In this paper we generalize LLL lattice basis reduction defined by Lenstra,
Lenstra, and Lov´asz. We consider OF -lattice, where OF is the ring of integers in
algebraic number field F. We can prove that basic properties of reduced basis can hold
over imaginary quadratic fields. We can reveal existence of a least positive element
over other algebraic number fields.
With respect to the sufficient statistics in the transformed exponential
family based on a continuous probability distribution, we examine a misuse that the
sufficient statistics ought to be distributed with a k-dimensional exponential family
where k is the dimension of the sufficient statistics. Under the irreducibility of the
sufficient statistics, we define two types of the transformed exponential family, i.e.,
regular and pseudo, so that the misuse is made explicit.
Ochi(1983) proposed an estimator for the autoregressive coefficient of the first-order autoregressive model (AR(1)) by using two constants for the end points of the process. Classical estimators for AR(1) , such as the least squares estimator, Burg's estimator, and Yule- Walker estimator are obtained as special cases by choice of the constants in Ochi's estimator. By writing the first-order autoregressive conditional heteroskedastic model, ARCH(1), in a form similar to that of AR(1), we extend Ochi ' s estimator to ARCH(1) models. This allows introducing analogues of the least squares estimator, Burg's estimator and Yule- Walker estimator, and we compare the relations of these with Ochi's estimator for ARCH(1) models. We then provide a simulation for AR(1) models and examine the performance of Ochi ' s estimator. Also, we simulate Ochi's estimator for ARCH(1) with different parameter values and sample sizes.
The pairwise comparisons in AHP (Analytic Hierarchy Process) are made
using a scale list that indicates the importance of one entity over another entity with
respect to a given criteria. Moreover, the pairwise comparison matrix represents the
intensities of the decision maker’s preference between individual pairs of alternatives.
The matrix is usually determined from the 1-point to 9-point scale. Various methods
for paired comparison method have been proposed, making more intuitive and highly
accurate decision making possible. However, the number of pairwise comparisons increases
as the number of criteria increases. Therefore, the burden of decision makers
would become heavier.
In this paper, we propose an algorithm for the allocation problem of the burden
and verify the algorithm by using a programming language called Haskell1, which
is specialized in the functional programming. This research contributes not only to
allocation algorithm, but also aids researchers and decision makers in applying the