A note on a consistent estimator of a mixing distribution function

Summary In this note, we will study a consistent estimator of a mixing distribution function (mixing d.f.). The estimator discussed in this note is that of Choi and Bulgren [4]. Since there is some doubt about the way of proving Lemma in [4] which is used for showing the consistency of the estimator in [2], [3] and [4], we will give different lemmas. We will show that their result (which is still true by using our lemmas) holds under a weaker assumption than theirs. The existence of the estimator is not discussed in [4]. So, we will give conditions under which the existence is guaranteed. The research was supported in part by Scientific Research Fund from the Ministry of Education of Japan. No. 564076.     

A heuristic approach for deriving the priority vector in AHP

This paper proposes an approach for deriving the priority vector from an inconsistent pair-wise comparison matrix through the nearest consistent matrix and experts judgments, which enables balancing the consistency and experts judgments. The developed algorithm for achieving a nearest consistent matrix is based on a logarithmic transformation of the pair-wise comparison matrix, and follows an iterative feedback process that identifies an acceptable level of consistency while complying with experts preferences. Three numerical examples are examined to illustrate applications and advantages of the developed approach.     

A cosine maximization method for the priority vector derivation in AHP

The derivation of a priority vector from a pair-wise comparison matrix (PCM) is an important issue in the Analytic Hierarchy Process (AHP). The existing methods for the priority vector derivation from PCM include eigenvector method (EV), weighted least squares method (WLS), additive normalization method (AN), logarithmic least squares method (LLS), etc. The derived priority vector should be as similar to each column vector of the PCM as possible if a pair-wise comparison matrix (PCM) is not perfectly consistent. Therefore, a cosine maximization method (CM) based on similarity measure is proposed, which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a PCM. An optimization model for the CM is proposed to derive the reliable priority vector. Using three numerical examples, the CM is compared with the other prioritization methods based on two performance evaluation criteria: Euclidean distance and minimum violation. The results show that the CM is flexible and efficient.     

A distance-metric methodology for the derivation of weights from a pairwise comparison matrix

A variety of approaches exist for the determination of a weighting scheme from a pairwise comparison matrix describing a scale-relation between objectives or alternatives. The most common context for such an algorithm is that of the analytic hierarchy process (AHP), although uses in other areas of the field of multicriteria decision making (MCDM) can also be found. Typically, the eigenvalue method is the standard method employed in the AHP to determine weights, as in the ExpertChoice software. However, another class of techniques are the distance-metric-based approaches, which are frequently proposed as alternatives to the eigenvalue method. This paper evaluates such distance-metric-based approaches comparing their effectiveness, using the eigenvalue method as a benchmark. A common framework is introduced to establish an efficient frontier for method comparison.     

A note on consistency improvements of AHP paired comparison data

The Analytic Hierarchy Process (AHP) is a popular multicriteria decision-making approach but the ease of AHP paired comparison data collection entails the problem that consistency restrictions have to be fulfilled for the data evaluation task. Quite a lot of consistency improvement techniques are available, however, this note explains why consistency adjustments are not necessarily helpful for computing acceptable weights for the determination of the underlying overall objective function.     

A statistical approach to measure the consistency level of the pairwise comparison matrix

The consistency of the pairwise comparison matrix (PCM) has been extensively studied in the Analytic Hierarchy Process. Most of the existing approaches for the consistency test of the PCM are based on the consistency ratio threshold (0.1) introduced by Saaty (1977). To accurately measure the consistency level of the PCM, a statistical approach based on the significance level is proposed in this paper by combining the hypothesis test and the random consistency index. The proposed statistical approach is applied to an education evaluation problem in order to demonstrate the rationality and reliability of it.     

A note on the structure of pairwise balanced designs

We show that if a pairwise balanced design with λ = 1 has ν objects and block sizes between 2 and ν − 2, and it is not a symmetric balanced design, then each of its largest blocks has a nonempty intersection with at least ν other blocks     

On pairwise comparison matrices that can be made consistent by the modification of a few elements

Pairwise comparison matrices are often used in Multi-attribute Decision Making for weighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications.     

A note on the pseudo-conservation law for a multi-queue with local priority

The pseudo-conservation law for a multi-queue system with non-preemptive priority (local priority) has recently been presented in Fournier and Rosberg [9]. This note points out that the result has independently been derived in Shimogawa and Takahashi [17]. The derivation of a mean waiting time approximation based on the pseudo-conservation law is also discussed.     

A note on a new matrix decomposition

It is shown that if det A=±1, then A=±^q_i=1B_i, where B_i² = I. This decomposition is used to find the Jacobian of the linear matrix transformation: Y=AX.     

A note on the median of a distribution

Summary LetF be a distribution function over the real line. DefineR _p(y)=∫|x−y|^pdF(x) forp≧1. Forp>1 there is a unique minimizer ofR _p(·), sayγ _p. Under mild conditions onF it is shown that exists and that the limit is a median. Thus, one could use this limit as a definition of a unique median. Also it is shown that whereR is the right extremity ofF andL is the left extremity ofF provided that −∞<L≦R<∞. A similar result is available ifL=−∞,R=∞, yetF has symmetric tails.     

两两NQD列大数定律的一个注记

利用两两NQD列三级数定理的思想和Chebyshev不等式,研究了两两NQD列在一类广泛条件下的弱大数定律和一类加强条件下的强大数定律,得到了与独立情形一致的结果,还特别讨论了同分布情形,推广了相关文献的结果．     

A simple formula to find the closest consistent matrix to a reciprocal matrix

Achieving consistency in pair-wise comparisons between decision elements given by experts or stakeholders is of paramount importance in decision-making based on the AHP methodology. Several alternatives to improve consistency have been proposed in the literature. The linearization method (Benítez et al., 2011 [10]), derives a consistent matrix based on an original matrix of comparisons through a suitable orthogonal projection expressed in terms of a Fourier-like expansion. We propose a formula that provides in a very simple manner the consistent matrix closest to a reciprocal (inconsistent) matrix. In addition, this formula is computationally efficient since it only uses sums to perform the calculations. A corollary of the main result shows that the normalized vector of the vector, whose components are the geometric means of the rows of a comparison matrix, gives the priority vector only for consistent matrices.     

A note on the comparison of Co-semigroups

If S(.) and T(.) are two C_o-semigroups of linear operators on a Banach space X, we derive a characterization of their generators so that ||S(t)−T(t)||=O(t) as t→O⁺.     

A note on unimodular congruence of the Laplacian matrix of a graph

Let G be a simple connected graph and L(G) be its Laplacian matrix. In this note, we prove that L(G) is congruent by a unimodular matrix to its Smith normal form if and only if G is a tree.     

A note on the resolvent of a nonnegative matrix and its applications

We prove a conjecture of Dubey et al. on the change in the resolvent of a nonnegative matrix if its entries are decreased, and discuss applications to mathematical economics.