Parameter estimation for the stochastic SIS epidemic model

In this paper we estimate the parameters in the stochastic SIS epidemic model by using pseudo-maximum likelihood estimation (pseudo-MLE) and least squares estimation. We obtain the point estimators and $100 (1-\alpha )\%$ confidence intervals as well as $100 (1-\alpha )\%$ joint confidence regions by applying least squares techniques. The pseudo-MLEs have almost the same form as the least squares case. We also obtain the exact as well as the asymptotic $100 (1-\alpha )\%$ joint confidence regions for the pseudo-MLEs. Computer simulations are performed to illustrate our theory.     

Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization

Let \(A\) be a compact \(d\) -rectifiable set embedded in Euclidean space \({\mathbb R}^p, d\le p\) . For a given continuous distribution \(\sigma (x)\) with respect to a \(d\) -dimensional Hausdorff measure on \(A\) , our earlier results provided a method for generating \(N\) -point configurations on \(A\) that have an asymptotic distribution \(\sigma (x)\) as \(N\rightarrow \infty \) ; moreover, such configurations are “quasi-uniform” in the sense that the ratio of the covering radius to the separation distance is bounded independently of \(N\) . The method is based upon minimizing the energy of \(N\) particles constrained to \(A\) interacting via a weighted power-law potential \(w(x,y)|x-y|^{-s}\) , where \(s>d\) is a fixed parameter and \(w(x,y)=\left( \sigma (x)\sigma (y)\right) ^{-({s}/{2d})}\) . Here we show that one can generate points on \(A\) with the aforementioned properties keeping in the energy sums only those pairs of points that are located at a distance of at most \(r_N=C_N N^{-1/d}\) from each other, with \(C_N\) being a positive sequence tending to infinity arbitrarily slowly. To do this, we minimize the energy with respect to a varying truncated weight \(v_N(x,y)=\Phi (|x-y|/r_N)\cdot w(x,y)\) , where \(\Phi :(0,\infty )\rightarrow [0,\infty )\) is a bounded function with \(\Phi (t)=0, t\ge 1\) , and \(\lim _{t\rightarrow 0^+}\Phi (t)=1\) . Under appropriate assumptions, this reduces the complexity of generating \(N\) -point “low energy” discretizations to order \(N C_N^d\) computations.     

On Estimating the Asymptotic Variance of Stationary Point Processes

We investigate a class of kernel estimators $\widehat{\sigma}^2_n$ of the asymptotic variance σ ² of a d-dimensional stationary point process $\Psi = \sum_{i\ge 1}\delta_{X_i}$ which can be observed in a cubic sampling window $W_n = [-n,n]^d\,$ . σ ² is defined by the asymptotic relation $Var(\Psi(W_n)) \sim \sigma^2 \,(2n)^d$ (as n →? ∞) and its existence is guaranteed whenever the corresponding reduced covariance measure $\gamma^{(2)}_{red}(\cdot)$ has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of $\gamma^{(2)}_{red}(\cdot)$ outside of an expanding ball centered at the origin, we determine optimal bandwidths b _n (up to a constant) minimizing the mean squared error of $\widehat{\sigma}^2_n$ . The case when $\gamma^{(2)}_{red}(\cdot)$ has bounded support is of particular interest. Further we suggest an isotropised estimator $\widetilde{\sigma}^2_n$ suitable for motion-invariant point processes and compare its properties with $\widehat{\sigma}^2_n$ . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of $\widehat{\sigma}^2_n$ for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b _n .     

Model selection for density estimation with \mathbb L _2-loss

We consider here estimation of an unknown probability density $s$ belonging to $\mathbb L _2(\mu )$ where $\mu $ is a probability measure. We have at hand $n$ i.i.d. observations with density $s$ and use the squared $\mathbb L _2$ -norm as our loss function. The purpose of this paper is to provide an abstract but completely general method for estimating $s$ by model selection, allowing to handle arbitrary families of finite-dimensional (possibly non-linear) models and any $s\in \mathbb L _2(\mu )$ . We shall, in particular, consider the cases of unbounded densities and bounded densities with unknown $\mathbb L _\infty $ -norm and investigate how the $\mathbb L _\infty $ -norm of $s$ may influence the risk. We shall also provide applications to adaptive estimation and aggregation of preliminary estimators. One major technical tool of our approach is a proof of the existence of suitable tests between $\mathbb L _2$ -balls with centers belonging to $\mathbb L _\infty $ . Although of a purely theoretical nature, our method leads to results that cannot presently be reached by more concrete ones.     

Empirical likelihood bivariate nonparametric maximum likelihood estimator with right censored data

This article considers the estimation for bivariate distribution function (d.f.) \(F_0(t, z)\) of survival time \(T\) and covariate variable \(Z\) based on bivariate data where \(T\) is subject to right censoring. We derive the empirical likelihood-based bivariate nonparametric maximum likelihood estimator \(\hat{F}_n(t,z)\) for \(F_0(t,z)\) , which has an explicit expression and is unique in the sense of empirical likelihood. Other nice features of \(\hat{F}_n(t,z)\) include that it has only nonnegative probability masses, thus it is monotone in bivariate sense. We show that under \(\hat{F}_n(t,z)\) , the conditional d.f. of \(T\) given \(Z\) is of the same form as the Kaplan–Meier estimator for the univariate case, and that the marginal d.f. \(\hat{F}_n(\infty ,z)\) coincides with the empirical d.f. of the covariate sample. We also show that when there is no censoring, \(\hat{F}_n(t,z)\) coincides with the bivariate empirical d.f. For discrete covariate \(Z\) , the strong consistency and weak convergence of \(\hat{F}_n(t,z)\) are established. Some simulation results are presented.     

Canonical subgroups via Breuil–Kisin modules

Let $p>2$ be a rational prime and $K/ \mathbb Q _p$ be an extension of complete discrete valuation fields. Let $\mathcal G $ be a truncated Barsotti–Tate group of level $n$ , height $h$ and dimension $d$ over $\mathcal{O }_K$ with $0<d<h$ . In this paper, we show that if the Hodge height of $\mathcal G $ is less than $1/(p^{n-2}(p+1))$ , then there exists a finite flat closed subgroup scheme of $\mathcal G $ of order $p^{nd}$ over $\mathcal{O }_K$ with standard properties as the canonical subgroup.     

Asymptotically Spirallike Mappings in Reflexive Complex Banach Spaces

In this paper we consider the notion of asymptotic spirallikeness in reflexive complex Banach spaces $X$ , and the connection with univalent subordination chains. Poreda initially introduced the notion of asymptotic starlikeness to characterize biholomorphic mappings on the unit polydisc in $\mathbb{C }^{n}$ which have parametric representation in the sense of Loewner theory. The authors introduced the notions of $A$ -asymptotic spirallikeness and $A$ -parametric representation on the Euclidean unit ball of $\mathbb{C }^{n}$ , where $A\in L(\mathbb{C }^{n})$ with $m(A)>0$ . They showed that these notions are equivalent whenever $k_+(A)<2m(A)$ . In this paper we prove that if $k_+(A)<2m(A)$ and $f\in S(B)$ has $A$ -parametric representation, then $f$ is also $A$ -asymptotically spirallike on the unit ball $B$ of $X$ . For the converse, we need the additional assumption that $f$ is a smooth $A$ -asymptotically spirallike mapping, except in the finite-dimensional case $X=\mathbb{C }^{n}$ with an arbitrary norm. The notion of asymptotic spirallikeness involves differential equations and may be regarded as giving a geometric characterization of certain domains in $X$ . That is one of the motivations for considering this notion in the case of reflexive complex Banach spaces.     

Tree metrics and edge-disjoint S-paths

Given an undirected graph \(G=(V,E)\) with a terminal set \(S \subseteq V\) , a weight function on terminal pairs, and an edge-cost \(a: E \rightarrow \mathbf{Z}_+\) , the \(\mu \) -weighted minimum-cost edge-disjoint \(S\) -paths problem ( \(\mu \) -CEDP) is to maximize \(\sum \nolimits _{P \in \mathcal{P}} \mu (s_P,t_P) - a(P)\) over all edge-disjoint sets \(\mathcal{P}\) of \(S\) -paths, where \(s_P,t_P\) denote the ends of \(P\) and \(a(P)\) is the sum of edge-cost \(a(e)\) over edges \(e\) in \(P\) . Our main result is a complete characterization of terminal weights \(\mu \) for which \(\mu \) -CEDP is tractable and admits a combinatorial min–max theorem. We prove that if \(\mu \) is a tree metric, then \(\mu \) -CEDP is solvable in polynomial time and has a combinatorial min–max formula, which extends Mader’s edge-disjoint \(S\) -paths theorem and its minimum-cost generalization by Karzanov. Our min–max theorem includes the dual half-integrality, which was earlier conjectured by Karzanov for a special case. We also prove that \(\mu \) -EDP, which is \(\mu \) -CEDP with \(a = 0\) , is NP-hard if \(\mu \) is not a truncated tree metric, where a truncated tree metric is a weight function represented as pairwise distances between balls in a tree. On the other hand, \(\mu \) -CEDP for a truncated tree metric \(\mu \) reduces to \(\mu '\) -CEDP for a tree metric \(\mu '\) . Thus our result is best possible unless P = NP. As an application, we obtain a good approximation algorithm for \(\mu \) -EDP with “near” tree metric \(\mu \) by utilizing results from the theory of low-distortion embedding.     

On constrained and regularized high-dimensional regression

High-dimensional feature selection has become increasingly crucial for seeking parsimonious models in estimation. For selection consistency, we derive one necessary and sufficient condition formulated on the notion of degree of separation. The minimal degree of separation is necessary for any method to be selection consistent. At a level slightly higher than the minimal degree of separation, selection consistency is achieved by a constrained $L_0$ -method and its computational surrogate—the constrained truncated $L_1$ -method. This permits up to exponentially many features in the sample size. In other words, these methods are optimal in feature selection against any selection method. In contrast, their regularization counterparts—the $L_0$ -regularization and truncated $L_1$ -regularization methods enable so under slightly stronger assumptions. More importantly, sharper parameter estimation/prediction is realized through such selection, leading to minimax parameter estimation. This, otherwise, is impossible in the absence of a good selection method for high-dimensional analysis.     

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.     

Stability of contact discontinuity for steady Euler system in infinite duct

In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct ${\mathcal{N}_0}$ of infinite length in ${\mathbb{R}^2}$ with width W ₀ and consider two uniform subsonic flow ${{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}$ with different horizontal velocity in ${\mathcal{N}_0}$ divided by a flat contact discontinuity ${\Gamma_{cd}}$ . And, we slightly perturb the boundary of ${\mathcal{N}_0}$ so that the width of the perturbed duct converges to ${W_0+\omega}$ for ${|\omega| < \delta}$ at ${x=\infty}$ for some ${\delta >0 }$ . Then, we prove that if the asymptotic state at left far field is given by ${{U_l}^{\pm}}$ , and if the perturbation of boundary of ${\mathcal{N}_0}$ and ${\delta}$ is sufficiently small, then there exists unique asymptotic state ${{U_r}^{\pm}}$ with a flat contact discontinuity ${\Gamma_{cd}^*}$ at right far field( ${x=\infty}$ ) and unique weak solution ${U}$ of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to ${{U_l}^{\pm}}$ and ${{U_r}^{\pm}}$ at ${x=-\infty}$ and ${x=\infty}$ respectively. For that purpose, we establish piecewise C ¹ estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of ${\partial\mathcal{N}_0}$ and ${\delta}$ .     

Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space

We consider positive solutions of $\varDelta u=0$ in $\mathbf{R}_+^n$ , $\partial _{\nu }u=u^q$ on $\partial \mathbf{R}_+^n$ , where $n\ge 3$ and $q>n/(n-2)$ . We investigate the qualitative property of positive $x_n$ -axial symmetric solutions. In particular, we are concerned with the asymptotic expansion and the intersection property of positive $x_n$ -axial symmetric solutions.     

On lower bounds for cohomology growth in p-adic analytic towers

Let \(p\) and \(\ell \) be two distinct prime numbers and let \(\Gamma \) be a group. We study the asymptotic behaviour of the mod- \(\ell \) Betti numbers in \(p\) -adic analytic towers of finite index subgroups. If \(\Theta \) is a finite \(\ell \) -group of automorphisms of \(\Gamma \) , our main theorem allows to lift lower bounds for the mod- \(\ell \) cohomology growth in the fixed point group \(\Gamma ^\Theta \) to lower bounds for the growth in \(\Gamma \) . We give applications to \(S\) -arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.     

On a convolution series attached to a Siegel Hecke cusp form of degree 2

We prove that the “naive” convolution Dirichlet series $D_{2}(s)$ attached to a degree 2 Siegel Hecke cusp form $F$ , has a pole at $s=1$ . As an application, we write down the asymptotic formula for the partial sums of the squares of the eigenvalues of $F$ with an explicit error term. Further, as a corollary, we are able to show that the abscissa of absolute convergence of the (normalized) spinor-zeta function attached to $F$ is $s = 1$ .     

Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

Counting nonsingular matrices with primitive row vectors

We give an asymptotic expression for the number of nonsingular integer $n\times n$ -matrices with primitive row vectors, determinant $k$ , and Euclidean matrix norm less than $T$ , as $T\rightarrow \infty $ . We also investigate the density of matrices with primitive rows in the space of matrices with determinant $k$ , and determine its asymptotics for large $k$ .     

Scattered linear sets generated by collineations between pencils of lines

Let \(A\) and \(B\) be two points of \(\mathrm{{PG}}(2,q^n)\) , and let \(\Phi \) be a collineation between the pencils of lines with vertices \(A\) and \(B\) . In this paper, we prove that the set of points of intersection of corresponding lines under \(\Phi \) is either the union of a scattered \(\mathrm{{GF}}(q)\) -linear set of rank \(n+1\) with the line \(AB\) or the union of \(q-1\) scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n\) with \(A\) and \(B\) . We also determine the intersection configurations of two scattered \(\mathrm{{GF}}(q)\) -linear sets of rank \(n+1\) of \(\mathrm{{PG}}(2,q^n)\) both meeting the line \(AB\) in a \(\mathrm{{GF}}(q)\) -linear set of pseudoregulus type with transversal points \(A\) and \(B\) .     

Sharp local estimates for the Szegö–Weinberger profile in Riemannian manifolds

We study the local Szegö–Weinberger profile in a geodesic ball \(B_g(y_0,r_0)\) centered at a point \(y_0\) in a Riemannian manifold \(({\mathcal {M}},g)\) . This profile is obtained by maximizing the first nontrivial Neumann eigenvalue \(\mu _2\) of the Laplace–Beltrami Operator \(\Delta _g\) on \({\mathcal {M}}\) among subdomains of \(B_g(y_0,r_0)\) with fixed volume. We derive a sharp asymptotic bounds of this profile in terms of the scalar curvature of \({\mathcal {M}}\) at \(y_0\) . As a corollary, we deduce a local comparison principle depending only on the scalar curvature. Our study is related to previous results on the profile corresponding to the minimization of the first Dirichlet eigenvalue of \(\Delta _g\) , but additional difficulties arise due to the fact that \(\mu _2\) is degenerate in the unit ball in \(\mathbb {R}^N\) and geodesic balls do not yield the optimal lower bound in the asymptotics we obtain.     

Integer-valued polynomials over matrices and divided differences

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$ , the $k$ th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$ . If in addition $D$ has zero Jacobson radical then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$ , that is, conjugate integral elements of degree $n$ over $D$ .     

Generalized derivations with annihilating and centralizing Engel conditions on Lie ideals

Let $R$ be a non-commutative prime ring, with center $Z(R)$ , extended centroid $C$ and let $F$ be a non-zero generalized derivation of $R$ . Denote by $L$ a non-central Lie ideal of $R$ . If there exists $0\ne a\in R$ such that $a[F(x),x]_k\in Z(R)$ for all $x\in L$ , where $k$ is a fixed integer, then one of the followings holds: (1) either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$ , (2) or $R$ satisfies $s_4$ , the standard identity in $4$ variables, and $char(R)=2$ ; (3) or $R$ satisfies $s_4$ and there exist $q\in U, \gamma \in C$ such that $F(x)=qx+xq+\gamma x$ .