Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model, based on the Gaussian likelihood conditional on initial values. We give conditions on the parameters such that the process X_t is fractional of order d and cofractional of order d−b; that is, there exist vectors β for which β^′X_t is fractional of order d−b and no other fractionality order is possible. For b=1, the model nests the I(d−1) vector autoregressive model. We define the statistical model by 0 < bd, but conduct inference when the true values satisfy 0d₀−b₀<1/2 and b₀≠1/2, for which β₀^′X_t is (asymptotically) a stationary process. Our main technical contribution is the proof of consistency of the maximum likelihood estimators. To this end, we prove weak convergence of the conditional likelihood as a continuous stochastic process in the parameters when errors are independent and identically distributed with suitable moment conditions and initial values are bounded. Because the limit is deterministic, this implies uniform convergence in probability of the conditional likelihood function. If the true value b₀>1/2, we prove that the limit distribution of is mixed Gaussian, while for the remaining parameters it is Gaussian. The limit distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion of type II. If b₀<1/2, all limit distributions are Gaussian or chi‐squared. We derive similar results for the model with d = b, allowing for a constant term.     

Efficient point coverage in wireless sensor networks

We study minimum-cost sensor placement on a bounded 3D sensing field, R, which comprises a number of discrete points that may or may not be grid points. Suppose we have ℓ types of sensors available with different sensing ranges and different costs. We want to find, given an integer σ ≥ 1, a selection of sensors and a subset of points to place these sensors such that every point in R is covered by at least σ sensors and the total cost of the sensors is minimum. This problem is known to be NP-hard. Let k_i denote the maximum number of points that can be covered by a sensor of the ith type. We present in this paper a polynomial-time approximation algorithm for this problem with a proven approximation ratio . In applications where the distance of any two points has a fixed positive lower bound, each k_i is a constant, and so we have a polynomial-time approximation algorithms with a constant guarantee. While γ may be large, we note that it is only a worst-case upper bound. In practice the actual approximation ratio is small, even on randomly generated points that do not have a fixed positive minimum distance between them. We provide a number of numerical results for comparing approximation solutions and optimal solutions, and show that the actual approximation ratios in these examples are all less than 3, even though γ is substantially larger. This research was supported in part by NSF under grant CCF-04080261 and by NSF of China under grant 60273062.     

Selective support vector machines

In this study we introduce a generalized support vector classification problem: Let X _i , i=1,…,n be mutually exclusive sets of pattern vectors such that all pattern vectors x _i,k , k=1,…,|X _i | have the same class label y _i . Select only one pattern vector $x_{i,k^{*}}In this study we introduce a generalized support vector classification problem: Let X _i , i=1,…,n be mutually exclusive sets of pattern vectors such that all pattern vectors x _i,k , k=1,…,|X _i | have the same class label y _i . Select only one pattern vector from each set X _i such that the margin between the set of selected positive and negative pattern vectors are maximized. This problem is formulated as a quadratic mixed 0-1 programming problem, which is a generalization of the standard support vector classifiers. The quadratic mixed 0-1 formulation is shown to be -hard. An alternative approach is proposed with the free slack concept. Primal and dual formulations are introduced for linear and nonlinear classification. These formulations provide flexibility to the separating hyperplane to identify the pattern vectors with large margin. Iterative elimination and direct selection methods are developed to select such pattern vectors using the alternative formulations. These methods are compared with a na?ve method on simulated data. The iterative elimination method is also applied to neural data from a visuomotor categorical discrimination task to classify highly cognitive brain activities.     

Testing for Common Conditionally Heteroskedastic Factors

This paper proposes a test for common conditionally heteroskedastic (CH) features in asset returns. Following Engle and Kozicki (1993), the common CH features property is expressed in terms of testable overidentifying moment restrictions. However, as we show, these moment conditions have a degenerate Jacobian matrix at the true parameter value and therefore the standard asymptotic results of Hansen (1982) do not apply. We show in this context that Hansen's (1982) J‐test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution. If two assets are considered, this asymptotic distribution is a fifty–fifty mixture of χ²_H−1 and χ²_H, where H is the number of moment conditions, as opposed to a χ²_H−1. With more than two assets, this distribution lies between the χ²_H−p and χ²_H (p denotes the number of parameters). These results show that ignoring the lack of first‐order identification of the moment condition model leads to oversized tests with a possibly increasing overrejection rate with the number of assets. A Monte Carlo study illustrates these findings.     

New algorithms for k-center and extensions

The problem of interest is covering a given point set with homothetic copies of several convex containers C ₁,…,C _k , while the objective is to minimize the maximum over the dilatation factors. Such k-containment problems arise in various applications, e.g. in facility location, shape fitting, data classification or clustering. So far most attention has been paid to the special case of the Euclidean k-center problem, where all containers C _i are Euclidean unit balls. Recent developments based on so-called core-sets enable not only better theoretical bounds in the running time of approximation algorithms but also improvements in practically solvable input sizes. Here, we present some new geometric inequalities and a Mixed-Integer-Convex-Programming formulation. Both are used in a very effective branch-and-bound routine which not only improves on best known running times in the Euclidean case but also handles general and even different containers among the C _i .     

A dynamic programming approach of finding an optimal broadcast schedule in minimizing total flow time

We study the problem of (off-line) broadcast scheduling in minimizing total flow time and propose a dynamic programming approach to compute an optimal broadcast schedule. Suppose the broadcast server has k pages and the last page request arrives at time n. The optimal schedule can be computed in O(k³(n+k)^k−1) time for the case that the server has a single broadcast channel. For m channels case, i.e., the server can broadcast m different pages at a time where m < k, the optimal schedule can be computed in O(n^k−m) time when k and m are constants. Note that this broadcast scheduling problem is NP-hard when k is a variable and will take O(n^k−m+1) time when k is fixed and m ≥ 1 with the straightforward implementation of the dynamic programming approach. The preliminary version of this paper appeared in Proceedings of the 11th Annual International Computing and Combinatorics Conference as “Off-line Algorithms for Minimizing the Total Flow Time in Broadcast Scheduling”.     

Finding a Length-Constrained Maximum-Density Path in a Tree

Let T = (V,E,w) be an undirected and weighted tree with node set V and edge set E, where w(e) is an edge weight function for e E. The density of a path, say e₁, e₂,..., e_k, is defined as ^k_{i = 1} w(e_i)/k. The length of a path is the number of its edges. Given a tree with n edges and a lower bound L where 1 L n, this paper presents two efficient algorithms for finding a maximum-density path of length at least L in O(nL) time. One of them is further modified to solve some special cases such as full m-ary trees in O(n) time.     

The total {k}-domatic number of a graph

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,…,k} such that for any vertex v∈V(G), the condition ∑ _u∈N(v) f(u)≥k is fulfilled, where N(v) is the open neighborhood of v. A set {f ₁,f ₂,…,f _d } of total {k}-dominating functions on G with the property that ?_i=1^df_i(v) ￡ k\sum_{i=1}^{d}f_{i}(v)\le k for each v∈V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d_t^{k}(G)d_{t}^{\{k\}}(G). Note that d_t^{1}(G)d_{t}^{\{1\}}(G) is the classic total domatic number d _t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for d_t^{k}(G)d_{t}^{\{k\}}(G). Many of the known bounds of d _t (G) are immediate consequences of our results.     

The Independence Number of Graphs with a Forbidden Cycle and Ramsey Numbers

Let k 5 be a fixed integer and let m = (k – 1)/2. It is shown that the independence number of a C _k-free graph is at least c ₁[ d(v)^{1/(m – 1)}]^{(m – 1)/m} and that, for odd k, the Ramsey number r(C _k, K _n) is at most c ₂(n ^{m + 1}/log n)^1/m , where c ₁ = c ₁(m) > 0 and c ₂ = c ₂(m) > 0.     

On the equitable k *-laceability of hypercubes

Let G be a finite undirected bipartite graph. Let u, v be two vertices of G from different partite sets. A collection of k internal vertex disjoint paths joining u to v is referred as a k-container C _k (u,v). A k-container is a k ^*-container if it spans all vertices of G. We define G to be a k ^*-laceable graph if there is a k ^*-container joining any two vertices from different partite sets. A k ^*-container C _k ^*(u,v)={P ₁,…,P _k } is equitable if ||V(P _i )|−|V(P _j )||≤2 for all 1≤i,j≤k. A graph is equitably k ^*-laceable if there is an equitable k ^*-container joining any two vertices in different partite sets. Let Q _n be the n-dimensional hypercube. In this paper, we prove that the hypercube Q _n is equitably k ^*-laceable for all k≤n−4 and n≥5. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. The work of H.-M. Huang was supported in part by the National Science Council of the Republic of China under NSC94-2115-M008-013.     

An Econometric Model of Network Formation With Degree Heterogeneity

I introduce a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics (homophily) as well as unrestricted agent‐level heterogeneity in link surplus (degree heterogeneity). Like in fixed effects panel data analyses, the joint distribution of observed and unobserved agent‐level characteristics is left unrestricted. Two estimators for the (common) homophily parameter, β₀, are developed and their properties studied under an asymptotic sequence involving a single network growing large. The first, tetrad logit (TL), estimator conditions on a sufficient statistic for the degree heterogeneity. The second, joint maximum likelihood (JML), estimator treats the degree heterogeneity {A_i0}_{i = 1}^N as additional (incidental) parameters to be estimated. The TL estimate is consistent under both sparse and dense graph sequences, whereas consistency of the JML estimate is shown only under dense graph sequences.     

Edge-colouring of joins of regular graphs, I

We prove that the edges of every even graph G=G ₁+G ₂ that is the join of two regular graphs G _i =(V _i ,E _i ) can be coloured with Δ(G) colours, whenever Δ(G)=Δ(G ₂)+|V ₁|. The proof of this result yields a combinatorial algorithm to optimally colour the edges of this type of graphs.     

Strongly chordal and chordal bipartite graphs are sandwich monotone

A graph class is sandwich monotone if, for every pair of its graphs G ₁=(V,E ₁) and G ₂=(V,E ₂) with E ₁⊂E ₂, there is an ordering e ₁,…,e _k of the edges in E ₂∖E ₁ such that G=(V,E ₁∪{e ₁,…,e _i }) belongs to the class for every i between 1 and k. In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono (Czechoslov. Math. J. 46:577–583, 1997). So far, very few classes have been proved to be sandwich monotone, and the most famous of these are chordal graphs. Sandwich monotonicity of a graph class implies that minimal completions of arbitrary graphs into that class can be recognized and computed in polynomial time. For minimal completions into strongly chordal or chordal bipartite graphs no polynomial-time algorithm has been known. With our results such algorithms follow for both classes. In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently.     

The subdivision-constrained routing requests problem

We are given a digraph D=(V,A;w), a length (delay) function w:A→R ⁺, a positive integer d and a set $\mathcal{P}=\{(s_{i},t_{i};B_{i}) | i=1,2,\ldots,k\}$ of k requests, where s _i ∈V is called as the ith source node, t _i ∈V is called the ith sink node and B _i is called as the ith length constraint. For a given positive integer d, the subdivision-constrained routing requests problem (SCRR, for short) is to find a directed subgraph D′=(V′,A′) of D, satisfying the two constraints: (1) Each request (s _i ,t _i ;B _i ) has a path P _i from s _i to t _i in D′ with length $w(P_{i})=\sum_{e\in P_{i}} w(e)$ no more than B _i ; (2) Insert some nodes uniformly on each arc e∈A′ to ensure that each new arc has length no more than d. The objective is to minimize the total number of the nodes inserted on the arcs in A′. We obtain the following three main results: (1) The SCRR problem is at least as hard as the set cover problem even if each request has the same source s, i.e., s _i =s for each i=1,2,…,k; (2) For each request (s,t;B), we design a dynamic programming algorithm to find a path from s to t with length no more than B such that the number of the nodes inserted on such a path is minimized, and as a corollary, we present a k-approximation algorithm to solve the SCRR problem for any k requests; (3) We finally present an optimal algorithm for the case where $\mathcal{P}$ contains all possible requests (s _i ,t _i ) in V×V and B _i is equal to the length of the shortest path in D from s _i to t _i . To the best of our knowledge, this is the first time that the dynamic programming algorithm within polynomial time in (2) is designed for a weighted optimization problem while previous optimal algorithms run in pseudo-polynomial time.     

Edgeworth Expansions for Semiparametric Averaged Derivatives

A valid Edgeworth expansion is established for the limit distribution of density‐weighted semiparametric averaged derivative estimates of single index models. The leading term that corrects the normal limit varies in magnitude, depending on the choice of bandwidth and kernel order. In general this term has order larger than the n^−1/2 that prevails in standard parametric problems, but we find circumstances in which it is O(n^−1/2), thereby extending the achievement of an n^−1/2 Berry‐Esseen bound in Robinson (1995a). A valid empirical Edgeworth expansion is also established. We also provide theoretical and empirical Edgeworth expansions for a studentized statistic, where some correction terms are different from those for the unstudentized case. We report a Monte Carlo study of finite sample performance.     

A Fractional Dickey–Fuller Test for Unit Roots

This paper presents a new test for fractionally integrated (FI) processes. In particular, we propose a testing procedure in the time domain that extends the well–known Dickey–Fuller approach, originally designed for the I(1) versus I(0) case, to the more general setup of FI(d₀) versus FI(d₁), with d₁<d₀. When d₀=1, the proposed test statistics are based on the OLS estimator, or its t–ratio, of the coefficient on Δ^d1y_t−1 in a regression of Δy_t on Δ^d1y_t−1 and, possibly, some lags of Δy_t. When d₁ is not taken to be known a priori, a pre–estimation of d₁ is needed to implement the test. We show that the choice of any T^1/2–consistent estimator of d₁∈[0 ,1) suffices to make the test feasible, while achieving asymptotic normality. Monte–Carlo simulations support the analytical results derived in the paper and show that proposed tests fare very well, both in terms of power and size, when compared with others available in the literature. The paper ends with two empirical applications.     

Restricted domination parameters in graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ _m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r _k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r _k (G,γ _m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r _k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r _k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

The Polymatroid Steiner Problems

The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S⊂eqV in a weighted graph G = (V,E,c), c:E→ R⁺. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P(V) is defined on V and the Steiner tree is required to span at least one base of P (in particular, there may be a single base S⊂eqV). This formulation is motivated by the following application in sensor networks – given a set of sensors S = {s₁,…,s_k}, each sensor s_i can choose to monitor only a single target from a subset of targets X_i, find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X₁ ∪ … ∪ X_k. The Polymatroid Steiner Problem generalizes many known Steiner tree problem formulations including the group and covering Steiner tree problems. We show that this problem can be solved with the polylogarithmic approximation ratio by a generalization of the combinatorial algorithm of Chekuri et al. (2002).We also define the Polymatroid directed Steiner problem which asks for a minimum cost arborescence connecting a given root to a base of a polymatroid P defined on the terminal set S. We show that this problem can be approximately solved by algorithms generalizing methods of Chekuri et al. (2002).A preliminary version of this paper appeared in ISAAC 2004     

Generalized perfect domination in graphs

Let k be a positive integer and G=(V,E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G, if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γ _kp (G). In this paper, we give characterizations of graphs for which γ _kp (G)=γ(G)+k?2 and prove that the perfect k-domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs. Also, by using dynamic programming techniques, we obtain an algorithm to determine the perfect k-domination number of trees.     

Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree

Given a simple, undirected graph G=(V,E) and a weight function w:E→ℤ⁺, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1,k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1+1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2−1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1,k} within a ratio of 3/2 for k=2 (note that this matches the inapproximability bound above), and (2−2/(k+1)) for any k≥3, respectively, in polynomial time.