On the stability of the risk hull method for projection estimators

We consider in this paper the regularization by projection of a linear inverse problem Y=Af+εξ$Y=\mathit{Af}+\epsilon \xi$ where ξ$\xi$ denotes a Gaussian white noise, A   a compact operator and ε>0$\epsilon >0$ a noise level. Compared to the standard unbiased risk estimation (URE) method, the risk hull minimization (RHM) procedure presents a very interesting numerical behavior. However, the regularization in the singular value decomposition setting requires the knowledge of the eigenvalues of A$A$. Here, we deal with noisy eigenvalues: only observations on this sequence are available. We study the efficiency of the RHM method in this situation. More generally, we shed light on some properties usually related to the regularization with a noisy operator.     

Skew Dyck paths,area, and superdiagonal bargraphs

Skew Dyck paths are a generalization of ordinary Dyck paths, defined as paths using up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, starting and ending on the x-axis, never going below it, and so that up and left steps never overlap. In this paper we study the class of these paths according to their area, extending several results holding for Dyck paths. Then we study the class of superdiagonal bargraphs, which can be naturally defined starting from skew Dyck paths.     

Simultaneous estimation of Poisson means of the selected subset

Let _π1,_π2,…,_πp${\pi }_1,{\pi }_2,\dots ,{\pi }_p$ be p   independent Poisson populations with means _λ1,…,_λp${\lambda }_1,\dots ,{\lambda }_p$, respectively. Let {X₁,…,X_p} denote the set of observations, where X_i is from _πi${\pi }_i$. Suppose a subset of populations is selected using Gupta and Huang's (1975) selection rule which selects _πi${\pi }_i$ if and only if _Xi+1?_cX(1)$X_i+1?{\mathit{cX}}_{(1)}$, where X₍₁₎=max{X₁,…,X_p}, and 0<c<1$0<c<1$. In this paper, the simultaneous estimation of the Poisson means associated with the selected populations is considered for the k-normalized squared error loss function. It is shown that the natural estimator is positively biased. Also, a class of estimators that are better than the natural estimator is obtained by solving certain difference inequalities over the sample space. A class of estimators which dominate the UMVUE is also obtained. Monte carlo simulations are used to assess the percentage improvements and an application to a real-life example is also discussed.     

Exact D-optimal designs for a linear log contrast model with mixture experiment for three and four ingredients

This study investigates the exact D-optimal designs of the linear log contrast model using the mixture experiment suggested by Aitchison and Bacon-Shone (1984) and the design space restricted by Lim (1987) and Chan (1988). Results show that for three ingredients, there are six extreme points that can be divided into two non-intersect sets S₁ and S₂. An exact N-point D  -optimal design for N=3p+q,p≥1,1≤q≤2$N=3p+q,p\ge 1,1\le q\le 2$ arranges equal weight n/N,0≤n≤p$n/N,0\le n\le p$ at the points of S₁ (S₂) and puts the remaining weight (N−3n)/N$(N-3n)/N$ on the points of S₂ (S₁) as evenly as possible. For four ingredients and N=6p+q,p≥1,1≤q≤5$N=6p+q,p\ge 1,1\le q\le 5$, an exact N-point design that distributes the weights as evenly as possible among the six supports of the approximate D-optimal design is exact D-optimal.     

Optimal and efficient semi-Latin squares

An (n×n)/k$(n\times n)/k$semi-Latin square is an n×n square array in which nk distinct symbols (representing treatments) are placed in such a way that there are exactly k   symbols in each cell (row–column intersection) and each symbol occurs once in each row and once in each column. Semi-Latin squares form a class of row–column designs generalising Latin squares, and have applications in areas including the design of agricultural experiments, consumer testing, and via their duals, human–machine interaction. In the present paper, new theoretical and computational methods are developed to determine optimal or efficient (n×n)/k$(n\times n)/k$ semi-Latin squares for values of n and k for which such semi-Latin squares were previously unknown. The concept of subsquares of uniform semi-Latin squares is studied, new applications of the DESIGN package for GAP are developed, and exact algebraic computational techniques for comparing efficiency measures of binary equireplicate block designs are described. Applications include the complete enumeration of the (4×4)/k$(4\times 4)/k$ semi-Latin squares for k=2,…,10$k=2,\dots ,10$, and the determination of those that are A-, D-, and E-optimal, the construction of efficient (6×6)/k$(6\times 6)/k$ semi-Latin squares for k=4,5,6$k=4,5,6$, and counterexamples to a long-standing conjecture of R.A. Bailey and to a similar conjecture of D. Bedford and R.M. Whitaker.     

Mantel-Haenszel estimators for irregular sparse K2×J tables

This paper deals with sparse K2×J(J>2)$K2\times J(J>2)$ tables. Projection-method Mantel–Haenszel (MH) estimators of the common odds ratios have been proposed for K2×J$K2\times J$ tables, which include Greenland's generalized MH estimator as a special case. The method projects log-transformed MH estimators for all K2×2$K2\times 2$ subtables, which were called naive MH estimators, onto a linear space spanned by log odds ratios. However, for sparse tables it is often the case that naive MH estimators are unable to be computed. In this paper we introduce alternative naive MH estimators using a graph that represents K2×J$K2\times J$ tables, and apply the projection to these alternative estimators. The idea leads to infinitely many reasonable estimators and we propose a method to choose the optimal one by solving a quadratic optimization problem induced by the graph, where some graph-theoretic arguments play important roles to simplify the optimization problem. An illustration is given using data from a case–control study. A simulation study is also conducted, which indicates that the MH estimator tends to have a smaller mean squared error than the MH estimator previously suggested and the conditional maximum likelihood estimator for sparse tables.     

Asymptotic results in canonical discriminant analysis when the dimension is large compared to the sample size

This paper deals with the distributions of test statistics for the number of useful discriminant functions and the characteristic roots in canonical discriminant analysis. These asymptotic distributions have been extensively studied when the number p   of variables is fixed, the number q+1$q+1$ of groups is fixed, and the sample size N tends to infinity. However, these approximations become increasingly inaccurate as the value of p increases for a fixed value of N. On the other hand, we encounter to analyze high-dimensional data such that p is large compared to n. The purpose of the present paper is to derive asymptotic distributions of these statistics in a high-dimensional framework such that q   is fixed, p→∞$p\to \infty$, m=n-p+q→∞$m=n-p+q\to \infty$, and p/n→c∈(0,1)$p/n\to c\in (0,1)$, where n=N-q-1$n=N-q-1$. Numerical simulation revealed that our new asymptotic approximations are more accurate than the classical asymptotic approximations in a considerably wide range of (n,p,q)$(n,p,q)$.     

Skew Dyck paths

In this paper we study the class S$\mathbb{S}$ of skew Dyck paths, i.e. of those lattice paths that are in the first quadrant, begin at the origin, end on the x-axis, consist of up steps  U=(1,1)$U=(1,1)$, down steps  D=(1,-1)$D=(1,-1)$, and left steps  L=(−1,-1)$L=(-1,-1)$, and such that up steps never overlap with left steps. In particular, we show that these paths are equinumerous with several other combinatorial objects, we describe some involutions on this class, and finally we consider several statistics on S$\mathbb{S}$.     

Results for two-level fractional factorial designs of resolution IV or more

The main theorem of this paper shows that foldover designs are the only (regular or nonregular) two-level factorial designs of resolution IV (strength 3) or more for n   runs and n/3?m?n/2$n/3?m?n/2$ factors. This theorem is a generalization of a coding theory result of Davydov and Tombak [1990. Quasiperfect linear binary codes with distance 4 and complete caps in projective geometry. Problems Inform. Transmission 25, 265–275] which, under translation, effectively states that foldover (or even) designs are the only regular two-level factorial designs of resolution IV or more for n   runs and 5n/16?m?n/2$5n/16?m?n/2$ factors. This paper also contains other theorems including an alternative proof of Davydov and Tombak's result.     

Optimal block designs for three treatments when observations are correlated

E$E$-optimal designs for comparing three treatments in blocks of size three are identified, where intrablock observations are correlated according to a first order autoregressive error process with parameter ρ∈(0,1)$\rho \in (0,1)$. For number of blocks b   of the form b=3n+1$b=3n+1$, there are two distinct optimal designs depending on the value of ρ$\rho$, with the best design being unequally replicated for large ρ$\rho$. For other values of b$b$, binary, equireplicate designs with specified within-block assignment patterns are best. In many cases, the stronger majorization optimality is established.     

Tools for constructing optimal two-level factorial designs for a linear model containing main effects and one two-factor interaction

In Hedayat and Pesotan [1992, Two-level factorial designs for main effects and selected two-factor interactions. Statist. Sinica 2, 453–464.] the concepts of a g(n,e)$g(n,e)$-design and a g(n,e)$g(n,e)$-matrix are introduced to study designs of n$n$ factor two-level experiments which can unbiasedly estimate the mean, the n$n$ main effects and e$e$ specified two-factor interactions appearing in an orthogonal polynomial model and it is observed that the construction of a g-design is equivalent to the construction of a g  -matrix. This paper deals with the construction of D-optimal g(n,1)$g(n,1)$-matrices. A standard form for a g(n,1)$g(n,1)$-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)$g(n,1)$-matrix in the class of all g(n,1)$g(n,1)$-matrices are obtained and an approach to construct D-optimal g(n,1)$g(n,1)$-matrices is given for 2?n?8$2?n?8$. For two specific subclasses, namely a certain class of g(n,1)$g(n,1)$-matrices within the class of g(n,1)$g(n,1)$-matrices of index one and the class C(H)$C(H)$ of g(8t+2,1)$g(8t+2,1)$-matrices constructed from a normalized Hadamard matrix H   of order 8t+4(t?1)$8t+4(t?1)$ two techniques for the construction of the restricted D-optimal matrices are given.     

Classification of orthogonal arrays by integer programming

The problem of classifying all isomorphism classes of OA(N,k,s,t)$\mathrm{OA}(N,k,s,t)$'s is shown to be equivalent to finding all isomorphism classes of non-negative integer solutions to a system of linear equations under the symmetry group of the system of equations. A branch-and-cut algorithm developed by Margot [2002. Pruning by isomorphism in branch-and-cut. Math. Programming Ser. A 94, 71–90; 2003a. Exploiting orbits in symmetric ILP. Math. Programming Ser. B 98, 3–21; 2003b. Small covering designs by branch-and-cut. Math. Programming Ser. B 94, 207–220; 2007. Symmetric ILP: coloring and small integers. Discrete Optim., 4, 40–62] for solving integer programming problems with large symmetry groups is used to find all non-isomorphic OA(24,7,2,2)$\mathrm{OA}(24,7,2,2)$'s, OA(24,k,2,3)$\mathrm{OA}(24,k,2,3)$'s for 6?k?11$6?k?11$, OA(32,k,2,3)$\mathrm{OA}(32,k,2,3)$'s for 6?k?11$6?k?11$, OA(40,k,2,3)$\mathrm{OA}(40,k,2,3)$'s for 6?k?10$6?k?10$, OA(48,k,2,3)$\mathrm{OA}(48,k,2,3)$'s for 6?k?8$6?k?8$, OA(56,k,2,3)$\mathrm{OA}(56,k,2,3)$'s, OA(80,k,2,4)$\mathrm{OA}(80,k,2,4)$'s, OA(112,k,2,4)$\mathrm{OA}(112,k,2,4)$'s, for k=6,7$k=6,7$, OA(64,k,2,4)$\mathrm{OA}(64,k,2,4)$'s, OA(96,k,2,4)$\mathrm{OA}(96,k,2,4)$'s for k=7,8$k=7,8$, and OA(144,k,2,4)$\mathrm{OA}(144,k,2,4)$'s for k=8,9$k=8,9$. Further applications to classifying covering arrays with the minimum number of runs and packing arrays with the maximum number of runs are presented.     

A counterexample to Beder's conjectures about Hadamard matrices

In this note we provide a counterexample which resolves conjectures about Hadamard matrices made in this journal. Beder [1998. Conjectures about Hadamard matrices. Journal of Statistical Planning and Inference 72, 7–14] conjectured that if H$\mathit{H}$ is a maximal m×n$m\times n$ row-Hadamard matrix then m is a multiple of 4; and that if n   is a power of 2 then every row-Hadamard matrix can be extended to a Hadamard matrix. Using binary integer programming we obtain a maximal 13×32$13\times 32$ row-Hadamard matrix, which disproves both conjectures. Additionally for n being a multiple of 4 up to 64, we tabulate values of m   for which we have found a maximal row-Hadamard matrix. Based on the tabulated results we conjecture that a m×n$m\times n$ row-Hadamard matrix with m?n-7$m?n-7$ can be extended to a Hadamard matrix.     

A new estimation method for Weibull-type tails based on the mean excess function

Studying the right tail of a distribution, one can classify the distributions into three classes based on the extreme value index γ$\gamma$. The class γ>0$\gamma >0$ corresponds to Pareto-type or heavy tailed distributions, while γ<0$\gamma <0$ indicates that the underlying distribution has a finite endpoint. The Weibull-type distributions form an important subgroup within the Gumbel class with γ=0$\gamma =0$. The tail behaviour can then be specified using the Weibull tail index. Classical estimators of this index show severe bias. In this paper we present a new estimation approach based on the mean excess function, which exhibits improved bias and mean squared error. The asserted properties are supported by simulation experiments and asymptotic results. Illustrations with real life data sets are provided.