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1.
Let f be an unknown possibly multimodal density on Rd and let X1, X2, … be a sequence of independent random vectors with density f. Several recursive estimates of the mode of f are proposed, and sufficient conditions ensuring their weak and strong consistency are established.  相似文献   

2.
Consider a set of r+1 independently and identically and uniformly distributed random points X0, X1,…,Xr in RnThese points determine almost surely via their convex hull a unique r-simplex in Re This article deals with the exact density of the r-content of this random r-simplex when the points are such that p of them are in the interior and r+l?p of them are on the surface of a unit n-ball. This problem is transformed into a distribution problem connected with multivariate test statistics. Various possible representations of the exact density in the general case, are also pointed out.  相似文献   

3.
Let X1 be a strictly stationary multiple time series with values in Rd and with a common density f. Let X1,.,.,Xn, be n consecutive observations of X1. Let k = kn, be a sequence of positive integers, and let Hni be the distance from Xi to its kth nearest neighbour among Xj, j i. The multivariate variable-kernel estimate fn, of f is defined by where K is a given density. The complete convergence of fn, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.  相似文献   

4.
The need to simulate from a univariate density arises in several settings, particularly in Bayesian analysis. An especially efficient algorithm which can be used to sample from a univariate density, f X , is the adaptive accept–reject algorithm. To implement the adaptive accept–reject algorithm, the user has to envelope T ° f X , where T is some transformation such that the density g(x) ∝ T ?1 (α+β x) is easy to sample from. Successfully enveloping T ° f X , however, requires that the user identify the number and location of T ° f X ’s inflection points. This is not always a trivial task. In this paper, we propose an adaptive accept–reject algorithm which relieves the user of precisely identifying the location of T ° f X ’s inflection points. This new algorithm is shown to be efficient and can be used to sample from any density such that its support is bounded and its log is three-times differentiable.  相似文献   

5.
The problem of estimation of the derivative of a probability density f is considered, using wavelet orthogonal bases. We consider an important kind of dependent random variables, the so-called mixing random variables and investigate the precise asymptotic expression for the mean integrated error of the wavelet estimators. We show that the mean integrated error of the proposed estimator attains the same rate as when the observations are independent, under certain week dependence conditions imposed to the {X i }, defined in {Ω, N, P}.  相似文献   

6.
Let F = {F0: 0 ϵ Θ} denote the class of natural exponential family of distributions having power variance function, (NEF-PVF). We consider the problem of sequentially estimating the mean μ of F0 ϵ F, based on i.i.d. observations from F0. We propose an appropriate sequential estimation procedure under a combined loss of estimation error and sampling cost. We provide expansion for the regret Ra and study its asymptotic properties. We show that Ra = cv2(μ) + o(1) as a → ∞, where c > 0 is a known constant and v(μ) denotes the coefficient of variation of F0.  相似文献   

7.
Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and ci(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[11100121], c1=[10], c2=[02].) Let Ti (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the sn?r solutions of the equations At=ci, wheret′=(t1,…,tn) is a vector of variables.(Thus, EG(4, 3) consists of the 34=81 points of the form (t1,t2,t3,t4), where t's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=ci is sn?r, where r=Rank(A), and the set of such solutions is said to form an (n?r)-flat, i.e. a flat of (n?r) dimensions. In our example, both T1 and T2 are 2-flats consisting of 34?2=9 points each. The flats T1,T2,…,Tf are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T1 are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T2 consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sn symmetric factorial experiment obtained by taking T1,T2,…,Tf together. (Thus, in the example, 34=81 treatments of the 34 factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T1 and T2 enumerated above.)In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X (n×v), occuring in the corresponding linear model, into components X.j(j=0,1,…,g), such that the information matrix X is the direct sum of the X′.jX.j. Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying (s?1) degrees of freedom as usual). For j≠0, we show that the spectrum of X′.jX.j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X.j into the Xij (i=1,…,f). Furthermore, the Xij are in turn partitioned into smaller matrices (which we shall here call the) Xijh. We show that each Xijh can be factored into a product of 3 matrices J, ζ (not depending on i,j, and h) and Q(j,h,i)where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF(s), and show that the Q(j,h,i) belong to a ring isomorphic to R. When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X.j and X′.jX.j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R.  相似文献   

8.
Consider the problem of pointwise estimation of f in a multivariate isotonic regression model Z=f(X1,…,Xd)+ϵ, where Z is the response variable, f is an unknown nonparametric regression function, which is isotonic with respect to each component, and ϵ is the error term. In this article, we investigate the behavior of the least squares estimator of f. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multivariate isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.  相似文献   

9.
EMPIRICAL BAYES ESTIMATION WITH NON-IDENTICAL COMPONENTS. CONTINUOUS CASE.   总被引:3,自引:0,他引:3  
In this paper a variant of the standard empirical Bayes estimation problem is considered where the component problems in the sequence are not identical in that the conditional distribution of the observations may vary with the component problems. Let {(Θn, Xn)} be a sequence of independent random vectors where Θn? G and, given Θnn, Xn -PΘ,m(n) with {m(n)} a sequence of positive integers where m(n)≤m? < ∞ for all n. With PΘ,m in a continuous exponential family of distributions, asymptotically optimal empirical Bayes procedures are exhibited which depend on uniform approximations of certain functions on compact sets by polynomials in eΘ. Such approximations have been explicitly calculated in the case of normal and gamma families. In the case of normal families, asymptotically optimal linear empirical Bayes estimators in the class of all linear estimators are derived and shown to have rate O(n-1/2).  相似文献   

10.
Let X1,X2,…,Xp be p random variables with cdf's F1(x),F2(x),…,Fp(x)respectively. Let U = min(X1,X2,…,Xp) and V = max(X1,X2,…,Xp).In this paper we study the problem of uniquely determining and estimating the marginal distributions F1,F2,…,Fp given the distribution of U or of V.

First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X1's are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X1's are dependent is considered next. Finally the problem of estimation is considered.  相似文献   

11.
12.
Let (X,Y) be a pair of random variables with supp(X)⊆[0,1] and EY2<∞. Let m be the corresponding regression function. Estimation of m from i.i.d. data is considered. The L2 error with integration with respect to the design measure μ (i.e., the distribution of X) is used as an error criterion.Estimates are constructed by estimating the coefficients of an orthonormal expansion of the regression function. This orthonormal expansion is done with respect to a family of piecewise polynomials, which are orthonormal in L2(μn), where μn denotes the empirical design measure.It is shown that the estimates are weakly and strongly consistent for every distribution of (X,Y). Furthermore, the estimates behave nearly as well as an ideal (but not applicable) estimate constructed by fitting a piecewise polynomial to the data, where the partition of the piecewise polynomial is chosen optimally for the underlying distribution. This implies e.g., that the estimates achieve up to a logarithmic factor the rate n−2p/(2p+1), if the underlying regression function is piecewise p-smooth, although their definition depends neither on the smoothness nor on the location of the discontinuities of the regression function.  相似文献   

13.
The problem of estimating ordered parameters is encountered in biological, agricultural, reliability and various other experiments. Consider two populations with densities f1(x11) and f2(x22) where ω12. The estimation of ω12) with the loss function, the sum of squared errors, is studied. when fi is the fi(,i,,i 2) density with ,i known, i=1,2; we obtain a class of minimax estimators. When ω12 we show some of these estimators are improved by the maximum likelihood estimator. For a general fi we give sufficient conditions for the minimaxity of the analogue of the Pitman estimator.  相似文献   

14.
Consider a sequence of independent random variables X 1, X 2,…,X n observed at n equally spaced time points where X i has a probability distribution which is known apart from the values of a parameter θ i R which may change from observation to observation. We consider the problem of estimating θ = (θ1, θ2,…,θ n ) given the observed values of X 1, X 2,…,X n . The paper proposes a prior distribution for the parameters θ for which sets of parameter values exhibiting no change, or no change apart from a few sudden large changes, or lots of small changes, all have positive prior probability. Markov chain sampling may be used to calculate Bayes estimates of the parameters. We report the results of a Monte Carlo study based on Poisson distributed data which compares the Bayes estimator with estimators obtained using cubic splines and with estimators derived from the Schwarz criterion. We conclude that the Bayes method is preferable in a minimax sense since it never produces the disastrously large errors of the other methods and pays only a modest price for this degree of safety. All three methods are used to smooth mortality rates for oesophageal cancer in Irish males aged 65–69 over the period 1955 through 1994.  相似文献   

15.
Let X1,., Xn, be i.i.d. random variables with distribution function F, and let Y1,.,.,Yn be i.i.d. with distribution function G. For i = 1, 2,.,., n set δi, = 1 if Xi ≤ Yi, and 0 otherwise, and Xi, = min{Xi, Ki}. A kernel-type density estimate of f, the density function of F w.r.t. Lebesgue measure on the Borel o-field, based on the censored data (δi, Xi), i = 1,.,.,n, is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher-order differentiability assumption on f. A procedure for relaxing such assumptions is also proposed.  相似文献   

16.
In the literature, assuming independence of random variables X and Y, statistical estimation of the stress–strength parameter R = P(X > Y) is intensively investigated. However, in some real applications, the strength variable X could be highly dependent on the stress variable Y. In this paper, unlike the common practice in the literature, we discuss on estimation of the parameter R where more realistically X and Y are dependent random variables distributed as bivariate Rayleigh model. We derive the Bayes estimates and highest posterior density credible intervals of the parameters using suitable priors on the parameters. Because there are not closed forms for the Bayes estimates, we will use an approximation based on Laplace method and a Markov Chain Monte Carlo technique to obtain the Bayes estimate of R and unknown parameters. Finally, simulation studies are conducted in order to evaluate the performances of the proposed estimators and analysis of two data sets are provided.  相似文献   

17.
We show that sup, completely as, where f is a uniformly continuous density on are independent random vectors with common density f, and fn is the variable kernel estimate Here Hni is the distance between Xi and its kth nearest neighbour, K is a given density satisfying some regularity conditions, and k is a sequence of integers with the property that log asn  相似文献   

18.
We derive an asymptotic theory of nonparametric estimation for a time series regression model Zt=f(Xt)+Wt, where {Xt} and {Zt} are observed nonstationary processes, and {Wt} is an unobserved stationary process. The class of nonstationary processes allowed for {Xt} is a subclass of the class of null recurrent Markov chains. This subclass contains the random walk, unit root processes and nonlinear processes. The process {Wt} is assumed to be linear and stationary.  相似文献   

19.
We consider moving average processes, {Xs, s ∈ ??}, where ?? is a triangular lattice in the plane R2. To estimate the parameters of such processes, Adjengue & Moore (1993) have considered likelihood and gaussian pseudo-likelihood methods. We consider here two other methods. The first one is based on the estimation of the correlations and the relation between these correlations and the parameters of the process. The second relies on a linear approximation of the process. The asymptotic properties of the proposed estimators are analyzed and compared. A simulation study allows us to compare the estimators for fixed sample sizes.  相似文献   

20.
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   

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