On the robustness of balanced fractional 2m factorial designs of resolution 2l+1 in the presence of outliers

Summary By use of the algebraic structure, we obtain a simplified expression for the outlier-insensitivity factor for balanced fractional 2^m factorial (2^m-BFF) designs of resolution 2l+1 derived from simple arrays (S-arrays), whose measure has been introduced by Ghosh and Kipnegeno (1985,J. Statist. Plann. Inference,11, 119–129). It is defined by use of the measure suggested by Box and Draper (1975,Biometrika, 62 (2), 347–352). As examples, we study the sensitivity ofA-optimal 2^m-BFF designs of resolution VII (i.e.,l=3) given by Shirakura (1976,Ann. Statist.,4, 515–531; 1977,Hiroshima Math. J.,7, 217–285). We observe that these designs are robust in the sense that they have low sensitivities. Research supported in part by Grant 59530012 (C) and 60530014 (C), Japan.     

A formula for deficiency (L-andR-tests)

In this paper an analogue of the formulas [D. M. Chibisov,Teor. Veroyatn. Primen.,30, 269–288 (1985);Izv. Akad. Nauk UzSSR,6, 23–30 (1982)] for the difference between the power of a given asymptotically efficient test and that of the most powerful test is justified for one-sample L-and R-tests, i.e., tests based on linear combinations of order statistics and linear rank statistics. This formula directly yields the Hodges-Lehmann deficiency of corresponding tests. A general theorem is stated which is applied to L-and R-tests. The explicit expressions given by this formula for L- and R-tests are also presented. The expression related to R-tests agrees with the one obtained in [W. Albers, P. J. Bickel, and W. R. Van Zwet,Ann. Statist.,4, 108–156 (1976);6, 1170–1171 (1978)]. We present here a nontechnical (heuristic) proof of these results. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-1446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

A study of redundancy of some variables in covariate discriminant analysis

The redundancy of some variables in discriminant analysis and its tests were developed by Rao (1946,Sankhyā,7, 407–414; 1948,Biometrika,35, 58–79, 1970,Essays in Probability and Statistics, (eds. R. C. Boseet al.), 587–602, Univ. of North Carolina Press, Chapel Hill), and were further studied by McKay (1977,J. Roy. Statist. Soc. Ser. B,39, 371–380) and Fujikoshi (1982,Ann. Inst. Statist. Math.,34, 523–530). These are now extended to the most general situation which includes redundancy in covariate as well as main variables in discrimination between two or more groups. The likelihood ratio test is derived in a closed form. An alternative test is also suggested.     

Block plan for a fractional 2 m factorial design derived from a 2 r factorial design

Summary For a given fractional 2 ^m factorial (2 ^m -FF) designT, the constitution of a block plan to divideT intok (2 ^r−1<k≦2 ^r ) blocks withr block factors each at two levels is proposed and investigated. The well-known three norms of the confounding matrix are used as measures for determining a “good” block plan. Some theorems concerning the constitution of a block plan are derived for a 2 ^m -FF design of odd or even resolution. Two norms which may be preferred over the other norm are slightly modified. For each value ofN assemblies with 11≦N≦26, optimum block plans fork=2 blocks with block sizes [N/2] andN−[N/2] minimizing the two norms are presented forA-optimal balanced 2⁴-FF designs of resolutionV given by Srivastava and Chopra (Technometrics,13, 257–269).     

Comparison of ‘Tail Method’ Exact Confidence Limits for the Difference of Binomial Probabilities

Suppose that Y ₁ and Y ₂ are independent and have Binomial(n ₁,p ₁) and Binomial (n ₂,p ₂) distributions respectively. Also suppose that θ=p ₁−p ₂ is the parameter of interest. We consider the problem of finding an exact confidence limit (either upper or lower) for θ. The solution to this problem is very important for statistical practice in the health and life sciences. The ‘tail method’ provides a solution to this problem. This method finds the exact confidence limit by exact inversion of a hypothesis test based on a specified test statistic. Buehler (J. Am. Stat. Assoc. 52, 482–493, 1957) described, for the first time, a finite-sample optimality property of this confidence limit. Consequently, this confidence limit is sometimes called a Buehler confidence limit. An early tail method confidence limit for θ was described by Santner and Snell (J. Am. Stat. Assoc. 75, 386–394, 1980) who used the maximum likelihood estimator of θ as the test statistic. This confidence limit is known to be very inefficient (see e.g. Cytel Software, StatXact, version 6, vol. 2, 2004). The efficiency of the confidence limit resulting from the tail method depends greatly on the test statistic on which it is based. We use the results of Kabaila (Stat. Probab. Lett. 52, 145–154, 2001) and Kabaila and Lloyd (Aust. New Zealand J. Stat. 46, 463–469, 2004, J. Stat. Plan. Inference 136, 3145–3155, 2006) to provide a detailed explanation for the dependence of this efficiency on the test statistic. We consider test statistics that are estimators, Z-statistics and approximate upper confidence limits. This explanation is used to find the situations in which the tail method exact confidence limits based on test statistics that are estimators or Z-statistics are least efficient.     

Likelihood ratio tests for comparingk populations —The two-parameter nonregular models

Summary The null and nonnull distributions of the likelihood ratio statistics for testing the homogeneity ofk given populations, each associated with a nonregular density depending on two truncation parameters, are investigated. This generalizes to the two-parameter case the work of Hogg (1956,Ann. Math. Statist.,27, 529–532), Barr (1966,J. Amer. Statist. Assoc.,61, 856–864) and Khatri and Jaiswal (1969,Aust. J. Statist.,11, 79–84; 1969, 1971,Ann. Inst. Statist. Math.,21, 127–136;23, 199–210).     

The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold

In this paper, we study a sharp lower bound of the first eigenvalue of the sublaplacian on a 3-dimensional pseudohermitian manifold with the CR Paneitz operator positive. In general cases, S.-Y. Li and H.-S. Luk ({Proc. Am. Math. Soc.} 132(3), 789–798) (2004) proved the lower bound under a condition on a covariant derivative of the torsion as well as the Ricci curvature and the torsion. We show that if the CR Paneitz operator is positive, then the sharp lower bound is obtained under one simpler condition on only the Ricci curvature and the torsion itself; which is similar to the condition given in high-dimensional cases in ({Commun. Partial Differential Equations}, 10(2/3), 191–217) (1985). We also show examples where our theorem applies, but Theorem 1.2 in ({Proc. Am. Math. Soc.} 132(3), 789–798) (2004) does not. Mathematics Subject Classifications (2000). Primary 32V05, 32V20, Secondary 53C56.     

An alternating iterative method and its application in statistical inference

This paper studies non-convex programming problems. It is known that, in statistical inference, many constrained estimation problems may be expressed as convex programming problems. However, in many practical problems, the objective functions are not convex. In this paper, we give a definition of a semi-convex objective function and discuss the corresponding non-convex programming problems. A two-step iterative algorithm called the alternating iterative method is proposed for finding solutions for such problems. The method is illustrated by three examples in constrained estimation problems given in Sasabuchi et al. （Biometrika, 72, 465472 （1983））, Shi N. Z. （J. Multivariate Anal., 50, 282-293 （1994）） and El Barmi H. and Dykstra R. （Ann. Statist., 26, 1878 1893 （1998））.     

The geometry of compact homogeneous spaces with two isotropy summands

We give a complete list of all homogeneous spaces M = G/H where G is a simple compact Lie group, H a connected, closed subgroup, and G/H is simply connected, for which the isotropy representation of H on T _p M decomposes into exactly two irreducible summands. For each homogeneous space, we determine whether it admits a G-invariant Einstein metric. When there is an intermediate subgroup H <  K <  G, we classify all the G-invariant Einstein metrics. This is an extension of the classification of isotropy irreducible spaces, given independently by Manturov (Dokl. Akad. Nauk SSSR 141, (1961), 792–795 1034–1037, Tr. Semin. Vector Tensor Anal. 13, (1966), 68–145) and J Wolf (Acta Math. 120, (1968), 59–148 152, (1984) 141–142).      

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Maximum principle for fully nonlinear equations via the iterated comparison function method

We present various versions of generalized Aleksandrov–Bakelman–Pucci (ABP) maximum principle for L ^p -viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the “iterated” comparison function method, which was introduced in our previous paper (Koike and Święch in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) for fully nonlinear elliptic equations. Our results extend those of (Koike and Święch in Nonlin. Diff. Eq. Appl. 11, 491–509, 2004) and (Fok in Comm. Partial Diff. Eq. 23(5–6), 967–983) in the elliptic case, and of (Crandall et al. in Indiana Univ. Math. J. 47(4), 1293–1326, 1998; Comm. Partial Diff. Eq. 25, 1997–2053, 2000; Wang in Comm. Pure Appl. Math. 45, 27–76, 1992) and (Crandall and Święch in Lecture Notes in Pure and Applied Mathematics, vol. 234. Dekker, New York, 2003) in the parabolic case. Dedicated to Hitoshi Ishii on the occasion of his 60th birthday.     

Optimal partially balanced fractional 2 m 1+m 2 factorial designs of resolution IV

Summary This paper investigates some partially balanced fractional 2 ^m ₁+m ₂ factorial designs of resolution IV derived from partially balanced arrays, which permit estimation of the general mean, all main effects, all two-factor interactions within each set of them _k factors (k=1, 2) and some linear combinations of the two-factor interactions between the sets of them _k ones. In addition, optimal designs with respect to the generalized trace criterion defined by Shirakura (1976,Ann. Statist.,4, 723–735) are presented for each pair (m ₁,m ₂) with 2≦m ₁≦m ₂ andm ₁+m ₂≦6, and for values ofN (the number of observations) in a reasonable range. Partially supported in part by Grants 56530009 (C) and 57530010 (C).     

Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function

 We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd–Omojokun class of algorithms, each step is composed of a quasi-normal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach. Received: December 14, 2000 / Accepted: August 30, 2001 Published online: September 27, 2002 Key words. nonmonotone trust-region methods – sequential quadratic programming – penalty function – global convergence – equality constraints – local convergence – large-scale optimization Mathematics Subject Classification (2000): 65K05, 90C30     

Dynamic allocation indices for restless projects and queueing admission control: a polyhedral approach

 This paper develops a polyhedral approach to the design, analysis, and computation of dynamic allocation indices for scheduling binary-action (engage/rest) Markovian stochastic projects which can change state when rested (restless bandits (RBs)), based on partial conservation laws (PCLs). This extends previous work by the author [J. Ni?o-Mora (2001): Restless bandits, partial conservation laws and indexability. Adv. Appl. Probab. 33, 76–98], where PCLs were shown to imply the optimality of index policies with a postulated structure in stochastic scheduling problems, under admissible linear objectives, and they were deployed to obtain simple sufficient conditions for the existence of Whittle's (1988) RB index (indexability), along with an adaptive-greedy index algorithm. The new contributions include: (i) we develop the polyhedral foundation of the PCL framework, based on the structural and algorithmic properties of a new polytope associated with an accessible set system -extended polymatroid}); (ii) we present new dynamic allocation indices for RBs, motivated by an admission control model, which extend Whittle's and have a significantly increased scope; (iii) we deploy PCLs to obtain both sufficient conditions for the existence of the new indices (PCL-indexability), and a new adaptive-greedy index algorithm; (iv) we interpret PCL-indexability as a form of the classic economics law of diminishing marginal returns, and characterize the index as an optimal marginal cost rate; we further solve a related optimal constrained control problem; (v) we carry out a PCL-indexability analysis of the motivating admission control model, under time-discounted and long-run average criteria; this gives, under mild conditions, a new index characterization of optimal threshold policies; and (vi) we apply the latter to present new heuristic index policies for two hard queueing control problems: admission control and routing to parallel queues; and scheduling a multiclass make-to-stock queue with lost sales, both under state-dependent holding cost rates and birth-death dynamics. Received: April 2000 / Accepted: October 2002 Published online: December 9, 2002 RID="★" ID="★" Work partly supported by the Spanish Ministry of Science and Technology (grant BEC2000-1027), NATO (Collaborative Linkage Grant PST.CLG.976568), and the Joint Spanish-US (Fulbright) Commission for Scientific and Technical Exchange (project 2000-20132) Key words. Markov decision process – restless bandits – polyhedral combinatorics – extended polymatroid – adaptive-greedy algorithm – dynamic allocation index – stochastic scheduling – threshold policy – index policy – Gittins index – Klimov index – Whittle index – control of queues – admission control – routing – make-to-stock – multiclass queue – finite buffers – conservation laws – achievable performance Mathematics Subject Classification (1991): (AMS 2000 Subject Classification): 90B36, 90B22, 90C40, 90C57, 90C08     

Local behavior of an iterative framework for generalized equations with nonisolated solutions

 An iterative framework for solving generalized equations with nonisolated solutions is presented. For generalized equations with the structure , where is a multifunction and F is single-valued, the framework covers methods that, at each step, solve subproblems of the type . The multifunction approximates F around s. Besides a condition on the quality of this approximation, two other basic assumptions are employed to show Q-superlinear or Q-quadratic convergence of the iterates to a solution. A key assumption is the upper Lipschitz-continuity of the solution set map of the perturbed generalized equation . Moreover, the solvability of the subproblems is required. Conditions that ensure these assumptions are discussed in general and by means of several applications. They include monotone mixed complementarity problems, Karush-Kuhn-Tucker systems arising from nonlinear programs, and nonlinear equations. Particular results deal with error bounds and upper Lipschitz-continuity properties for these problems. Received: November 2001 / Accepted: November 2002 Published online: December 9, 2002 Key Words. generalized equation – nonisolated solutions – Newton's method – superlinear convergence – upper Lipschitz-continuity – mixed complementarity problem – error bounds Mathematics Subject Classification (1991): 90C30, 65K05, 90C31, 90C33     

Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in \mathbbRⁿ {\mathbb{R}^n} . We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the (k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang–Xia inequality (J. Funct. Anal., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.     

Maximum stable set formulations and heuristics based on continuous optimization

 The stability number α(G) for a given graph G is the size of a maximum stable set in G. The Lovász theta number provides an upper bound on α(G) and can be computed in polynomial time as the optimal value of the Lovász semidefinite program. In this paper, we show that restricting the matrix variable in the Lovász semidefinite program to be rank-one and rank-two, respectively, yields a pair of continuous, nonlinear optimization problems each having the global optimal value α(G). We propose heuristics for obtaining large stable sets in G based on these new formulations and present computational results indicating the effectiveness of the heuristics. Received: December 13, 2000 / Accepted: September 3, 2002 Published online: December 19, 2002 RID="★" ID="★" Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009. Key Words. maximum stable set – maximum clique – minimum vertex cover – semidefinite program – semidefinite relaxation – continuous optimization heuristics – nonlinear programming Mathematics Subject Classification (2000): 90C06, 90C27, 90C30     

On the Mean Distance in Scale Free Graphs

We consider a graph, where the nodes have a pre-described degree distribution F, and where nodes are randomly connected in accordance to their degree. Based on a recent result (R. van der Hofstad, G. Hooghiemstra and P. Van Mieghem, “Random graphs with finite variance degrees,” Random Structures and Algorithms, vol. 17(5) pp. 76–105, 2005), we improve the approximation of the mean distance between two randomly chosen nodes given by M. E. J. Newman, S. H. Strogatz, and D. J. Watts, “Random graphs with arbitrary degree distribution and their application,” Physical Review. E vol. 64, 026118, pp. 1–17, 2001. Our new expression for the mean distance involves the expectation of the logarithm of the limit of a super-critical branching process. We compare simulations of the mean distance with the results of Newman et al. and with our new approach. AMS 2000 Subject Classification: 05C80, 60F05     

One-regular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer

A graph G is one-regular if its automorphism group Aut（G） acts transitively and semiregularly on the arc set. A Cayley graph Cay（Г, S） is normal if Г is a normal subgroup of the full automorphism group of Cay（Г, S）. Xu, M. Y., Xu, J. （Southeast Asian Bulletin of Math., 25, 355-363 （2001）） classified one-regular Cayley graphs of valency at most 4 on finite abelian groups. Marusic, D., Pisanski, T. （Croat. Chemica Acta, 73, 969-981 （2000）） classified cubic one-regular Cayley graphs on a dihedral group, and all of such graphs turn out to be normal. In this paper, we classify the 4-valent one-regular normal Cayley graphs G on a dihedral group whose vertex stabilizers in Aut（G） are cyclic. A classification of the same kind of graphs of valency 6 is also discussed.