18 results
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2. A Graph Formulation of a School Scheduling Algorithm.
- Author
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Salazar, A. and Oakford, R. V.
- Subjects
- *
ALGORITHMS , *SCHOOL schedules , *GRAPH theory , *APPROXIMATION theory , *SET theory , *CURRICULUM - Abstract
The problem classically titled "The Examination Schedule Problem" takes various forms in the literature. Most of these formulations can be presented in the terminology of classical Network Theory. One such formulation is: Given a nondirected network, partition its nodes into a minimal number of subsets such that no two members of the same subset are connected by an An obvious lower limit to this number is the size of the largest strongly connected subgraph. Kirchgassner proved that an upper limit is this size plus one. One logical extension of the previous work is the introduction of variable length examinations where W(I) is the number of periods for exam I. The object of this paper is to generalize the definition of largest strongly connected subgraph to include the weighting of nodes, to present an approximate algorithm which usually finds the largest strongly connected subgraph, and to discuss the application of this algorithm to the solution of school scheduling and exam scheduling problems. [ABSTRACT FROM AUTHOR]
- Published
- 1974
- Full Text
- View/download PDF
3. MOMENTS OF THE DISTRIBUTION OF SAMPLE SIZE IN A SPRT.
- Author
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Ghosh, B. K.
- Subjects
- *
MOMENTS method (Statistics) , *STATISTICAL sampling , *ARITHMETIC , *DISTRIBUTION (Probability theory) , *DIFFERENTIABLE functions , *PROBABILITY theory , *APPROXIMATION theory , *EQUATIONS , *STATISTICS - Abstract
The article discusses moments of the distribution of sample size, N in a sequential probability ratio test (SPRT). The present paper provides variance, the third and the fourth moments of N. The details are worked out in five common applications of the SPRIT. The relation of the variance of N to the truncation of a SPRT is discussed is also discussed in the paper. Scholar A. Wald indicated in passing how one can obtain the moments of N, but the only published literature where the author encountered a general expression for the variance of N. However, their expression is incorrect. Using scholar J. Wolfowitz's results, which they do, or differentiating Wald's, fundamental identity twice one gets provided. In many practical applications of the SPRT, μ and moments in an equation derived are differentiable functions of a real-valued parameter. The limiting expressions for the moments can then be determined by standard methods of mathematical analysis. However, for the third and fourth moments the actual technique may involve an excessive amount of arithmetic.
- Published
- 1969
- Full Text
- View/download PDF
4. RATIOS OF NORMAL VARIABLES AND RATIOS OF SUMS OF UNIFORM VARIABLES.
- Author
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Marsaglia, George
- Subjects
- *
RANDOM variables , *DISTRIBUTION (Probability theory) , *PROBABILITY theory , *DENSITY functionals , *MATHEMATICAL variables , *GAUSSIAN distribution , *APPROXIMATION theory - Abstract
The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the vivariate normal distribution and Nicholson's V function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal. The last part of the paper discusses the distribution of the ratio (u[sub 1] + ... + u[sub n])/(v[sub I] + ... + v[sub m]) where the u's and v's are independent, uniform variables. The exact distribution for all n and m is given, and some approximations discussed. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
5. Optimal Designs for Estimating the Slope of a Polynomial Regression.
- Author
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Murty, V. N. and Studden, W. J.
- Subjects
- *
POLYNOMIALS , *REGRESSION analysis , *ANALYSIS of variance , *APPROXIMATION theory , *EXPERIMENTAL design , *LEAST squares , *OPTIMAL designs (Statistics) , *MATHEMATICAL statistics - Abstract
The problem of estimating the slope of a polynomial regression at a fixed point of the experimental region such that (a) the variance of the least-square estimate of the slope at the fixed point is a minimum and {b) the average variance of the least-square estimate of the slope is a minimum is discussed in this paper. In general these designs can be obtained using Kiefer-Wolfowitz [5] characterization of c-optimal designs, Federov [2] characterization of L-optimal designs, and Studden's [10] generalization of the Elfving Theorem [1]. After presenting a brief review of these characterization theorems, specific illustrations for the quadratic and cubic regressions are presented in detail. [ABSTRACT FROM AUTHOR]
- Published
- 1972
- Full Text
- View/download PDF
6. BOUNDS AND APPROXIMATIONS FOR THE MOMENTS OF ORDER STATISTICS.
- Author
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Joshi, Prakash C.
- Subjects
- *
APPROXIMATION theory , *NONPARAMETRIC statistics , *MOMENTS method (Statistics) , *MATHEMATICAL statistics , *STATISTICS , *DISTRIBUTION (Probability theory) , *NUMERICAL calculations , *PROBABILITY theory - Abstract
In this paper methods for obtaining approximations and bounds for the moments of order statistics from a continuous parent distribution are discussed. These bounds and approximations depend on the distribution function only through certain moments of order statistics in small samples. It is shown that for the Cauchy distribution bounds and approximations of all finite moments can be obtained. Some numerical calculations for normal and Cauchy distributions are also given. [ABSTRACT FROM AUTHOR]
- Published
- 1969
- Full Text
- View/download PDF
7. FIRST AND SECOND MOMENTS OF THE RANDOMIZATION TEST IN TWO-ASSOCIATE PBIB DESIGNS.
- Author
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Cléroux, Robert
- Subjects
- *
MOMENTS method (Statistics) , *MATHEMATICS , *STATISTICS , *DISTRIBUTION (Probability theory) , *THEORY , *APPROXIMATION theory , *ERROR , *THEORY of knowledge , *PROBLEM solving - Abstract
In this paper the first two moments of the Statistic (treatment sum of squares)/(treatment +error sums of squares) over all possible random assignments of treatments to the experimental plots are obtained for two associate PBIB designs. They are compared with the corresponding moments of a central beta distribution to study the extent to which the normal theory test may serve as an approximation to the randomization test. It is found that the approximation is reasonable for some classes of PBIB designs. [ABSTRACT FROM AUTHOR]
- Published
- 1969
- Full Text
- View/download PDF
8. CURVE FITTING BY SEGMENTED STRAIGHT LINES.
- Author
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Bellman, Richard and Roth, Robert
- Subjects
- *
APPROXIMATION theory , *POLYGONAL numbers , *CURVE fitting , *POLYNOMIALS , *DYNAMIC programming , *FUNCTIONAL analysis , *MATHEMATICAL optimization , *NUMERICAL analysis - Abstract
In many situations, approximation of a set of data by a polygonal curve is more advantageous than approximation by a polynomial. If the join points of the polygonal curve are known, the problem is quite simple. If, however, they are to be chosen in some expeditious fashion, considerable numerical difficulties can arise if the curve-fitting problem is approached directly. In this paper it is shown that dynamic programming offers a simple direct approach to the determination of an optimal fit. [ABSTRACT FROM AUTHOR]
- Published
- 1969
- Full Text
- View/download PDF
9. TESTING AND ESTIMATING RATIOS OF SCALE PARAMETERS.
- Author
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Shorack, Galen R.
- Subjects
- *
ROBUST statistics , *PERMUTATIONS , *RANDOM variables , *STATISTICAL sampling , *TESTING , *MONTE Carlo method , *SAMPLE size (Statistics) , *NUMERICAL analysis , *APPROXIMATION theory - Abstract
Let X[sub 1], ... , X[sub m] and Y[sub 1], ... , Y[sub n] be independent random samples from populations having continuous d.f.'s psi((x-micro)/sigma) and psi((y-nu)/tau) respectively. The classical F-test of a hypothesis concerning angle = tau/sigma is known to be non-robust. This paper examines several robust alternative procedures and compares them on the basis of Pitman a.r.e and Monte Carlo studies of power functions. An approximate permutation test [13] and a "jackknife" procedure [9] are found to be most satisfactory; while a class of "rank-like" tests [10] are found to be "useful inefficient statistics" [ABSTRACT FROM AUTHOR]
- Published
- 1969
- Full Text
- View/download PDF
10. TWO-SIDED TOLERANCE LIMITS FOR NORMAL POPULATIONS--SOME IMPROVEMENTS.
- Author
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Howe, W. G.
- Subjects
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GAUSSIAN distribution , *LAMBDA algebra , *DISTRIBUTION (Probability theory) , *APPROXIMATION theory , *LAMBDA calculus , *VARIANCES , *EQUATIONS , *CHI-squared test - Abstract
Ellison has shown that the Wald-Wolfowitz tolerance limits for a normal distribution, x + lambda s, are good only to 0(n/N[sup 2]), rather than to 0(1/N[sup 2]). Here x is distributed normally with mean mu and variance sigma[sup 2]/N while s[sup 2]/sigma[sup 2] is distributed as chi[sup 2, sub n]/n independently of x. Thus, for n much greater than N[sup 2] the usual values of lambda are incorrect; Ellison has proposed an alternative in this case. This paper derives new lambda's which have two advantages over the Wald-Wolfowitz and the Ellison limits. First, they are shown to be better approximations. Secondly, they are easily calculated in the sense that only tables of the normal and chi[sup 2] distributions are required and the solution of a non-linear equation is not required. [ABSTRACT FROM AUTHOR]
- Published
- 1969
- Full Text
- View/download PDF
11. A NORMAL APPROXIMATION FOR BINOMIAL, F, BETA, AND OTHER COMMON, RELATED TAIL PROBABILITIES, II.
- Author
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Pratt, John W.
- Subjects
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APPROXIMATION theory , *SQUARE root , *PROBABILITY theory , *DISTRIBUTION (Probability theory) , *MATHEMATICAL variables , *MATHEMATICAL statistics - Abstract
This paper describes and derives the asymptotic behavior of the new Normal approximation introduced in Part I and of a family of Normal approximations based on roots, including the square root approximations of Fisher, and Freeman and Tukey, and the cube root approximations of Wilson and Hilferty, Camp, and Paulson. Various asymptotic comparisons are made, all of which rank the new approximation first, the cube root approximations second, and the other root approximations (and the ordinary Normal approximation) third. For instance, in the binomial case, if the tail probability is fixed as n arrow right infinity, the errors resulting from the foregoing approximations are generally of order n[sup -3/2], n[sup -1], and n[sup -1/2] respectively, while for a tail probability approaching 0, the relative error in approximating it approaches 0 if the corresponding standard Normal deviate is of smaller order than n[sup 1/2] n[sup 1/4], or n[sup 1/6] respectively, but not generally otherwise. These comparisons are deduced from far more detailed results which are also given. [ABSTRACT FROM AUTHOR]
- Published
- 1968
- Full Text
- View/download PDF
12. ACCURACY OF AN APPROXIMATION TO THE POWER OF THE CHI-SQUARE GOODNESS OF FIT TEST WITH SMALL BUT EQUAL EXPECTED FREQUENCIES.
- Author
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Slakter, Malcolm J.
- Subjects
- *
ESTIMATION theory , *SAMPLE size (Statistics) , *MONTE Carlo method , *STATISTICAL sampling , *APPROXIMATION theory , *MATHEMATICAL models - Abstract
This paper presents the results of a Monte Carlo study of the accuracy of an approximation to the power of the chi-square goodness of fit test with small but equal expected frequencies. Various combinations of sample size, number of groups, and alpha level are considered, and in most instances the actual power of the test is estimated to be less than the nominal power. The degree of accuracy appears to be more related to the size of the sample than to the size of the expected frequencies. The following rule of thumb is offered for obtaining crude estimates of the actual power from the nominal power for sample sizes from 10 to 50: The actual power of the test equals about eight-tenths of the nominal power. [ABSTRACT FROM AUTHOR]
- Published
- 1968
- Full Text
- View/download PDF
13. SOME PROPERTIES OF SYMMETRIC STABLE DISTRIBUTIONS.
- Author
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Fama, Eugene F. and Roll, Richard
- Subjects
- *
APPROXIMATION theory , *MONTE Carlo method , *DISTRIBUTION (Probability theory) , *LINEAR statistical models , *MATHEMATICAL models , *FUNCTIONAL analysis - Abstract
This paper takes a few steps toward alleviating problems of data analysis that arise from the fact that elementary expressions for density and cumulative distribution functions (c.d.f.'s) for most stable distributions are unknown. In section 2 results of Bergstrom [3] are used to develop numerical approximations for the c.d.f.'s and the inverse functions of the c.d.f.'s of symmetric stable distributions. Tables of the c.d.f.'s and their inverse functions are presented for twelve values of the characteristic exponent. In section 3 the usefulness of the numerical c.d.f.'s and their inverse functions in estimating the parameters of stable distributions and testing linear models involving stable variables is discussed. Finally, section 4 presents a Monte Carlo study of truncated means as estimates of location. In every case but the Gaussian, some truncated mean is shown to have smaller sampling dispersion than the full mean. [ABSTRACT FROM AUTHOR]
- Published
- 1968
- Full Text
- View/download PDF
14. AN INVESTIGATION INTO THE SMALL SAMPLE PROPERTIES OF A TWO SAMPLE TEST OF LEHMANN'S'S.
- Author
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Afifi, A. A., Elashoff, R. M., and Langley, P. G.
- Subjects
- *
ASYMPTOTIC distribution , *STATISTICAL sampling , *GAUSSIAN distribution , *SAMPLE size (Statistics) , *DISTRIBUTION (Probability theory) , *APPROXIMATION theory , *MATHEMATICAL statistics , *STATISTICS - Abstract
In this paper we examine how well the asymptotic null distribution of a two sample test due to Lehmann approximates the small sample distribution of the test, compare the validity of this Lehman test with the validity of the two sample t test under the null hypothesis of equal means, and compare the power of this Lehmannn test with the power of the t test. Our general conclusion is that experimenters will prefer to use the t test when the underlying distribution is the scale contaminated compound normal distribution and the sample sizes are less than thirty. [ABSTRACT FROM AUTHOR]
- Published
- 1968
- Full Text
- View/download PDF
15. CONFIDENCE LIMITS FOR THE RELIABILITY OF SERIES SYSTEMS.
- Author
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El Mawaziny, A. H. and Buehler, R. J.
- Subjects
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CONFIDENCE intervals , *CONFIDENCE , *PROBABILITY theory , *EXPONENTIAL families (Statistics) , *APPROXIMATION theory , *FUNCTIONAL analysis , *POLYNOMIALS - Abstract
It is desired to set confidence limits for the probability of successful operation at least until time x[sub 0] of a series system of k dissimilar components. The components follow exponential failure laws, and data are available on failure times of components of each type. Exact confidence limits for k=2 have previously been given by Lentner and Buehler. The present paper deals with a large-sample approximation to the exact solution for arbitrary k. [ABSTRACT FROM AUTHOR]
- Published
- 1967
- Full Text
- View/download PDF
16. THE MOMENTS OF A DOUBLY NONCENTRAL t-DISTRIBUTION.
- Author
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Krishnan, Marakatha
- Subjects
- *
FUNCTIONAL analysis , *APPROXIMATION theory , *MATHEMATICAL functions , *DISTRIBUTION (Probability theory) , *STATISTICAL reliability , *PROBABILITY theory - Abstract
This paper gives analytic expressions for the moments and recurrence relations for the first four raw moments of a doubly noncentral t-distribution with v degrees of freedom and noncentrality parameters δ and λ. Table I provides numerical values of these moments for v = 2 (I) 20, λ = 2 (2) 8 (4) 20 and any suitable σ. Two approximations to the t"-distribution, involving the moments, are considered. [ABSTRACT FROM AUTHOR]
- Published
- 1967
- Full Text
- View/download PDF
17. APPLICATIONS OF PROBABILITY THEORY IN CRIMINALISTICS--II.
- Author
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Kingston, Charles R.
- Subjects
- *
FORENSIC sciences , *PROBABILITY theory , *BAYESIAN analysis , *APPROXIMATION theory , *STATISTICAL correlation , *ERROR , *THEORY of knowledge , *EVALUATION - Abstract
A general model for the evaluation of partial transfer evidence is developed in this article. This model is less restrictive than were the original models presented in an earlier paper, thus allowing it to be applied to a more general class of situations. A Bayesian approach to the problem of evidence evaluation is explored, and the probability of error calculated in the Bayesian model is shown to be a reasonable approximation to the probability of possible error calculated in the general model when these probabilities are small. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
18. ON THE F-TEST IN THE INTRABLOCK ANALYSIS OF A CLASS OF TWO ASSOCIATE PBIB DESIGNS.
- Author
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Giri, N.
- Subjects
- *
DISTRIBUTION (Probability theory) , *MOMENT problems (Mathematics) , *CHARACTERISTIC functions , *F-distribution , *APPROXIMATION theory , *PROBABILITY theory , *STATISTICAL sampling , *STATISTICS - Abstract
In this paper the first two moments of the ratio (treatment sum of squares)/(treatment sum of squares + error sum of squares)over all possible random assignment of treatments to the experimental plots, for a class of 2 associate PBIBD has been obtained. These two moments are compared with the corresponding moments of a continuous beta distribution to settle the question of approximating the randomization test by the usual F-test. It has been shown that a reasonable approximation to the randomization test based on the statistic F is equivalent to modifying the normal theory test by multiplying the numbers of d.f. of the F-distribution by a factor depending on the heterogeneity of the blocks. [ABSTRACT FROM AUTHOR]
- Published
- 1965
- Full Text
- View/download PDF
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