Simultaneous confidence bands for expectile functions

Expectile regression, as a general M smoother, is used to capture the tail behaviour of a distribution. Let (X ₁,Y ₁),…,(X _n ,Y _n ) be i.i.d. rvs. Denote by v(x) the unknown τ-expectile regression curve of Y conditional on X, and by v _n (x) its kernel smoothing estimator. In this paper, we prove the strong uniform consistency rate of v _n (x) under general conditions. Moreover, using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup_0≤x≤1|v _n (x)?v(x)|. According to the asymptotic theory, we construct simultaneous confidence bands around the estimated expectile function. Furthermore, we apply this confidence band to temperature analysis. Taking Berlin and Taipei as an example, we investigate the temperature risk drivers to these two cities.     

Nonparametric statistical procedures for the changepoint problem

Let X₁,…,X_r?1,X_r,X_r+1,…,X_n be independent, continuous random variables such that X_i, i = 1,…,r, has distribution function F(x), and X_i, 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.     

Zur abschätzung von lösungen linearer integralgleichungen mit hilfe der monte-carlo-methode

Let X₁, X₂, … be i.i.d.r.v. and write (X₁+…X_n?A_n)/B_n?F_n, where B_n >0.A_nER₁, n≥1. It is known that solely one–sided asymptotic assumptions imposed on F_n imply F_n0. In the present note we show that stronger one–sided assumptions lead even to the existence of EX₁ ³ so that the BERRY-ESSEEN inequalities hold true.     

Theory of factorial designs of the parallel flats type. I: the coefficient matrix

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 c_i(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 T_i (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the s^n?r solutions of the equations At=c_i, wheret′=(t₁,…,t_n) 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=c_i is s^n?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 3^4?2=9 points each. The flats T₁,T₂,…,T_f 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,…,T_f 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 X_ij (i=1,…,f). Furthermore, the X_ij are in turn partitioned into smaller matrices (which we shall here call the) X_ijh. We show that each X_ijh 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.     

Linear representation of M-estimates in linear models

Consider the linear regression model, y_i = x_iβ₀ + e_i, i = l,…,n, and an M-estimate β of β_o obtained by minimizing Σρ(y_i — x_iβ), where ρ is a convex function. Let S_n = ΣX_iX_iX_i and r_n = S_n^½ (β — β₀) — S_n ² Σx_ih(e_i), where, with a suitable choice of h(.), the expression Σ x_ix(e,) provides a linear representation of β. Bahadur (1966) obtained the order of r_n as n→ ∞ when β_o is a one-dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of r_n as n → ∞ for a general M-estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them.     

Perfect octagon quadrangle systems with upper C4-systems

An octagon quadrangle is the graph consisting of an 8-cycle (x₁, x₂,…, x₈) with two additional chords: the edges {x₁, x₄} and {x₅, x₈}. An octagon quadrangle system of order v and index ρ [OQS] is a pair (X,H), where X is a finite set of v vertices and H is a collection of edge disjoint octagon quadrangles (called blocks) which partition the edge set of ρK_v defined on X. An octagon quadrangle systemΣ=(X,H) of order v and index λ is said to be upper C₄-perfect if the collection of all of the upper4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v; it is said to be upper strongly perfect, if the collection of all of the upper4-cycles contained in the octagon quadrangles form a μ-fold 4-cycle system of order v and also the collection of all of the outside8-cycles contained in the octagon quadrangles form a ?-fold 8-cycle system of order v. In this paper, the authors determine the spectrum for these systems.     

A stochastic Newton-Raphson method

A stochastic approximation procedure of the Robbins-Monro type is considered. The original idea behind the Newton-Raphson method is used as follows. Given n approximations X₁,…, X_n with observations Y₁,…, Y_n, a least squares line is fitted to the points (X_m, Y_m),…, (X_n, Y_n) where m<n may depend on n. The (n+1)st approximation is taken to be the intersection of the least squares line with y=0. A variation of the resulting process is studied. It is shown that this process yields a strongly consistent sequence of estimates which is asymptotically normal with minimal asymptotic variance.     

Exponential and polynomial tailbounds for change-point estimators

Let X_1n,…,X_nn be independent random elements with an unknown change point θ∈(0,1), that is X_in has a distribution ν₁ or ν₂, respectively, according to i⩽[nθ] or i>[nθ]. We propose an estimator θ_n of θ, which is defined as the maximizer of a weighted empirical process on (0,1). Finding upper bounds of polynomial and exponential type for the tails of nθ_n−[nθ], we are able to derive rates of almost sure convergence, of distributional convergence, of L_p-convergence and of convergence in the Ky-Fan- and in the Prokhorov-metric.     

BAHADUR SLOPES FOR COMPARISONS OF TESTS BASED ON RESAMPLING

Let X₁,…, X_n be random variables symmetric about θ from a common unknown distribution F_θ(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.     

Bootstrap test of goodness of fit to a linear model when errors are correlated

Given the regression model Y_i = m(x_i) +ε_i (x_i ε C, i = l,…,n, C a compact set in R) where m is unknown and the random errors {ε_i} present an ARMA structure, we design a bootstrap method for testing the hypothesis that the regression function follows a general linear model: H_o : m ε {mθ(.) = A^t(.)θ : θ ε ? ? R^q} with A a functional from R to R^q. The criterion of the test derives from a Cramer-von-Mises type functional distance D = d²([mcirc]_n, A^t(.)θ_n), between [mcirc]_n, a Gasser-Miiller non-parametric estimator of m, and the member of the class defined in H_o that is closest to m_n in terms of this distance. The consistency of the bootstrap distribution of D and θ_n is obtained under general conditions. Finally, simulations show the good behavior of the bootstrap approximation with respect to the asymptotic distribution of D = d².     

On the Influence of Record Terms in the Addition of Independent Random Variables

We discuss some problems connected with the role of record values and maximal values generated by sequences of random variables X₁, X₂,…, X _n in the process of the growth of sums X₁ +···+ X_n, n = 1, 2,….     

Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative

Let X_i, 1 ≤ i ≤ n, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of □{ρ_α(F_n, G) - _α(F, G)}, where F_n is the empirical distribution function based on X₁,…,X_n and _α is a Kolmogorov-Lévy-type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p_α is given by p_α(F, G) = inf {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}.     

Empirical Bayes regression analysis with many regressors but fewer observations

In this paper, we consider the prediction problem in multiple linear regression model in which the number of predictor variables, p, is extremely large compared to the number of available observations, n  . The least-squares predictor based on a generalized inverse is not efficient. We propose six empirical Bayes estimators of the regression parameters. Three of them are shown to have uniformly lower prediction error than the least-squares predictors when the vector of regressor variables are assumed to be random with mean vector zero and the covariance matrix (1/n)X^tX$(1/n){\mathit{X}}^{\mathrm{t}}\mathit{X}$ where X^t=(_x1,…,_xn)${\mathit{X}}^{\mathrm{t}}=({\mathit{x}}_1,\dots ,{\mathit{x}}_n)$ is the p×n$p\times n$ matrix of observations on the regressor vector centered from their sample means. For other estimators, we use simulation to show its superiority over the least-squares predictor.     

Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics

Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a₁<a₂<a₃<…. In particular, a_i could be equal to i for all i. Let X_1n≦X_2n≦?≦X_nn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦k≦n), independence of the event {X_kn=X_1n} and X_1n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{X_kn = X_1n | X_1n = a_i} is equivalent to X having the IFR (DFR) property. Let a_i = i and $\text{G(i) = P(X≧i), i = 1, 2, …}$. We prove that the independence of {X_2n ? X_1n ∈B} and X_1n for all i is equivalent to X being geometric, where B = {m} (B = {m,m+1,…}), provided G(i) = q^i?1, 1≦i≦m+2 (1≦i≦m+1), where 0<q<1.     

Lois limites pour les statistiques d'ordre dans le cas non identiquement distribue

Let X₁, X₂,… be a sequence of independent random variables with distribution functions F₁, where 1 ≤ i ≤ n, and for each n ≥ 1 let X_1,n ≤… ≤ X_n,n denote the order statistics of the first n random variables. Under suitable hypotheses about the F₁, we characterize the limit distribution functions H(x) for which P(X_k,n ? a_nx + b_n) → H(x), where a_n > 0 and b_n are real constants. We consider the cases where κ = κ(n) satisfies √n {κ(n)/n — λ} → 0 and √n {κ(n)/n — λ} → ∞ separately.     

Moments of Order Statistics from Weibull Distribution in the Presence of Multiple Outliers

In this article, we derive exact expressions for the single and product moments of order statistics from Weibull distribution under the contamination model. We assume that X₁, X₂, …, X_{n ? p} are independent with density function f(x) while the remaining, p observations (outliers) X_{n ? p + 1}, …, X_n are independent with density function arises from some modified version of f(x), which is called g(x), in which the location and/or scale parameters have been shifted in value. Next, we investigate the effect of the outliers on the BLUE of the scale parameter. Finally, we deduce some special cases.     

Rate of consistency of one sample tests of location

Let X₁,…,X_n be a sample from a population with continuous distribution function F(x?θ) such that F(x)+F(-x)=1 and 0<F(x)<1, x?R¹. It is shown that the power- function of a monotone test of H: θ=θ₀ against K: θ>θ₀ cannot tend to 1 as θ?θ₀ → ∞ more than n times faster than the tails of F tend to 0. Some standard as well as robust tests are considered with respect to this rate of convergence.     

Estimation of an autoregressive semiparametric model with exogenous variables

In many autoregressive relationships, there are observed external influences. This paper deals with the estimation of the multivariate model X_t+1= φ(X_t,…,X_t−r+1) + ψ(Y_t) + ε_t, where φ(·) is an unknown nonlinear function, ∫ the exogenous variable concerning ψ(·). Two cases are considered: ψ(·) is linear ψ(Y_t) = AY_t, where A is an unknown parameter, and ψ(·) the nonlinear function corresponding to a series expansion. In the latter situation, the method of estimation is ‘seminonparametric’. We first isolate and estimate parametrically the exogenous part, and then estimate nonparametrically the endogenous part ψ(·).