Showing papers in "Siam Review in 1969"
TL;DR: In this article, the authors give liberal conditions on the steps of a "descent" method for finding extrema of a function and show that most known results are special cases.
Abstract: Liberal conditions on the steps of a “descent” method for finding extrema of a function are given; most known results are special cases.
833 citations
185 citations
180 citations
93 citations
TL;DR: In this article, a nonnegative order for a single product is placed at the beginning of each of an infinite sequence of equally spaced periods into the future, labeled 0, 1, 2,.. Demands depleting the inventory form a non-negative sequence of independent identically distributed random variables 40, 1 '2, *.
Abstract: These equations depict the following inventory situation. A nonnegative order for a single product is to be placed at the beginning of each of an infinite sequence of equally spaced periods into the future, labeled 0, 1, 2, .. Demands depleting the inventory form a nonnegative sequence of independent identically distributed random variables 40, 1, '2, * . For analytic convenience the distribution is assumed to possess a continuous positive density function 0(4) on [0, oo). An order zi placed at the beginning of period i arrives after a fixed Aperiod lag at the beginning of period i + A, and is relabeled Y(i + ), which explains (4). The pipeline vector qi at the beginning of period i consists of the inventory on hand at that point xi together with the sequence of outstanding orders to arrive in succeeding periods Y(i+ 1), Y(i+ 2)' ... , Y(i+ -1) If demand in a given period exceeds inventory on hand, two extreme cases are considered. In model I, also called the no backlogging or the lost sales case, no customer will wait, so the inventory level is conveniently thought of as being nonnegative; it is recursively generated by (5a). In model II, also called the backlogging case, all customers
88 citations
76 citations
TL;DR: In this article, special numerical integration formulas are developed which transform a differential equation into a difference equation, such that the differential equation and the corresponding difference equation are both stable or else they are both unstable.
Abstract: A differential equation is stable if the roots of the characteristic polynomial are in the interior of the left half plane. Likewise a difference equation is stable if the roots of the characteristic polynomial are in the interior of the unit circle. This paper concerns algorithms which test polynomials for these properties. Also of concern is the relationship between the two problems. In particular, special numerical integration formulas are developed which transform a differential equation into a difference equation. These formulas are such that the differential equation and the corresponding difference equation are both stable or else they are both unstable.
58 citations
TL;DR: In this article, it was shown that the maximum likelihood estimators of the location and scale parameters of the distributions have pivotal functions with distributions independent of the locations and scales of both parameters.
Abstract: It is shown that the maximum likelihood estimators (m.l.e.'s) $\hat \alpha $ and $\hat \beta $ of the location and scale parameters $\alpha $ and $\beta $ have the property that ${{\hat \beta } / \beta }$, ${{(\hat \alpha - \alpha )} / \beta }$ and ${{(\hat \alpha - \alpha )} / {\hat \beta }}$ are pivotal functions with distributions independent of both parameters. Similar results are seen to hold for other types of distributions by making suitable transformations.
TL;DR: Theorem 1 and 2 of the Neyman-Pearson lemma as mentioned in this paper provide sufficiency criteria for constrained extrema of nonlinear functionals with continuous or discrete variation, and the use of ratio contours is described as a technique for producing a trial solution to be tested against Theorem 1 or 2.
Abstract: Versions of the Neyman–Pearson lemma are given (Theorems 1 and 2) which provide sufficiency criteria for constrained extrema of nonlinear functionals with continuous or discrete variation. The use of ratio contours is described as a technique for producing a trial solution to be tested against Theorem 1 or 2. Examples of applications include improvements over previously published methods of solutions of certain problems. Relationship to previous versions of the lemma and to the method of Lagrange multipliers is discussed.
TL;DR: In this article, the authors presented two characterizations of the lack of uniqueness of G: (i) in terms of a relation among elements of G and (ii) in any other generalized inverse of A. The first characterization provides results concerning the rank and symmetry of G; the latter characterization provides an alternate form for the general solution to the consistent equations.
Abstract: Any matrix G which satisfies $AGA = A$ is a generalized inverse of A. Unless A is square and nonsingular, G is not unique. This paper presents two characterizations of the lack of uniqueness of G: (i) in terms of a relation among elements of G and (ii) in terms of any other generalized inverse of A. The first characterization provides results concerning the rank and symmetry of G; the latter characterization provides an alternate form for the general solution to the consistent equations ${\bf Ax} = {\bf y}$, i.e., as G varies over all matrices satisfying $AGA = A$ then ${\bf x} = {\bf Gy}$ generates all solutions to ${\bf Ax} = {\bf y}$ provided ${\bf y}
e 0$.