The behavioral assumptions which economists call âperfect competition,â imply that decentralized decision making under certain conditions leads to a social optimum. This is a central result of classical economic theory. The author discusses the result, and shows that it cannot be expected to hold when uncertainty is introduced. The point is illustrated by a simple example from business finance.
On behalf of the American Association of Neurological Surgeons/Congress of Neurological Surgeons Joint Section on Disorders of the Spine and Peripheral Nerves, it is my great privilege to introduce these Guidelines for the Management of Acute Cervical Spine and Spinal Cord Injuries. These guidelines represent the initial installment of a more comprehensive guidelines initiative from the Joint Section on behalf of all practicing neurosurgeons and their patients. The Section is grateful to the small working group who devoted considerable time and effort to the generation of this outstanding document. We would like to formally recognize the Joint Section on Trauma for their important collaboration on this project. The Section would also like to acknowledge and thank the parent organizations, the American Association of Neurological Surgeons and the Congress of Neurological Surgeons, for their guidance of and support for this project, most notably through the efforts of the American Association of Neurological Surgeons/ Congress of Neurological Surgeons Guidelines Committee. The Section is also deeply indebted to Michael Apuzzo and the staff of Neurosurgery for their advice and editorial assistance in preparing this document for publication. The application of Neurosurgeryâ s rigorous peer-reviewed editorial process has clearly enhanced the quality, balance, and stature of this document. Perhaps most importantly, Neurosurgery has provided an extraordinary vehicle for the widespread dissemination and ultimate incorporation of these guidelines to improve the care and enhance the outcomes of patients with traumatic cervical spine and spinal cord injuries. One of the truly important functions of organized neurosurgery is the generation of evidenced-based clinical practice guidelines. Properly developed, such guidelines can answer important questions, resolve uncertainty, identify areas of deficient knowledge and opportunities for future scientific investigation, standardize treatment, and improve the quality of care and the outcomes for patients. The now widely disseminated head trauma guidelines, for example, have clearly made a difference in the outcomes of patients with severe head injury. Guidelines development is a highly structured process with rigorous methodological criteria and exacting standards. It is a time-, labor-, and resource-intensive process that has served as a significant obstacle to more widespread guidelines development throughout neurosurgery. In the past, clinical practice guidelines have been developed by publicly supported epidemiologists and methodologists who understood study design, data analysis, and the guidelines process, but not the disease. This absence of context and clinical perspective significantly limited the value and relevance of their results. Alternatively, clinician-generated guidelines often took the form of methodologically flawed consensus panels and expert opinion, also of limited value. The Joint Spine Section recognized the importance of evidence-based clinical practice guidelines and the challenges of their development. The appropriate clinical expertise, strict adherence to established methodological standards for guidelines development, and considerable resource investment for the development, dissemination, and maintenance of the guidelines documents were deemed crucial to our guidelines initiative. Cervical spine and spinal cord injury was chosen as the initial guidelines topic because of the personal, social, and economic devastation of these injuries, their complex nature, and the high level of uncertainty, as reflected in wide practice variations, as to the value and indications for many of the aspects of evaluation and treatment. The clinical practice guidelines contained in this supplement to Neurosurgery represent a remarkable effort. They address the key issues related to the evaluation and management of these complex conditions that are relevant to the treating physician. In every chapter, the pertinent issues are succinctly stated, the published data are comprehensively presented in the evidentiary tables, and the evidence is thoroughly discussed and critically evaluated throughout the text. The linkage between the quality of the evidence and the strength of the recommendations was not a âblack boxâ process but an open, deliberative exercise by skilled experts guided by a rigorous set of standards. Despite the strength and potential value of this document, it is important to acknowledge the inherent limitations of clinical practice guidelines. This, or any other, evidence-based clinical practice guidelines document does not represent the definitive source of knowledge on the stated topic. Rather, it represents recommendations of varying strength and certainty based on an analysis of the best available published data. These data, however, are often conflicting, flawed, or incomplete, and there are unavoidable elements of potential bias from subjectivity, perspective, and experience of the individuals and group involved in the analysis and interpretation of these data. In essence, proof is a relative term based on the interpretation of evidence. Furthermore, it is subject to different standards. A relevant example comes from the field of jurisprudence, where the standard of proof (i.e., guilt) for criminal trials is âbeyond a reasonable doubt,â whereas the standard for civil courts must simply reflect âa preponderance of evidenceâ or âmore likely than not.â These different standards evolved because of the perceived different consequences of a wrongful verdict. Moreover, as Stephen Haines likes to note, the verdict ânot guiltyâ does not mean innocent; it merely says not proved. Such are the vagaries associated with the interpretation of even scientific evidence. Principled people can look at the same evidence and come to different conclusions subject to their own personal perspective, experience, and stake in the result. Nevertheless, the Joint Section and Guidelines Development Group went to great lengths to identify and avoidâor at least minimizeâthese potential problems. The working group adopted the most widely recognized and rigorous standards for guidelines development. A diverse panel of experts with expertise in spine, trauma, and epidemiology brought relevant clinical, scientific, and methodological competence to enhance both the analytical and the deliberative aspects of this process. Periodic outside reviews were routinely obtained for topics or areas of contention or uncertainty to add additional perspective and balance. Above all, the process was accountable and transparent at every stage. Ultimately, we offer these guidelines as a living document to those professionals who treat patients with traumatic spinal injury. We hope each practitioner will critically evaluate these guidelines and come to his or her own conclusion on how, whether, and when to implement its recommendations. It may be used either as a reference or as a basis for standardized protocols of evaluation and management of the patients with traumatic spinal injury. For clinical and basic science researchers, we hope that it will identify and catalyze scientific investigation in areas of deficient knowledge. As a Section, we stand firmly behind this important document and will continuously update the recommendations as new knowledge and understanding is developed. We sincerely believe that these guidelines can improve the care and enhance the outcomes of patients with traumatic injuries to the cervical spine and spinal cord.
1. Introduction.The classical Sturmian theorem of ordinary differential equations deals with functions u(x) and v(x) which are, respectively, solutions of differential equations(1) Um-^^j+tumO,(2) Mv=-l{J^+yv = 0.Under the assumption that "F is larger than M" (in the sense that a(x) Ă€ a(x) > 0 and c(x) ÂŁ y(x)) one can infer information about all solutions of (2) from knowledge about a particular nontrivial solution of (1)-i.e. if u(xx) = u(x2)=0 then every solution of (2) has a zero in [xx, x2].These ideas have been generalized to second order elliptic equations by several authors ([l]-[4]) considering elliptic operators and also by Protter [5] and Swanson [6] considering the nonselfadjoint case.Given a proper relation among the coefficients of F and M and that Lu=0 has a nontrivial solution with nodal domain Ă, then it can be shown that every solution of Mv = 0 has a zero in Ă2.While all the above proofs of this fact make essential use of some sort of ordering among elliptic operators, the nature of this ordering is never defined in operator-theoretic terms.The results of §2 below suggest that it is an order relationship between certain resolvents of the differential operators F and M which underlies the separation properties characteristic of Sturmian theorems.It will be shown that quite general operator equations in a Banach space 3S satisfy a type of Sturmian theorem if the operators' resolvents satisfy prescribed positivity requirements with respect to a cone SP.In order to apply this theory to differential operators, one must first establish the corresponding positivity properties for their resolvents.This is done in §3 for sufficiently regular nonselfadjoint second order elliptic operators, and the general theory of §2 is then applied in the proof of two Sturmian theorems and the establishment of criteria for certain Green's functions to be positive.
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Spectral Theory in Mathematical Physics
Quantum chaos and dynamical systems
Stability and Controllability of Differential Equations
Historians have recently given much attention to the active, formative role of state governments in the American economy before the Civil War.1 The states exercised nearly exclusive control over many aspects of economic life, and in such areas as labor, banking, and corporation policy the federal government interfered relatively little. The consequence was considerable decentralization of power in policymaking, together with variations in policy from state to state.2 Perhaps in no policy area were variations so dramatic as in state legislation on banking. In some states banking was prohibited outright, while in others the state government itself established and operated banks, sometimes on a monopoly basis. Elsewhere, safety funds were established and banks required to meet minimum standards of specie reserve and the like; and in a few states, stringent regulatory policies were pursued, with public commissioners given considerable discretion in administering policy.3
Let $\{X(t), t = \cdots -1, 0, 1, \cdots\}$ be a $P$ dimensional zero mean stationary Gaussian time series, $X(t) = \begin{pmatrix}X_1(t)\\X_2(t)\\\vdots\\X_P(t)\end{pmatrix}$ we let $R(\tau) = EX(t)X' (t + \tau)$, where $R(\tau) = \{R_{ij}(\tau), i,j = 1, 2, \cdots P\}$, and $F(\omega) = (2\pi)^{-1} \sum^\infty_{\tau=-\infty}e^{-i\omega\tau}R(\tau)$. It is assumed that $\sum^P_{i,j=1} \sum^\infty_{\tau=-\infty} |\tau| |R_{ij}(\tau)| < \infty$, and hence $F(\omega)$ exists and the elements possess bounded derivatives. It is further assumed that $F(\omega)$ is strictly positive definite, all $\omega$. Knowledge of $F(\omega)$ serves to specify the process. $F(\omega)$, and $S$, the covariance matrix of $x = \begin{pmatrix}x_1 \\ x_2\ \\ vdots\\x_P\end{pmatrix}$, a Normal $(0, S)$ random vector are known to enjoy many analogous properties. (See [7].) To cite two examples, the hypothesis that $X_i(s)$ is independent of $X_j(t)$ for $i \neq j = 1, 2, \cdots P$, any $s, t$, is equivalent to the hypothesis that $F(\omega)$ is diagonal, all $\omega$, while the hypothesis that $x_i$ is independent of $x_j$, for $i \neq j = 1, 2, \cdots P$ is equivalent to the hypothesis that $S$ is diagonal. The conditional expectation of $x_1$, given $x_2, \cdots x_P$ is \begin{equation*}E(x_1\mid x_2, \cdots x_P) = S_{12}S^{-1}_{22}\begin{pmatrix}x_2 \\ \vdots \\ x_P\end{pmatrix}, S = \bigg(\begin{array}{c|c} S_{11} & S_{12} \\ \hline S_{21} & S_{22}\end{array} \bigg)\end{equation*}. The corresponding regression problem for stationary Gaussian time series goes as follows. If \begin{equation*}E\{X_1(t)\mid X_2(s), \cdots X_P(s), s = \cdots -1, 0, 1, \cdots\} = \sum^P_{j=2} \sum^\infty_{s=-\infty} b_j(t - s)X_j(s)\end{equation*} then $B(\omega)$, defined by $B(\omega) = (B_2(\omega), \cdots B_P(\omega)), B_j(\omega) = \sum^\infty_{s=-\infty} b_j(s)e^{i\omega s}$ satisfies \begin{equation*}B(\omega) = F_{12}(\omega)F_{22}^{-1}(\omega), \quad F(\omega) = \bigg(\begin{array}{c|c}f_{11}(\omega) & F_{12}(\omega) \\ \hline F_{21}(\omega) & F_{22}(\omega)\end{array} \bigg).\end{equation*} It is interesting to ask how well these and similar analogies carry over to sampling theory and hypothesis testing. Goodman [3] gave a heuristic argument to support the conclusion that $\hat{F}_X(\omega_k)$, a suitably formed estimate of the spectral density matrix $F(\omega_k)$ has the complex Wishart distribution. The question is met here by the following results. Firstly if $\hat{F}_X(\omega_l), l = 1, 2, \cdots M$ are estimates of the spectral density matrix, each consisting of averages of $(2n + 1)$ periodograms based on a record of length $T$, with the $\omega_l$ equally spaced and $(2n + 1)M \leqq \frac{1}{2} T$, then it is possible to construct, on the same sample space as $X(t), M$ independent complex Wishart matrices $\hat{F}{\bar{\bar{X}}}(\omega_l), l = 1, 2, \cdots M$ such that $\{\hat{F}_X(\omega_l), l = 1, 2, \cdots M\}$ converge simultaneously in mean square to $\{\hat{F}_{\bar{\bar{X}}}(\omega_l), l = 1, 2,\cdots M\}$, as $n, M$ get large. Secondly, it is legitimate to use the natural analogies from multivariate analysis to test hypotheses about time series. One example is presented, as follows. The likelihood ratio test statistic for testing $S$ diagonal is $|\hat{S}|/\mathbf{\prod}^P_{i=1} \hat{s}_{ii}$ where $\hat{S} = \{\hat{s}_{ij}$ is the sample covariance matrix. The analogous statistic $\psi$ for testing $X_i(s), X_j(t)$ independent, $i,j 1 = 2, \cdots P$ from a record of length $T$ is $\psi = \prod^M_{l=1} \lbrack|\hat{F}_X(\omega_l)|/\prod^P_{i=1} \hat{f}_{ii}(\omega_l)\rbrack$ where $\hat{F}_X(\omega_l) = \{\hat{f}_{ij}(\omega_l)\}$ are the sample spectral density matrices as above. Letting ${\bar{dbar{\psi}}} = \prod^M_{l=1} \lbrack|\hat{F}_{\bar{\bar{x}}}(\omega_l)|/\prod^P_{i=1} \hat{h}_{ii}(\omega_l)\rbrack$ where $\hat{F}_{\bar{\bar{x}}}(\omega_l) = \{\hat{h}_{ij}(\omega_l)\}$ are the independent complex Wishart matrices referred to above, we show $EC_{n,M} |\log \psi - \log {\bar{\bar{\psi}}} \rightarrow 0$ for large $n, M$, where $C_{n,M}$ are chosen to make the result non-trival. The method of proof applies to any statistic which is a product over $l$ of sufficiently smooth functions of the entries of $\hat{F}_X(\omega_l)$. Applications to estimation and testing in the regression problem will appear elsewhere [8]. The distribution theory of functions of complex Wishart matrices has been well investigated by a number of authors [3] [5] [6], and hence can be easily applied here to statistics like ${\bar{\bar{\psi}}}$. The results above are shown for $P = 2$, it is clear that the proofs extended to any (fixed) finite $P$. The proofs proceed as follows, via a theorem which has somewhat more general application. For each $T$, let $X$ be the $2 \times T$ random matrix $X = \binom{X_1}{X_2} = \begin{pmatrix}X_1(1), \cdots, X_1(T)\\X_2(1), \cdots, X_2(T)\end{pmatrix}$ and let the $2T \times 2T$ covariance matrix $\Sigma$ be given by $\Sigma = \begin{pmatrix}\sum_{11} \sum_{12} \\ \sum_{21} \sum_{22}\end{pmatrix}$ where $\Sigma_{ij} = EX_i'X_j. \{\hat{F}_X(\omega_l)\}$, the sample spectral density matrices described above based on a record of length $T$, are each of the form $\hat{F}_X(\omega_l) = T^{-1}XQX'$ where $Q$ is a $T \times T$ circulant matrix with largest eigenvalue $ = T(2n + 1)^{-1} \leqq \frac{1}{2}M < <T$. We define circulant matrices $\bar{\Sigma}_{ij}$ which approximate $\Sigma_{ij}$, and a random matrix $\bar{X}$ on the sample space of $X$, $\bar{X} = \binom{\bar{X}_1}{\bar{X}_2} = \begin{pmatrix}\bar{X}_1(1), \cdots, \bar{X}_1(T)\\\bar{X}_2(1), \cdots, \bar{X}_2(T)\end{pmatrix}$ with $E\bar{X}_i'\bar{X}_j = \bar\Sigma_{ij}$. The $2T$ eigenvalues of the block circulant matrix $\bar\Sigma = \begin{pmatrix}\bar\Sigma_{11} \bar\Sigma_{12} \\ \bar\Sigma_{21} \bar\Sigma_{22}\end{pmatrix}$ will be the $2T$ eigenvalues of the $T$ matrices $\{F(2\pi j/T),j = 1, 2, \cdots T\}$. The distribution of random matrices of the form $T^{-1}\bar{X}Q\bar{X}'$ where $Q$ is any circulant matrix are relatively simple to investigate due to the fact that all circulant matrices commute, and their eigenvalues may be exhibited as simple functions of the elements. Circulant quadratic forms in random vectors with circulant covariance matrices are well known in the literature, (See [1] and references cited there). Let $\hat{F}_{X,Q} = T^{-1}XQX'$ and $\hat{F}_{\bar{X},Q} = T^{-1}\bar{X}Q\bar{X}'$ where $Q$ is now any $T \times T$ (real or complex) quadratic form with largest absolute eigenvalue $\leqq q$. The main Theorem allows the replacement of $X$ by $\bar{X}$ in the analysis, and is, that under the assumptions on $F(\omega)$ and $R(\tau)$, for any $T$, \begin{equation*}\tag{1.1} E \operatorname{tr} (\hat{F}_{X,Q} - \hat{F}_{\bar{X},Q})(\hat{F}_{X,Q} - \hat{F}_{\bar{X}, Q})^{\ast'} \leqq cq^2/T^2\end{equation*} where $c$ is a constant depending only on $F(\omega)$ and $R(\tau)$. A lemma, essentially allowing the replacement of $F(\omega)$ by a suitably chosen step-function, together with the application of (1.1) gives the results concerning the $\{\hat{F}_X(\omega_l)\}$ an $\lambda$. Since $\hat{R}(\tau)$, the sample (circularized) autocorrelation function is also of the form $T^{-1}XQX'$ with $Q$ circulant we obtain an easy corollary on the distribution of $\{\hat{R}(\tau)\}$.
For a self-financing business enterprise (or for an underdeveloped economy subject to constraints on the availability of foreign investment funds), three theorems are presented. Each result is based upon the assumption that the firm's investment opportunities follow constant returns-to-scale, and are of the âpoint inputâstream outputâ type. Theorem 1 shows that if the enterprise is attempting to maximize a linear function of the cash dividends paid out, the optimization model cannot explain a readily observed phenomenon: both investment expenditures and also cash dividends at the same point in time. Theorems 2 and 3 explore the consequences of supposing that the maximand is a concave, nonlinear function of the cash dividends paid out, and that the optimal solution consists of positive investment expenditures over time. (The optimal policy may or may not call for positive dividends during each time period.) Then Theorem 2 shows that the optimal dual variable price ratios are determined uniquely by the set of investment opportunities available, and Theorem 3 shows that the optimal policy can be evaluated numerically through optimization of the original utility function subject to a specially constructed single linear equality constraint on the cash withdrawals. An economic decentralization interpretation is attached to this auxiliary maximization problem.
This study attempts to link the formal structure of bureaucratic organizations to decision-making processes, and in particular to centralization or decentralization of authority. Interview data were obtained from 254 city, county, and state departments of finance. These data show that, controlling for an organization's size, decision-making authority is more highly centralized as the number of subunits in an organization increases; but as the number of levels of supervision grows, there is greater decentralization and at the same time proliferation of rules that specify criteria to guide decisions. Marshall W. Meyer is lecturer on sociology in the department of social relations at Harvard University.
The Korean family planning program from 1964 to 1968 is considered in its entirety. The program is financed primarily by Korean government appropriations (averaging about $2 million in past years from central and local budgets) but also is supported substantially by foreign agencies such as the Population Council SIDA and AID. The Planned Parenthood Federation of Korea working in cooperation with the government is financed by about $200000 yearly from Korean government and foreign sources. The program has offered the IUD condom and vasectomy to every couple without charge. Accomplishments include adoption of the loop or vasectomy by 1/3 of couples with wife below age 45 elicitation of the best response among illiterate and low-income families and in rural areas reduction of crude birth rate by roughly 10% as of 1968 by IUDs inserted through mid-1968 and launching a pill program for IUD drop-outs starting with an unknown method a population of 30 million was informed of it and their approval won. Approaches used in the program have included strong governmental policy; implementation through the existing health structure; decentralization of functions to local and provincial government levels; payment by government of physicians and fieldworkers involved as well as vasectomy acceptors for work lost; high fieldworker density; fixation of target quotas with positive and negative sanctions on workers; and full use of mass media. Problems to be dealt with involve funding doctorless and remote areas and termination rates. The problems have been obtaining the funds; areas that are doctorless and that are remote; the training of workers; funds for maintenance and supply systems; shortcomings of the present contraceptive methods; and the changing of field-worker duties.
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Diverse Approaches in Healthcare and Education Studies
Abstract Capital and Management in Socialist Economies The idea, that an economy can be operated by methods of central planning to realize the economic optimum like the model of a private business corporation, proved to be successful only in the first periods of (âextensiveâ) industrialization. Growing product differentiation increased the danger of disallocation of resources. The solution of microeconomic allocation problems turned out to be most unsatisfactory. Since 1962, reforms try to decentralize decision making in the individual enterprise striving mostly for production aims planned centrally. The problem of appropriate investment and success criteria for the systemâs management remains unsolved. While in sectors of the economy, where large sums of capital are needed, ex ante central planning is necessary, the problem is different in manufacturing industry. Here, an efficient selection from the abundance of continually and acceleratingly renewing technical progress can be made only with decentralized rules of efficiency, which, however, must be based on market prices. But the choice of a soviet director is limited by problems of financing and supply. He may calculate approximately the technical effect of investment, but he cannot evaluate its economic effect, since prices have been set since long and do not reflect the permanently changing relations of scarcity of resources, Besides, the certain profit from selling established products is frequently preferred to the risk of innovation. The industrial reform of 1965 has focused on the relation of the profit from sold production and the invested capital stock. It emphasizes the importance of investment financing from enterprize profits and credit (in contrast to the usual allotment of means for investment by central authorities). A greater flexibility in fulfilling the plan can be observed, but managementâs investment choices are still very limited. The centrally planned supply of investment cannot keep pace with increased decentralized investment demands. The reason for the stated inferiority of socialist economies in realizing technical progress is ultimately the absence of a capital market. Its role for most investment decisions can be substituted only most imperfectly by central planning.
A gas or vapor bubble moving in translation in a liquid and varying its volume propels itself by a rocket effect, which occurs in foams moving in pressure or temperature fields with a non-zero gradient. It is notably observed in boiling and centrifugal pump operation, which latter features the following two characteristic bubble types : __ 1. Gas bubbles, which are apt to emerge faster from the pump than the liquid; 2. Vapor bubbles (cavitalion). When a bubble in translatory motion implodes, energy is transferred: potential pressure energy is converted info kinetic implosion energy, which in turn becomes kinetic translation energy. The instrument for this energy transfer process is a "micro-jet" following the bubble. It is shown that a considerable increase in kinetic translation energy density occurs, which is converted into potential pressure energy on impact against a solid obstacle. The resulting pressures (10,000 kg/sq.cm) explain the mechanical aspect of cavitation erosion. Several experimental results have confirmed these theoretical considerations. The development of a rotoscope is also described, this being a device enabling a centrifugal pump impeller (for example) to be "stopped" so that only the relative motion of the flow particles remains. Unlike with a stroboscope, time exposures can he taken with this device. With more thorough knowledge of foam mechanisms, and especially of cavitalion bubble behavior, it should be possible to design cavitation erosion-proof impeller blades.
The principles of unity and complexity introduced into the Polish State budget in 1951 are no longer fully observed. In particular, certain funds administered by local councils have appeared gradually in the last years, on the basis of the budget law of 1958 which admitted making some budget expenses dependent on definite budget revenues. Moreover, some funds have been created outside the local budgets. These deserve special attention as a new phenomenon which becomes increasingly important for the local councils activities and especially for councils of the lowest level. The development of decentralized funds seems to be desirable firstly in the economic micro-regions, such as villages and towns, in order to accelerate their development. The augmentation of the role of decentralized funds would also be advisable in counties and provinces provided they constitute regions with specific economic structure. Decentralized funds should be first of all used to finance local investments, the more so that this would be in accordance with the main destination of the funds already administered by local councils. In this way, decentralized funds could in the future be transformed into the local investment budgets. The considerable development of decentralized funds in this country, in particular of those of the lowest level of local administration, seems, moreover, to prove that local budgets at this level, which since 1951 have been incorporated into the State budget, are not fully adequate for financing local councils activities. This suggests that a reform of local budgets of the lowest level is advisable, which would make their administration more similar to that of decentralized funds. The author considers in detail the perspectives and conditions of s'uch a reform of the local budgets of the lowest level, as well as the prospects and conditions of further development of the decentralized funds if no budget reform takes place. He discusses especially the links between decentralized funds and local budgets in this last case. The other subject of the author's consideration is the advisability of financing the local councils activities by bank credit or by credit granted by central funds administered by budget authorities
Abstract This article focuses on multiple regression analysis to cost control of decentralized operations in the consumer finance industry. There are potential accounting applications of multiple regression analysis in control of decentralized operations. Moreover, multiple regression can be a useful empirical research tool in other areas of accounting and finance. It is essential, however, to know the hidden limitations and assumptions in the approach and to perform the necessary tests to see that these assumptions are met be- fore plunging head-first into a sea of regression formulae. In cost analysis, one feature of multiple regression is the ability to use dichotomous variables. The advantage herein arises when a given characteristic may or may not exist in decentralized units. Multiple regression may be applied without assuming the disturbance terms are normally distributed. Multiple regression may be used in testing structural relationships between operating costs and various factors which are thought to affect these costs. Analysis of variance procedures may be extended to statistical tests of single coefficients and to statistical tests of the contribution to explained variation of sub-groups of factors included in the model.
This paper is concerned with repetitive sequential play in finite statistical games (decision problems) from the statistician's point of view. We shall assume that the statistician's move at stage $k$ may depend on the previous $k - 1$ moves of Nature as well as the random variable $\mathbf{X}_k = (X_1, \cdots, X_k)$, where the $X_i$ are independent observations (r.v.'s) (possibly vector-valued) from the sequence of statistical games, $k = 1, 2, \cdots$. The play is repetitive in the sense that each component game is identical in structure, with only the moves of the statistician and Nature changing. Furthermore, we impose no assumptions regarding the behavior of the parameter sequence of Nature's moves. The statistician does have the added disadvantage that the finite class of distributions in the component game is not fully specified. However, he does know that class in question has: either (i) all members with discrete distributions or (ii) all members with $q$-dimensional a.e. continuous Lebesgue densities. This same problem when the distributions are fully known has been treated in [6] for statistical as well as more general games in which Nature's space is finite. In the case where the distributions are completely specified but the history of the past moves is unknown to the statistician, see [20], [22], [27], and [28]. The development in this paper is closely connected to and motivated by these results, particularly those of the preceding paper [27]. If for fixed $N$, the empirical distribution $p_N$ of Nature's moves is known, then the statistician could use as a rule for each of the $N$ component games a strategy Bayes against $p_N$ having risk $\phi(p_N)$. In all the papers cited in the previous paragraph, the aim was to construct for the statistician, when $p_N$ is unknown and $N$ not specified, a sequence of randomized decision functions whose $N$th average loss minus $\phi(p_N)$ approaches zero (or has an upper bound approaching zero) in a suitable sense as the number of repetitions of play, $N$, increases. However, in the case of statistical games, all of the above results require that the finite class of distributions be fully specified. In this paper we remove that assumption by estimating the distributions sequentially based on past moves and observations. Then in the present play of the component game the statistician substitutes these estimators into a procedure which is Bayes against the empirical distribution of Nature's previous moves. The resulting sequence of procedures is shown to be "asymptotically good" in the sense that the average loss over the $N$ games $W_N$ minus the Bayes risk $\phi(p_N)$ approaches zero (in an appropriate sense) as $N$, the number of games played, increases. In Section 2 we introduce notation and preliminaries. Section 3 discusses play in repetitive games and defines the proposed sequential procedures $\mathbf{t} = \{\mathbf{t}_k\}$. In Section 4 we prove preliminary results upon which all proofs are founded. Section 5 considers the discrete case giving uniform (in sequences of Nature's moves) convergence theorems (as $N \rightarrow \infty$) for the quantity $W_N - \phi(p_N)$. Theorem 5.1 is a uniform convergence theorem of $O(N^{-\frac{1}{2}})$ of the expected value of $W_N - \phi(p_N)$ for finite discrete classes, each member of which is non-degenerate and satisfies a certain tail probability condition. Under the same conditions, Theorem 5.2 gives uniform convergence to zero in probability for the quantity $N^{\frac{1}{2}} (\log N)^{-1} \{W_N - \phi(p_N)\} \text{as} N \rightarrow \infty$. Uniform convergence of $W_N - \phi(p_N) \rightarrow 0$ in probability for general non-degenerate finite discrete class is presented in Theorem 5.3. Section 6 treats the estimation problem for densities needed to form the randomized strategy sequences $\mathbf{t}$ in the continuous case. The results stated are based on a paper by Cacoullos [3] generalizing the univariate results of Parzen [15]. In Section 7, we present results for the continuous case. Theorem 7.1 and its corollary give uniform convergence of $W_N - \phi(p_N)$ to zero in probability and of its expectation to zero, respectively. The finite continuous classes of Theorem 7.1 are very general in the sense that each member is a continuous a.e. density. Finally, in Section 8 we draw certain conclusions and relate our results to similar results obtained elsewhere. The novelty of the paper rests in the fact that through the past history of Nature's moves and the observations connected with past play, one can construct a sequential strategy, $\mathbf{t} = \{\mathbf{t}_k\}$, with very little knowledge about the finite class of distributions, which approaches asymptotic "optimal" play. The lack of knowledge on the finite class of distributions distinguishes this work from the related "repetitive type" problems in games and/or decision theory treated in [1], [2], [4], [6], [7], [8], [9], [10], [12], [17], [18], [19], [20], [21], [22], [24], [25], [26], [27], [28], and [29]. For possible applications of this work see Neyman [14], especially his Example 3 and his discussion relating to the work of Blackwell [2].
Abstract The accounting fraternity has employed regression analysis rather infrequently. This article presents an application of multiple regression analysis to cost control. The context of the application is the consumer finance industry where extensive decentralization makes effective cost control extremely important. The consumer finance industry is made up of companies whose principal activity is making personal installment cash loans under state small loan laws. The cost behavior model employed in this article is developed from the results of multiple regression analysis of cost and other operating data of branch offices of a major consumer finance chain. While the consumer finance industry is used as the basis, it should be emphasized that the procedure outlined would be applicable to other types of businesses as well. The article shows that an important requirement for the applicability of the procedure is the existence of a relatively large number of homogeneous operating units. Consumer finance companies meet this requirement particularly well. However, other types of business also operate with large numbers of homogeneous units-food including service chains and lodging chains. The procedure outlined would, therefore, be applicable to them as well.
Previous article Next article Heuristic Methods for Location-Allocation ProblemsLeon CooperLeon Cooperhttps://doi.org/10.1137/1006005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Leon Cooper, Location-allocation problems, Operations Res., 11 (1963), 331â343, No. 3 MR0152365 0113.14201 CrossrefISIGoogle Scholar[2] Google Scholar[3] D. H. Lehmer, Teaching combinatorial tricks to a computerProc. Sympos. Appl. Math., Vol. 10, American Mathematical Society, Providence, R.I., 1960, 179â193, Combinatorial Analysis MR0113289 0096.00504 CrossrefGoogle Scholar[4] J. Heller, Some numerical experiments for an $M\times J$ flow shop and its decision-theoretical aspects, Operations Res., 8 (1960), 178â184, No. 2 MR0111626 0092.27910 CrossrefISIGoogle Scholar[5] J. Moshman, The Application of Sequential Estimation To Computer Simulation and Monte Carlo Procedures, J. Association for Computing Machinery, 5 (1958), 0086.11603 CrossrefISIGoogle Scholar[6] A. Wald, Sequential Analysis, John Wiley and Sons, New York, 1952 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails An elliptical cover problem in drone delivery network design and its solution algorithmsEuropean Journal of Operational Research, Vol. 304, No. 3 | 1 Feb 2023 Cross Ref Two lower-bounding algorithms for the p-center problem in an areaComputational Urban Science, Vol. 2, No. 1 | 24 January 2022 Cross Ref Optimal planar facility location with dense demands along a curveJournal of the Operational Research Society, Vol. 73, No. 8 | 8 May 2021 Cross Ref A continuous location and maintenance routing problem for offshore wind farms: Mathematical models and hybrid methodsComputers & Operations Research, Vol. 144 | 1 Aug 2022 Cross Ref Spatial autocorrelation informed approaches to solving locationâallocation problemsSpatial Statistics, Vol. 50 | 1 Aug 2022 Cross Ref Electric vehicle charging stations emplacement using genetic algorithms and 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Case reports of three residents of Ontario with clinical histoplasmic chorioretinitis are presented. The diagnosis was made on the basis of the clinical appearance, the presence of calcified lesions in the chest, a negative skin test to tuberculin, and a positive skin test to toxoplasmin. All patients were treated with intravenous amphotericin B. Except for transitory elevation of blood urea nitrogen, there were no serious complications from the drug and in all cases the lesions in the eyes were improved. Histologic or cultural proof of the presence of fungus in the eye is not available, but clinical and laboratory findings can combine to point to the diagnosis of histoplasmosis. In such cases, since vision is at stake, treatment with amphotericin B should be considered.