. We suppose that there exists an instrument W that is independent of U. The observable random variables are Y, Z, and W, all one-dimensional. We construct test statistics for the hypothesis that Z is exogenous, that is, that U is independent of Z. The test statistics are based on the observation that Z is exogenous if and only if V = FY|Z(Y|Z) is independent of W, and hence they do not require the estimation of the function ?. The asymptotic properties of the proposed tests are proved, and a bootstrap approximation of the critical values of the tests is shown to be consistent and to work for finite samples via simulations. An empirical example using the U.K. Family Expenditure Survey is also given. As a byproduct of our results we obtain the asymptotic properties of a kernel estimator of the distribution of V, which equals U when Z is exogenous. We show that this estimator converges to the uniform distribution at faster rate than the parametric n? 1/2-rate.  相似文献   

2.
Let U, V and W be independent random variables, U and V having a gamma distribution with respective shape parameters a and b, and W having a non-central gamma distribution with shape and non-centrality parameters c and δ, respectively. Define X = U/(U + W) and Y = V/(V + W). Clearly, X and Y are correlated each having a non-central beta type 1 distribution, X ~ NCB1 (a,c;d){X \sim {\rm NCB1} (a,c;\delta)} and Y ~ NCB1 (b,c;d){Y \sim {\rm NCB1} (b,c;\delta)} . In this article we derive the joint probability density function of X and Y and study its properties.  相似文献   

3.
The Riesz distributions on a symmetric cone are used to introduce a class of beta-Riesz distributions. Some fundamental properties of these distributions are established. In particular, we study the effect of a projection on a beta-Riesz distribution and we give some properties of independence. We also calculate the expectation of a beta-Riesz random variable. As a corollary, we give the regression on the mean of a Riesz random variable; that is, we determine the conditional expectation E(UU+V) where U and V are two independent Riesz random variables.  相似文献   

4.
The family of lp-norm symmetric distributions was proposed by Yue and Ma and is a natural generalization to the family of l1-norm symmetric distributions studied by Fang et al. In this article, we propose a stochastic representation for the lp-norm symmetric distribution for any constant p > 0. The stochastic representation is expressed through independent and identically distributed uniform U(0, 1) random variables. It is illustrated that the stochastic representation can be applied to statistical simulation and uniform experimental design.  相似文献   

5.
In this paper, we prove a Hoeffding-like inequality for the survival function of a sum of symmetric independent identically distributed random variables, taking values in a segment [?b, b] of the reals. The symmetric case is relevant to the auditing practice and is an important case study for further investigations. The bounds as given by Hoeffding in 1963 cannot be improved upon unless we restrict the class of random variables, for instance, by assuming the law of the random variables to be symmetric with respect to their mean, which we may assume to be zero. The main result in this paper is an improvement of the Hoeffding bound for i.i.d. random variables which are bounded and have a (upper bound for the) variance by further assuming that they have a symmetric law.  相似文献   

6.
LIMIT THEOREMS FOR STANDARDIZED PARTIAL SUMS OF WEAKLY EXCHANGEABLE ARRAYS   总被引:1,自引:1,他引:0  
A symmetric array of random variables is weakly exchangeable if, when the same arbitrary permutation is applied to rows and columns, the joint distribution remains the same. We consider the asymptotic distribution of the standardized sums of the elements of the upper left hand corner of the partitioned array, generalizing results on U-statistics. In general, the asymptotic distribution is normal, but if the array is first standardized by subtracting row and column means, then it is a linear form in a normal variable and independent squares of normal variables with coefficients depending on the limits of the eigenvalues of the array.  相似文献   

7.
Let FN(.) be the density function of X2N. Values of C1/N, i= 1, 2, satisfying the twin conditions Pr (C1≤X2N≤C2)=1-α and the conditional expectation of X2N given C1≤X2N≤C2 is N are tabulated for α=.2, .1, .05, .01, .005, .001, N=1(1)20(2)50(5)150(10)350. The second condition may be replaced by the condition fN+2(C1)=fN+2V(C2). The author has with him a bigger table giving C1 and C2 for α=.2, .1, .05, .01, .005, .001, N=1(1)350 to three decimals (to three significant digits, if some decimals are not significant). Several applications are mentioned. A practical application that is perhaps not obvious is to test whether two or more counts are distributed as independent Poisson variables. The new simple formulae used in the construction of the table are given and should prove useful in obtaining accurate values for omitted entries and in increasing the accuracy of entries.  相似文献   

8.
Admissibility of linear estimators is characterized in linear models E(Y)=Xβ, D(Y)=V, with an unknown multidimensional parameter (β, V) varying in the Cartesian product C × ν, where C is a subset of space and ν is a given set of non negative definite symmetric matrices. The relation between admissibility of inhomogeneous and homogeneous linear estimators is discussed, and some sufficient and necessary conditions for admissibility of an inhomogeneous linear estimator are given.  相似文献   

9.
For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω1 → Ω2 a continuous, strictly increasing function, such that Ω1∩Ω2?(a, b), but otherwise arbitrary, we establish that the random variables F(X) ? F(g(X)) and F(g? 1(X)) ? F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U ? ψ(U) = dψ? 1(U) ? U for UU(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.  相似文献   

10.
In this paper we consider a sequence of independent continuous symmetric random variables X1, X2, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages Sn = R(n)1X1 + ??? + R(n)nXn, where the random weights R(n)1, …, Rn(n) which are independent of X1, X2, …, Xn, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that cnSn converges in distribution to a symmetric α-stable random variable with cn = n1 ? 1/α1/α(α + 1).  相似文献   

11.
Let X1,…, Xn be random variables symmetric about θ from a common unknown distribution Fθ(x) =F(x–θ). To test the null hypothesis H0:θ= 0 against the alternative H1:θ > 0, permutation tests can be used at the cost of computational difficulties. This paper investigates alternative tests that are computationally simpler, notably some bootstrap tests which are compared with permutation tests. Of these the symmetrical bootstrap-f test competes very favourably with the permutation test in terms of Bahadur asymptotic efficiency, so it is a very attractive alternative.  相似文献   

12.
We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 Gz) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.  相似文献   

13.
It is shown that if V1 and V2 are two positive random variables such that V1 is “star-shaped” with respect to V2, then for any random variable X with a distribution F(x) such that F(ax) has the monotone likelihood-ratio property, XV1 is star-shaped with respect to XV2. This result is then used to prove that the stable laws are star-shaped ordered.  相似文献   

14.
ABSTRACT

The sum of independent exponential random variables – the hypoexponential random variables – plays an important role of modeling in many domains. Khuong and Kong in (2006) Khuong, H.V., Kong, H.Y. (2006). General expression for pdf of a sum of independent exponential random variables. IEEE Commun. Lett. 10: 159161.[Crossref], [Web of Science ®] [Google Scholar] were concerned in evaluating the performance of some diversity scheme, which deals with the problem of finding the probability density function of this hypoexponential random variable. They considered a particular case of m independent exponential random variable, when l random variables have the same mean and m ? l remaining random variables of different means and they found a closed expression of its probability density function. In this paper, we consider the general case of the hypoexponential random variable when the means do not have to be distinct. We find a more simple and general closed expression of its probability density function than that of Khuong and Kong. This expression is obtained using a new defined matrix called the Kad matrix, which is similar to the general Vandermonde matrix. Eventually, we present an application illustrating our work.  相似文献   

15.
This paper deals with √n-consistent estimation of the parameter μ in the RCAR(l) model defined by the difference equation Xj=(μ+Uj)Xj-l+ej (jε Z), where {ej: jε Z} and {Uj: jε Z} are two independent sets of i.i.d. random variables with zero means, positive finite variances and E[(μ+U1)2] < 1. A class of asymptotically normal estimators of μ indexed by a family of bounded measurable functions is introduced. Then an estimator is constructed which is asymptotically equivalent to the best estimator in that class. This estimator, asymptotically equivalent to the quasi-maximum likelihood estimator derived in Nicholls & Quinn (1982), is much simpler to calculate and is asymptotically normal without the additional moment conditions those authors impose.  相似文献   

16.
Egmar Rödel 《Statistics》2013,47(3):387-397
Let Xbe a bivariate exponential-type random vector (BIDLIKAR, PATIL (1968)), than it is proved:

1. If P(X ≥0) = 1 is valid, then Xhas linear regression to both directions if and only if Xpossesses a symmetric Γ-distribution.

2. Xpossesses linear regression to both directions with constant regression coefficients (independent of the parameter vector ? of the exponential-type distribution (BIDLIKAR, PATIL (1968)) if and only if Xis normal distributed.  相似文献   

17.
We develop the score test for the hypothesis that a parameter of a Markov sequence is constant over time, against the alternatives that it varies over time, i.e., θt = θ + Ut; t = 1,2,…, where {Ut; t = 1,2,...} is a sequence of independently and identically distributed random variables with mean zero and variance σz u and θ is a fixed constant. The asymptotic null distribution of the test statistic is proved to be normal. We illustrate our procedure by examples and a real life data analysis.  相似文献   

18.
Let X1X2,.be i.i.d. random variables and let Un= (n r)-1S?(n,r) h (Xi1,., Xir,) be a U-statistic with EUn= v, v unknown. Assume that g(X1) =E[h(X1,.,Xr) - v |X1]has a strictly positive variance s?2. Further, let a be such that φ(a) - φ(-a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S2n be the Jackknife estimator of n Var Un. Consider the stopping times N(d)= min {n: S2n: + n-12a-2},d > 0, and a confidence interval for v of length 2d,of the form In,d= [Un,-d, Un + d]. We assume that Var Un is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let IN(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. limd → 0P(v ∈ IN(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ IN(d),d) converges to α is investigated. We obtain that |P(v ∈ IN(d),d) - α| = 0 (d1/2-(1+k)/2(1+m)), d → 0, where K = max {0,4 - m}, under the condition that E|h(X1, Xr)|m < ∞m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).  相似文献   

19.
Abstract

In this short note, a very simple proof of the Chebyshev's inequality for random vectors is given. This inequality provides a lower bound for the percentage of the population of an arbitrary random vector X with finite mean μ = E(X) and a positive definite covariance matrix V = Cov(X) whose Mahalanobis distance with respect to V to the mean μ is less than a fixed value. The main advantage of the proof is that it is a simple exercise for a first year probability course. An alternative proof based on principal components is also provided. This proof can be used to study the case of a singular covariance matrix V.  相似文献   

20.
The process capability index C pk is widely used when measuring the capability of a manufacturing process. A process is defined to be capable if the capability index exceeds a stated threshold value, e.g. C pk >4/3. This inequality can be expressed graphically using a process capability plot, which is a plot in the plane defined by the process mean and the process standard deviation, showing the region for a capable process. In the process capability plot, a safety region can be plotted to obtain a simple graphical decision rule to assess process capability at a given significance level. We consider safety regions to be used for the index C pk . Under the assumption of normality, we derive elliptical safety regions so that, using a random sample, conclusions about the process capability can be drawn at a given significance level. This simple graphical tool is helpful when trying to understand whether it is the variability, the deviation from target, or both that need to be reduced to improve the capability. Furthermore, using safety regions, several characteristics with different specification limits and different sample sizes can be monitored in the same plot. The proposed graphical decision rule is also investigated with respect to power.  相似文献   

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