On the asymptotic efficiency of estimators in an autoregressive process

Summary Let {X _t } be defined recursively byX _t =θX _t−1+U _t (t=1,2, ...), whereX ₀=0 and {U _t } is a sequence of independent identically distributed real random variables having a density functionf with mean 0 and varianceσ ². We assume that |θ|<1. In the present paper we obtain the bound of the asymptotic distributions of asymptotically median unbiased (AMU) estimators of θ and the sufficient condition that an AMU estimator be asymptotically efficient in the sense that its distribution attains the above bound. It is also shown that the least squares estimator of θ is asymptotically efficient if and only iff is a normal density function. University of Electro-Communications     

Some approximations of stochastic θ-integrals

In this paper, we consider problems of approximation of stochastic θ-integrals (θ)∫ ₀ ^t f(B(s))dB(s) with respect to a Brownian motion by sums of the form ∑ _k=1 ^p f_n(B _n ^θ (t_k-1))[B _n ^θ (t_k)-B _n ^θ (t_k-1], where the sequences {f_n,n∈∕#x007D; and {[B _n ^θ ,n∈∕} are convolution-type approximations of the functionf and Brownian motionB. Belorussian State University, F. Skoryna ave. 4, 220050 Minsk, Belorus. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp. 248–256, April–June, 1999. Translated by V. Mackevičius     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Monotone maximum likelihood estimate for hazard function

A wide class of reliability theory models or lifetime data can be described as follows. Assume that the lifetime distribution function is F(t, θ)=F₀(λ(θ)t), where θ is the parameter characterizing some inner properties of a product and λ(θ) is an unknown increasing function. The paper deals with methods of estimation of λ(θ) from the sample (t _i ,θ _i ),i = 1, ...,n, for the case of exponentialF ₀. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 46–51, Perm, 1991.     

On the iteration of holomorphic self-maps of ℂ*

Letf be a holomorphic self-map of the punctured plane ℂ^*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI ₀(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞} andI _∞(f)={z ∈ ℂ^*:f ⁿ (z) → 0,n → ∞}. We try to find the relation betweenI ₀(f),I _∞(t) andJ(f). It is proved that both the boundary ofI ₀(f) and the boundary ofI _∞)f) equal toJ(f),I ₀(f) ∩J(f) ≠ θ andI _∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI ₀(f) andI _∞(f) are not doubly-bounded. Supported by the National Natural Science Foundation of China     

Cramér-type conditions and quadratic mean differentiability

Summary Let (Ω,A) be a measurable space, let Θ be an open set inR ^k , and let {P _θ; θ∈Θ} be a family of probability measures defined onA. Let μ be a σ-finite measure onA, and assume thatP _θ≪μ for each θ∈Θ. Let us denote a specified version ofdP _θ /d _μ byf(ω; θ). In many large sample problems in statistics, where a study of the log-likelihood is important, it has been convenient to impose conditions onf(ω; θ) similar to those used by Cramér [2] to establish the consistency and asymptotic normality of maximum likelihood estimates. These are of a purely analytical nature, involving two or three pointwise derivatives of lnf(ω; θ) with respect to θ. Assumptions of this nature do not have any clear probabilistic or statistical interpretation. In [10], LeCam introduced the concept of differentially asymptotically normal (DAN) families of distributions. One of the basic properties of such a family is the form of the asymptotic expansion, in the probability sense, of the log-likelihoods. Roussas [14] and LeCam [11] give conditions under which certain Markov Processes, and sequences of independent identically distributed random variables, respectively, form DAN families of distributions. In both of these papers one of the basic assumptions is the differentiability in quadratic mean of a certain random function. This seems to be a more appealing type of assumption because of its probabilistic nature. In this paper, we shall prove a theorem involving differentiability in quadratic mean of random functions. This is done in Section 2. Then, by confining attention to the special case when the random function is that considered by LeCam and Roussas, we will be able to show that the standard conditions of Cramér type are actually stronger than the conditions of LeCam and Roussas in that they imply the existence of the necessary quadratic mean derivative. The relevant discussion is found in Section 3. This research was supported by the National Science Foundation, Grant GP-20036.     

Hölder continuity for spatial and path processes via spectral analysis

For ν(dθ), a σ-finite Borel measure on R ^d , we consider L ²(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ ^t ₀ e ^{−λ(θ)( t − s )} dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ ^t ₀ g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process _t (τ)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

ψ-correct decision for selection and elimination

Summary The selection oft out ofk populations with parameters θ _i (i=1, ...,k) is said to result in an ψ-correct decision provided ψ (minimum selected θ)>maximum non-selected θ where ψ(θ) (>θ) is an increasing function. For the cases of location or scale parameters the minimum probability of ψ-correct decision over the entire parameter space is shown to be no less than the minimum probability of correct selection over a preference zone determined by ψ(θ). For other types of parameters this result is shown to be true under certain conditions linking the distribution function and the ψ function.     

Incompleteness and minimality of complex exponential system

A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A,M)in C_α,where C_αis the weighted Banach space consisting of all complex continuous functions f on the real axis R with f(t)exp(-α(t))vanishing at infinity,in the uniform norm‖f‖_α=sup{|f(t)e~(-α(t))|:t∈R}with respect to the weightα(t).If the incompleteness holds, then the complex exponential system E(?)is minimal and each function in the closure of the linear span of complex exponential system E(?)can be extended to an entire function represented by a Taylor-Dirichlet series.     

The Existence and Uniqueness of Nash Equilibrium Point in an m-player Game “Shoot Later, Shoot First!”

We consider the following “silent duel” of m players with a possible economic interpretation. Each player has one “bullet”, which she can shoot at any time during the time interval [0,1]. The probability that the i-th player hits the “target” at moment t is given by an increasing accuracy function f _i (t). The winner is the player who hits the target first. Under natural assumptions on the functions f _i (t) we prove the existence and uniqueness of a Nash equilibrium point in this game, and we provide an explicit construction of this equilibrium. This construction allows us to obtain exact solutions for many specific examples. Some of them are presented.This work was partly supported by RBRF grants 03-01-00479.     

The character table of the group GL 2(Q) when extended by a certain group of order two

LetG denote either of the groupsGL ₂(q) or SL₂(q). Then θ :G →G given by θ(A) = (A ^t)^t, whereA ^t denotes the transpose of the matrixA, is an automorphism ofG. Therefore we may form the groupG.θ> which is the split extension of the groupG by the cyclic group θ of order 2. Our aim in this paper is to find the complex irreducible character table ofG. θ.     

Weighted approximation of random functions

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Some limit results for probabilities estimates of Brownian motion with polynomial drift

Let (B _t + f(t)) _t∈[0,+∞) be a Brownian motion with polynomial drift f(t), where f(t) is a polynomial. Some Limit Results for Lower tail and large deviation probabilities estimates, and Level crossing probabilities estimates of (B _t + f(t)) _t∈[0,+∞) are given in this paper.     

Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval

For location families with densitiesf ₀(x−θ), we study the problem of estimating θ for location invariant lossL(θ,d)=ρ(d−θ), and under a lower-bound constraint of the form θ≥a. We show, that for quite general (f ₀, ρ), the Bayes estimator δ _U with respect to a uniform prior on (a, ∞) is a minimax estimator which dominates the benchmark minimum risk equivariant (MRE) estimator. In extending some previous dominance results due to Katz and Farrell, we make use of Kubokawa'sIERD (Integral Expression of Risk Difference) method, and actually obtain classes of dominating estimators which include, and are characterized in terms of δ _U . Implications are also given and, finally, the above dominance phenomenon is studied and extended to an interval constraint of the form θ∈[a, b]. Research supported by NSERC of Canada.     

On a Function Related to Multinomial Coefficients (I)

Let F _p,t (n) denote the number of the coefficients of (x ₁+1x ₂+...+x _t ) ^j , 0 ≤j≤n− 1, which are not divisible by the prime p. Define G _p,t (n) = F _p,t /n ^θ and β(p,t) = lim infF _p,t )(n)/n ^θ, where θ = (log)/(log p). In this paper, we mainly prove that G _p,t can be extended to a continuous function on ℝ⁺, and the function G _p,t is nowhere monotonic. Both the set of differential points of the function G _p,t and the set of non-differential points of the function G _p,t are dense in ℝ⁺. Received February 18, 2000, Accepted December 7, 2000     

Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation

Let X _t be a one-dimensional Harris recurrent diffusion, with a drift depending on an unknown parameter θ belonging to some metric compact Θ. We firstly show that all integrable additive functionals of X _t are asymptotically equivalent in probability to some deterministic process v _t . Then we use this result to study the behavior of the maximum likelihood estimator for the parameter θ. Under mild regularity assumptions, we find an upper rate of its convergence as a function of v _t , extending some recent results for ergodic diffusions.      

Filtration of components of processes of random evolution

The problem of estimation of a nonobservable component θ_t for a two-dimensional process (θ_t, ξ_t) of random evolution (θ _t,ξ_t);x_t, 0≤t≤T, is investigated on the basis of observations of ξ_s. s≤t, where x _t is a homogeneous Markov process with infinitesimal operator Q. Applications to stochastic models of a (B,S)-market of securities is described under conditions of incomplete market. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1701–1705, December, 1998.     

Asymptotic equivalence for nonparametric generalized linear models

Summary. We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t _i ) of a regression function f at a grid point t _i (nonparametric GLM). When f is in a H?lder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process. Received: 4 February 1997     

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).     

Stability of Solution of a Class of Quasilinear Elliptic Equations

In this paper, using capacity theory and extension theorem of Lipschitz functions we first discuss the uniqueness of weak solution of nonhomogeneous quasilinear elliptic equationsin space W(θ,p)(Ω), which is bigger than W1,p(Ω). Next, using revise reverse Holder inequality we prove that if ωc is uniformly p-think, then there exists a neighborhood U of p, such that for all t ∈U, the weak solutions of equation corresponding t are bounded uniformly. Finally, we get the stability of weak solutions on exponent p.