Discretized likelihood methods—asymptotic properties of discretized likelihood estimators (DLE's)

Suppose thatX ₁,X ₂, ...,X _n , ... is a sequence of i.i.d. random variables with a densityf(x, θ). Letc _n be a maximum order of consistency. We consider a solution \(\hat \theta _n \) of the discretized likelihood equation $$\sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n + rc_n^{ - 1} ) - } \sum\limits_{i = 1}^n {\log f(X_i ,\hat \theta _n ) = a_n (\hat \theta _n ,r)} $$ wherea _n (θ,r) is chosen so that \(\hat \theta _n \) is asymptotically median unbiased (AMU). Then the solution \(\hat \theta _n \) is called a discretized likelihood estimator (DLE). In this paper it is shown in comparison with DLE that a maximum likelihood estimator (MLE) is second order asymptotically efficient but not third order asymptotically efficient in the regular case. Further it is seen that the asymptotic efficiency (including higher order cases) may be systematically discussed by the discretized likelihood methods.     

Asymptotic Palm likelihood theory for stationary point processes

In the present paper, we propose a Palm likelihood approach as a general estimating principle for stationary point processes in $\mathbf{R}^d$ for which the density of the second-order factorial moment measure is available in closed form or in an integral representation. Examples of such point processes include the Neyman–Scott processes and the log Gaussian Cox processes. The computations involved in determining the Palm likelihood estimator are simple. Conditions are provided under which the Palm likelihood estimator is strongly consistent and asymptotically normally distributed.     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Parameter estimation for the discretely observed fractional Ornstein–Uhlenbeck process and the Yuima R package

This paper proposes consistent and asymptotically Gaussian estimators for the parameters $\lambda , \sigma $ and $H$ of the discretely observed fractional Ornstein–Uhlenbeck process solution of the stochastic differential equation $d Y_t = -\lambda Y_t dt + \sigma d W_t^H$ , where $(W_t^H, t\ge 0)$ is the fractional Brownian motion. For the estimation of the drift $\lambda $ , the results are obtained only in the case when $\frac{1}{2} < H < \frac{3}{4}$ . This paper also provides ready-to-use software for the R statistical environment based on the YUIMA package.     

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

On the Distribution of Atkin and Elkies Primes

Given an elliptic curve $E$ over a finite field $\mathbb {F}_q$ of $q$ elements, we say that an odd prime $\ell \not \mid q$ is an Elkies prime for $E$ if $t_E^2 - 4q$ is a square modulo  $\ell $ , where $t_E = q+1 - \#E(\mathbb {F}_q)$ and $\#E(\mathbb {F}_q)$ is the number of $\mathbb {F}_q$ -rational points on $E$ ; otherwise, $\ell $ is called an Atkin prime. We show that there are asymptotically the same number of Atkin and Elkies primes $\ell < L$ on average over all curves $E$ over $\mathbb {F}_q$ , provided that $L \ge (\log q)^\varepsilon $ for any fixed $\varepsilon >0$ and a sufficiently large $q$ . We use this result to design and analyze a fast algorithm to generate random elliptic curves with $\#E(\mathbb {F}_p)$ prime, where $p$ varies uniformly over primes in a given interval $[x,2x]$ .     

On Packing \mathbb {R}^3 with Thin Tori

We show that $\mathbb {R}^3$ can be packed at a density of $0.222\ldots $ with tori whose minor radius goes to zero. Furthermore, we show that the same torus arrangement yields an asymptotically optimal number of pairwise-linked tori.     

Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

Let $\Delta _{n-1}$ denote the $(n-1)$ -dimensional simplex. Let $Y$ be a random $d$ -dimensional subcomplex of $\Delta _{n-1}$ obtained by starting with the full $(d-1)$ -dimensional skeleton of $\Delta _{n-1}$ and then adding each $d$ -simplex independently with probability $p=\frac{c}{n}$ . We compute an explicit constant $\gamma _d$ , with $\gamma _2 \simeq 2.45$ , $\gamma _3 \simeq 3.5$ , and $\gamma _d=\Theta (\log d)$ as $d \rightarrow \infty $ , so that for $c < \gamma _d$ such a random simplicial complex either collapses to a $(d-1)$ -dimensional subcomplex or it contains $\partial \Delta _{d+1}$ , the boundary of a $(d+1)$ -dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant $\gamma _d< c_d <d+1$ such that for any $c>c_d$ and a fixed field $\mathbb{F }$ , asymptotically almost surely $H_d(Y;\mathbb{F }) \ne 0$ .     

Cops and Robber Game with a Fast Robber on Expander Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let ${c_{\infty}(G)}$ denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c _??(G) =? 1, and give an ${O( \mid V(G)\mid^2)}$ algorithm for their detection. We prove a lower bound for c _?? of expander graphs, and use it to prove three things. The first is that if ${np \geq 4.2 {\rm log}n}$ then the random graph ${G= \mathcal{G}(n, p)}$ asymptotically almost surely has ${\eta_{1}/p \leq \eta_{2}{\rm log}(np)/p}$ , for suitable positive constants ${\eta_{1}}$ and ${\eta_{2}}$ . The second is that a fixed-degree random regular graph G with n vertices asymptotically almost surely has ${c_{\infty}(G) = \Theta(n)}$ . The third is that if G is a Cartesian product of m paths, then ${n/4km^2 \leq c_{\infty}(G) \leq n/k}$ , where ${n = \mid V(G)\mid}$ and k is the number of vertices of the longest path.     

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by G?ba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant ${\omega(\nabla \Phi)}$ , expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of ${\Phi}$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here ${f : V \rightarrow \mathbb{R}}$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant ${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$ in the case n _k ≠ 0 (for maximal twisted orbit types (H _k )) guarantees the existence of at least |n _k | different G-orbits of periodic solutions with symmetries at least (H _k). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of ${\omega(\,\nabla \Phi)}$ evaluated.     

Computing fuzzy process efficiency in parallel systems

This paper deals with parallel process systems in which the input and output data are fuzzy. The $\upalpha $ -level based approach is used to compute the fuzzy system efficiency and a simple procedure is proposed to estimate the fuzzy efficiency of the different processes. The main contribution of the paper is estimating the latter taking into account the variability of the process efficiencies compatible with a given value of the system efficiency. This variability comes from the existence of alternative optimal weights in the system efficiency multiplier network DEA models. The computation of the fuzzy system efficiency involves one Linear and one Non-linear Program for each $\upalpha $ -cut while the computation of each process efficiency requires solving just a couple of related Linear Programs for each $\upalpha $ -cut. The proposed approach is illustrated with a parallel systems dataset extracted from the literature.     

Doubly periodic self-dual vortices in a relativistic non-Abelian Chern–Simons model

In this paper we establish a multiplicity result concerning the existence of doubly periodic solutions in a $2\times 2$ nonlinear elliptic system arising in the study of self-dual non-Abelian Chern–Simons vortices. We show that the system admits at least two solutions when the Chern–Simons coupling parameter $\kappa >0$ is sufficiently small; while no solution exists for $\kappa >0$ sufficiently large. As in Nolasco and Tarantello (Commun Math Phys 213:599–639, 2000), we use a variational formulation of the problem. Thus, we obtain a first solution via a constrained minimization method and show that it is asymptotically gauge-equivalent to the (broken) principal embedding vacuum of the system, as $\kappa \rightarrow 0$ . Then we obtain a second solution by a min-max procedure of “mountain pass” type.     

Reorthogonalization for the Golub–Kahan–Lanczos bidiagonal reduction

The Golub–Kahan–Lanczos (GKL) bidiagonal reduction generates, by recurrence, the matrix factorization of $X \in \mathbb{R }^{m \times n}, m \ge n$ , given by $$\begin{aligned} X = UBV^T \end{aligned}$$ where $U \in \mathbb{R }^{m \times n}$ is left orthogonal, $V \in \mathbb{R }^{n \times n}$ is orthogonal, and $B \in \mathbb{R }^{n \times n}$ is bidiagonal. When the GKL recurrence is implemented in finite precision arithmetic, the columns of $U$ and $V$ tend to lose orthogonality, making a reorthogonalization strategy necessary to preserve convergence of the singular values. The use of an approach started by Simon and Zha (SIAM J Sci Stat Comput, 21:2257–2274, 2000) that reorthogonalizes only one of the two left orthogonal matrices $U$ and $V$ is shown to be very effective by the results presented here. Supposing that $V$ is the matrix reorthogonalized, the reorthogonalized GKL algorithm proposed here is modeled as the Householder Q–R factorization of $\left( \begin{array}{c} 0_{n \times k} \\ X V_k \end{array}\right) $ where $V_k = V(:,1:k)$ . That model is used to show that if $\varepsilon _M $ is the machine unit and $$\begin{aligned} \bar{\eta }= \Vert \mathbf{tril }(I-V^T\!~V)\Vert _F, \end{aligned}$$ where $\mathbf{tril }(\cdot )$ is the strictly lower triangular part of the contents, then: (1) the GKL recurrence produces Krylov spaces generated by a nearby matrix $X + \delta X$ , $\Vert \delta X\Vert _F = \mathcal O (\varepsilon _M + \bar{\eta }) \Vert X\Vert _F$ ; (2) singular values converge in the Lanczos process at the rate expected from the GKL algorithm in exact arithmetic on a nearby matrix; (3) a new proposed algorithm for recovering leading left singular vectors produces better bounds on loss of orthogonality and residual errors.     

Sharpening the Norm Bound in the Subspace Perturbation Theory

Let $A$ be a (possibly unbounded) self-adjoint operator on a separable Hilbert space $\mathfrak H .$ Assume that $\sigma $ is an isolated component of the spectrum of $A$ , that is, $\mathrm{dist}(\sigma ,\Sigma )=d>0$ where $\Sigma =\mathrm spec (A)\setminus \sigma .$ Suppose that $V$ is a bounded self-adjoint operator on $\mathfrak H $ such that $\Vert V\Vert <d/2$ and let $L=A+V$ , $\mathrm{Dom }(L)=\mathrm{Dom }(A).$ Denote by $P$ the spectral projection of $A$ associated with the spectral set $\sigma $ and let $Q$ be the spectral projection of $L$ corresponding to the closed $\Vert V\Vert $ -neighborhood of $\sigma .$ Introducing the sequence $$\begin{aligned} \varkappa _n=\frac{1}{2}\left(1-\frac{(\pi ^2-4)^n}{(\pi ^2+4)^n}\right), \quad n\in \{0\}\cup {\mathbb N }, \end{aligned}$$ we prove that the following bound holds: $$\begin{aligned} \arcsin (\Vert P-Q\Vert )\le M_\star \left(\frac{\Vert V\Vert }{d}\right), \end{aligned}$$ where the estimating function $M_\star (x)$ , $x\in \bigl [0,\frac{1}{2}\bigr )$ , is given by $$\begin{aligned} M_\star (x)=\frac{1}{2}\,\,n_{_\#}(x)\,\arcsin \left(\frac{4\pi }{\pi ^2+4}\right) +\frac{1}{2}\,\arcsin \left(\frac{\pi ( x-\varkappa _{n_{_\#}(x)})}{1-2\varkappa _{n_{_\#}(x)})}\right), \end{aligned}$$ with $n_{_\#}(x)=\max \left\{ n\,\bigr |\,\,n\in \{0\}\cup {\mathbb N }\,, \varkappa _n\le x\right\} $ . The bound obtained is essentially stronger than the previously known estimates for $\Vert P-Q\Vert .$ Furthermore, this bound ensures that $\Vert P-Q\Vert <1$ and, thus, that the spectral subspaces $\mathrm{Ran }(P)$ and $\mathrm{Ran }(Q)$ are in the acute-angle case whenever $\Vert V\Vert <c_\star \,d$ , where $$\begin{aligned} c_\star =16\,\,\frac{\pi ^6-2\pi ^4+32\pi ^2-32}{(\pi ^2+4)^4}=0.454169\ldots . \end{aligned}$$ Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic $\sin 2\theta $ estimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.     

Approximation of values of the Gauss hypergeometric function by rational fractions

We consider a new approach to estimating the irrationality measure of numbers that are values of the Gauss hypergeometric function. Some of the previous results are improved, in particular, those concerning irrationalities of the form $ \sqrt k $ ln(( $ \sqrt k $ + 1)/( $ \sqrt k $ ? 1)) with k ∈ ?.     

On the dependency for asymptotically independent estimates

Suppose ${\widehat{\theta}_1}$ and ${\widehat{\theta}_2}$ are asymptotically independent non-lattice with a joint second order Edgeworth expansion in n ^?1/2. Then the ?? dependency coefficient is $$\alpha \left(\widehat{\theta}_1, \widehat{\theta}_2 \right) = n^{-1/2} C + O \left(n^{-1} \right),$$ where ${C = (4 \pi)^{-1}\exp (-1/2) (\tau^2_1 + \tau^2_2) ^{1/2}}$ for ${\tau_1, \tau_2}$ their joint skewness coefficients.     

On maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions

Maximum likelihood estimation of the concentration parameter of von Mises–Fisher distributions involves inverting the ratio \(R_\nu = I_{\nu +1} / I_\nu \) of modified Bessel functions and computational methods are required to invert these functions using approximative or iterative algorithms. In this paper we use Amos-type bounds for \(R_\nu \) to deduce sharper bounds for the inverse function, determine the approximation error of these bounds, and use these to propose a new approximation for which the error tends to zero when the inverse of \(R_\nu \) is evaluated at values tending to \(1\) (from the left). We show that previously introduced rational bounds for \(R_\nu \) which are invertible using quadratic equations cannot be used to improve these bounds.     

[1,1,t]-Colorings of Complete Graphs

Given non-negative integers $r, s,$ and $t,$ an $[r,s,t]$ -coloring of a graph $G = (V(G),E(G))$ is a mapping $c$ from $V(G) \cup E(G)$ to the color set $\{1,\ldots ,k\}$ such that $\left|c(v_i) - c(v_j)\right| \ge r$ for every two adjacent vertices $v_i,v_j, \left|c({e_i}) - c(e_j)\right| \ge s$ for every two adjacent edges $e_i,e_j,$ and $\left|c(v_i) - c(e_j)\right| \ge t$ for all pairs of incident vertices and edges, respectively. The $[r,s,t]$ -chromatic number $\chi _{r,s,t}(G)$ of $G$ is defined to be the minimum $k$ such that $G$ admits an $[r,s,t]$ -coloring. In this note we examine $\chi _{1,1,t}(K_p)$ for complete graphs $K_p.$ We prove, among others, that $\chi _{1,1,t}(K_p)$ is equal to $p+t-2+\min \{p,t\}$ whenever $t \ge \left\lfloor {\frac{p}{2}}\right\rfloor -1,$ but is strictly larger if $p$ is even and sufficiently large with respect to $t.$ Moreover, as $p \rightarrow \infty $ and $t=t(p),$ we asymptotically have $\chi _{1,1,t}(K_p)=p+o(p)$ if and only if $t=o(p).$