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1.
In this paper we study the class 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), down steps D=(1,-1), and left steps 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. 相似文献
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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)-design and a g(n,e)-matrix are introduced to study designs of n factor two-level experiments which can unbiasedly estimate the mean, the n main effects and 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)-matrices. A standard form for a g(n,1)-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)-matrix in the class of all g(n,1)-matrices are obtained and an approach to construct D-optimal g(n,1)-matrices is given for 2?n?8. For two specific subclasses, namely a certain class of g(n,1)-matrices within the class of g(n,1)-matrices of index one and the class C(H) of g(8t+2,1)-matrices constructed from a normalized Hadamard matrix H of order 8t+4(t?1) two techniques for the construction of the restricted D-optimal matrices are given. 相似文献
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The problem of classifying all isomorphism classes of 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)'s, OA(24,k,2,3)'s for 6?k?11, OA(32,k,2,3)'s for 6?k?11, OA(40,k,2,3)'s for 6?k?10, OA(48,k,2,3)'s for 6?k?8, OA(56,k,2,3)'s, OA(80,k,2,4)'s, OA(112,k,2,4)'s, for k=6,7, OA(64,k,2,4)'s, OA(96,k,2,4)'s for k=7,8, and OA(144,k,2,4)'s for 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. 相似文献
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Super-simple cyclic designs are useful on constructing codes and designs such as superimposed codes, perfect hash families and optical orthogonal codes with index two. In this paper, we show that there exists a super-simple cyclic (v,4,λ) for 7?v?41 and all admissible λ with two definite exceptions of (v,λ)=(9,3),(13,5) and one possible exception of (v,λ)=(39,18). Some useful algorithms are explained for computer search and new designs are displayed. 相似文献
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The enumeration of binary cyclic self-orthogonal codes of length 63 is used to prove that any cyclic quasi-symmetric 2-(63,15,35) design with block intersection numbers x=3 and y=7 is isomorphic to the geometric design having as blocks the three-dimensional subspaces in PG(5,2). 相似文献
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Joseph P.S. Kung Anna de Mier Xinyu Sun Catherine Yan 《Journal of statistical planning and inference》2009
We consider paths in the plane with (1,0), (0,1), and (a,b)-steps that start at the origin, end at height n, and stay strictly to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most b/a, then the ordinary generating function for the number of such paths ending at height n is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or “umbral” generating function, in which the power zn is replaced by a power series of the form znφn(z), where φn(0)=1. Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic. We give several concrete examples, including an alternative way to solve the tennis ball problem. 相似文献
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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). For number of blocks b of the form b=3n+1, there are two distinct optimal designs depending on the value of ρ, with the best design being unequally replicated for large ρ. For other values of b, binary, equireplicate designs with specified within-block assignment patterns are best. In many cases, the stronger majorization optimality is established. 相似文献
11.
A ridge function with shape function g in the horizontal direction is a function of the form g(x)h(y,0). Along each horizontal line it has the shape g(x), multiplied by a function h(y,0) which depends on the y-value of the horizontal line. Similarly a ridge function with shape function g in the vertical direction has the form g(y)h(x,π/2). For a given shape function g it may or may not be possible to represent an arbitrary function f(x,y) as a superposition over all angles of a ridge function with shape g in each direction, where h=hf=hf,g depends on the functions f and g and also on the direction, θ:h=hf,g(·,θ). We show that if g is Gaussian centered at zero then this is always possible and we give the function hf,g for a given f(x,y). For highpass or for odd shapes g , we show it is impossible to represent an arbitrary f(x,y), i.e. in general there is no hf,g. Note that our problem is similar to tomography, where the problem is to invert the Radon transform, except that the use of the word inversion is here somewhat “inverted”: in tomography f(x,y) is unknown and we find it by inverting the projections of f ; here, f(x,y) is known, g(z) is known, and hf(·,θ)=hf,g(·,θ) is the unknown. 相似文献
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In this paper, we investigate the estimation problem of the mixture proportion λ in a nonparametric mixture model of the form λF(x)+(1-λ)G(x) using the minimum Hellinger distance approach, where F and G are two unknown distributions. We assume that data from the distributions F and G as well as from the mixture distribution λF+(1-λ)G are available. We construct a minimum Hellinger distance estimator of λ and study its asymptotic properties. The proposed estimator is chosen to minimize the Hellinger distance between a parametric mixture model and a nonparametric density estimator. We also develop a maximum likelihood estimator of λ. Theoretical properties such as the existence, strong consistency, asymptotic normality and asymptotic efficiency of the proposed estimators are investigated. Robustness properties of the proposed estimator are studied using a Monte Carlo study. Two real data examples are also analyzed. 相似文献
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An (n×n)/ksemi-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 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 semi-Latin squares for k=2,…,10, and the determination of those that are A-, D-, and E-optimal, the construction of efficient (6×6)/k semi-Latin squares for 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. 相似文献
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We consider a linear regression model with regression parameter β=(β1,…,βp) and independent and identically N(0,σ2) distributed errors. Suppose that the parameter of interest is θ=aTβ where a is a specified vector. Define the parameter τ=cTβ-t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that τ=0. We present a new frequentist 1-α confidence interval for θ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about τ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×2 factorial experiment with 20 replicates. Suppose that the parameter of interest θ is a specified simple effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θ that utilizes this prior information. 相似文献
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Consider the model where there are I independent multivariate normal treatment populations with p×1 mean vectors μi, i=1,…,I, and covariance matrix Σ. Independently the (I+1)st population corresponds to a control and it too is multivariate normal with mean vector μI+1 and covariance matrix Σ. Now consider the following two multiple testing problems. 相似文献
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We consider the problem of estimating the mean θ of an Np(θ,Ip) distribution with squared error loss ∥δ−θ∥2 and under the constraint ∥θ∥≤m, for some constant m>0. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator δmle. We obtain for fixed (m,p) sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates δmle, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p. 相似文献
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This paper discusses a new perspective in fitting spatial point process models. Specifically the spatial point process of interest is treated as a marked point process where at each observed event x a stochastic process M(x;t), 0<t<r, is defined. Each mark process M(x;t) is compared with its expected value, say F(t;θ), to produce a discrepancy measure at x, where θ is a set of unknown parameters. All individual discrepancy measures are combined to define an overall measure which will then be minimized to estimate the unknown parameters. The proposed approach can be easily applied to data with sample size commonly encountered in practice. Simulations and an application to a real data example demonstrate the efficacy of the proposed approach. 相似文献
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We consider the estimation of smooth regression functions in a class of conditionally parametric co-variate-response models. Independent and identically distributed observations are available from the distribution of (Z,X), where Z is a real-valued co-variate with some unknown distribution, and the response X conditional on Z is distributed according to the density p(·,ψ(Z)), where p(·,θ) is a one-parameter exponential family. The function ψ is a smooth monotone function. Under this formulation, the regression function E(X|Z) is monotone in the co-variate Z (and can be expressed as a one–one function of ψ); hence the term “monotone response model”. Using a penalized least squares approach that incorporates both monotonicity and smoothness, we develop a scheme for producing smooth monotone estimates of the regression function and also the function ψ across this entire class of models. Point-wise asymptotic normality of this estimator is established, with the rate of convergence depending on the smoothing parameter. This enables construction of Wald-type (point-wise) as well as pivotal confidence sets for ψ and also the regression function. The methodology is extended to the general heteroscedastic model, and its asymptotic properties are discussed. 相似文献