Scheduling identical jobs on uniform parallel machines

We address the problem of scheduling n identical jobs on m uniform parallel machines to optimize scheduling criteria that are nondecreasing in the job completion times. It is well known that this can be formulated as a linear assignment problem, and subsequently solved in O ( n ³) time. We give a more concise formulation for minsum criteria, and show that general minmax criteria can be minimized in O ( n ²) time. We present faster algorithms, requiring only O ( n + m log m ) time for minimizing makespan and total completion time, O ( n log n ) time for minimizing total weighted completion time, maximum lateness, total tardiness and the weighted number of tardy jobs, and O ( n log² n ) time for maximum weighted tardiness. In the case of release dates, we propose an O ( n log n ) algorithm for minimizing makespan, and an O ( mn ^2m+1) time dynamic programming algorithm for minimizing total completion time.     

Functional laws of the iterated logarithm for small increments of empirical processes

Let F , denote the uniform empirical distribution based on the first n ≥ 1 observations from an i.i.d. sequence of uniform (0, 1) random variables. We describe the almost sure limiting behavior of the sets of increment functions {F_n(t + h_n.) - F_n(t): 0 ≤ t ≤ 1 - h_n}, when {h_n: n ≥ 1) is a nonincreasing sequence of constants such that nh_n /log n ← 0.     

On the existence of bivariate moments of lower order given the existence of moments of higher order

Dit artikel geeft een voorbeeld van een twee-dimensionale waarschijnlijkheidsdcht-heid met de eigenschap, dat het bestaan van momenten van de vorm E(g^ky^l ) niet het bestaan garandeert van alle momenten van lagere orde, E(g^myⁿ ) met 0myⁿ ) met 0 < m < k en 0 < n < l is, dat de drie momenten E(x^k), E(y^l ) en E(x^ky^l ) bestaan, alsmede een iets algemener resultaat.     

An improved empirical Bayes test for positive exponential families

We exhibit an empirical Bayes test δ^* _n for a decision problem using a linear error loss in a class of positive exponential families. This empirical Bayes test δ^* _n possesses the asymptotic optimality, and its associated regret converges to zero with rate n ⁻¹(ln n )⁶ This rate of convergence improves the previous results in the literature in the sense that a faster rate of convergence is achieved under much weaker conditions. Examples are presented to illustrate the performance of the empirical Bayes test δ^* _n     

Fourier inversion and the Hausdorff distance

This paper continues research done by F.H. Ruymgaart and the author. For a function f on R ^d we consider its Fourier transform F f and the functions f_M (M>0) derived from F f by the formula f_M(x) =( F( ε_M · F f ))(− x );, where the ε_M are suitable integrable functions tending to 1 pointwise as M →∞. It was shown earlier that, relative to a metric d _H , analogous to the Hausdorff distance between closed sets, one has d _H (f_M, f) = O( M ^−½) for all f in a certain class. We now show that, for such f , the estimate O( M ^−½) is optimal if and only if f has a discontinuity point.     

Thin layer chromatography, a case study

This paper gives an account of the collaboration between two mathematical statisticians and a toxicologist (the second author) interested in thin layer chromatography (TLC). A TLC "system" consists of a medium through which a solvent is transported. If a solution of some (toxic) sample is applied to the medium, then the components are carried forward by the solvent over different distances. Section 1 describes the concept of a data bank which provides standard values for the degrees of migration characteristic for each of m well-studied substances in each of n systems. Sections 2–5 are mainly devoted to the construction of The "best design(s)"{ j ₁*… j _k * } of k systems from the n available ones. The attention is restricted to the situation that an unidentified sample exclusively contains one of the m substances covered by the data bank and produces the scores xj … ^xj_k in the systems j,… j _krespectively. Three different approaches to the identification problem were successively considered. Each approach leads to a class of procedures and their performances. The performance of the optimum procedure can be used to define the performance of any of the ( ⁿ_k ) designs ( j ₁… j_k }. The latter performance is maximized in order to determine { j ₁*.,., j_k* }. In practice usually data is obtained for mixtures instead of single. pure substances. Section 6 gives some tentative theory for the evaluation of such data.     

First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties

Some properties of a first-order integer-valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self-decomposability and unimodality of the 1-dimensional marginals of the process {X_n} generated according to the scheme X_n=α° X _n-i +e_n, where α° X _n-1 denotes a sum of X_{n - 1}, independent 0 - 1 random variables Y^(n-1), independent of X _n-1 with Pr -( y ^{(n - 1)}= 1) = 1 - Pr ( y ^(n-i)= 0) =α. The distribution of the innovation process ( e _n) is obtained when the marginal distribution of the process ( X _n) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.     

Een indelingsprobleem

Summary  A branch and bound algorithm is given to solve the following problem: To each pair of elements (i,j) from a set X ={l,…, n } a number r _ij with r _ij≥ 0, r _ij= r _ij and r _ij= 0 has been assigned. Find a prescribed number of disjoint subsets P ₁…, P _m from X , such that

Experiments indicate that an optimal solution is usually found in a small number of iterations, but the verification may be rather time consuming.
The algorithm may be used to find the minimum value of m for which a partitioning of X with z = 0 exists. The algorithm appears to be efficient for finding this 'chromatic number of a graph'.     

Characteristic properties of order statistics based on random sample size from an exponential distribution

Suppose X₁, X₂, X_m is a random sample of size m from a population with probability density function f (x), x > 0), and let X_{1, m}< × 2, m <… < X_{m, m} be the corresponding order statistics.
We assume m is an integer-valued random variable with P( m = k ) = p (1- p )^k-1, k = 1,2,… and 0 < p < 1. Two characterizations of the exponential distribution are given based on the distributional properties of X_{l, m}.     

The effect of rounding off in samples

Dit artikel bevat een elementaire behandeling van afrondingseffecten in steekproeven.
Zoals men verwacht, neemt de variantie van het steekproefgermddelde door af ronding der waarnemingen toe met een factor 1 + b ²/12σ _x ², waarbij b het afrondings- interval en σ _x ² de stamvariantie is, in analogie met de bekende correcties van Sheppard. Verder wordt het effect van afronding op de variantie van de steek-proefvariantie onderzocht. Er wordt op gewezen, dat de afrondingsfout ε in één steekproef gecorreleerd is met de afwijking van de nauwkeurige — d.w.z. niet afgeronde — stochastische variabele x van zijn gemiddelde μ, hoewel deze correlatie gemiddeld nul is. Het gevolg is, dat de variantie van de steekproefvariantie toeneemt. Bij "gewone" waarschijnlijkheidsverdelingen, die continue naar nul gaan bij x →, is deze toename gelijk aan
1/ n (1/3 b ²σ _x ²+ 1/180 b ⁴)
waarbij n de grootte van de steekproef is. Bij normale verdelingen is het effect ongeveer twee maal zo groot als de toename volgens de Sheppard's correctie van de afgeronde variantie.     

On a crossroad of resampling plans: bootstrapping elementary symmetric polynomials

We investigate the validity of the bootstrap method for the elementary symmetric polynomials S ^{( k )} _n =( ⁿ _k )⁻¹Σ_{1≤ i 1}< ... < i _k ≤ n X _{i 1} ... X _{i k} of i.i.d. random variables X ₁, ..., X _n . For both fixed and increasing order k , as n→∞ the cases where μ=E X ₁[moe2]0, the nondegenerate case, and where μ=E X ₁=0, the degenerate case, are considered.     

The asymptotic distribution of the sum of weighted squared residuals in binary choice models

It is claimed by some authors that the distribution of the sum of weighted squared residuals, used as a goodness of fit measure in binary choice models, behaves for large n as a x²_n– k–1 distribution. This claim seems to be based on a false analogy with the well–known Pearson x² statistic for frequency tables with a fixed number of cells and cell sizes tending to infinity. We derive the asymptotic (normal) distribution and show that the approximation by the x² distribution in general will not be valid. A new x² test is proposed based on the asymptotic normality of the measure.     

Approximation for the cumulative and inverse gamma distribution

The gamma distribution function can be expressed in terms of the Normal distribution and density functions with sufficient accuracy for most practical purposes.
The distribution function for the density x^Λ-1e^-x/μ/μ^ΛΓ(A) on 0 -R(Λ){(1 + 1/1 2Λ) φ(z) + 11 -z/4Λ^1/2+2(z²+ 2)/45Λ] φ(z) /3 Λ^1/2} where φ(z)≅1/[1 +_e^{-2z(√2/π+z2 /28)}] and φ(z) = e^{-z2 /2}/√2π are the Normal distribution and density functions, y is the appropriate root of y-y²/6+y³/36-y⁴/270= In (x/Λμ), z= Λ^1/2 y, and R( Λ) is the remainder term in Stirling's approximation for In Γ(Λ).     

On the generalized Poisson distribution

We use Euler's difference lemma to prove that, for θ > 0 and 0 ≤λ < 1, the function P _n defined on the non-negative integers by
P _n (θ, λ) = [θ(θ + n λ) ^{n −1}/ n !]e^{− n λ−θ}
defines a probability distribution, known as the Generalized Poisson Distribution.     

A note on the upper bound for the distance in total variation between the binomal and the Poisson distribution

Abstract  The total variation distance between the binomial B ( n, p ) distribution and the Poisson P ( np ) distribution is smaller than 2 ^1/2 p (1- p )^-1/2 according to V ERVAAT [4], [5]. We shall sharpen this inequality by using a result due to K EMPERMAN [1], C SISZÁR [2] and K ULLBACK [3].     

De omvang van de steekproef bij een enkelvoudig steekproefsysteem

Summary  (Sample size for a single sampling scheme).
The operating characteristic of a sampling scheme may be specified by the producers 1 in 20 risk point ( p ₁), at which the probability of rejecting a batch is 0.05, and the consumers 1 in 20 risk point ( p ₂) at which the probability of accepting a batch of that quality is also 0.05.
A nomogram is given (fig. 2) to determine for single sampling schemes and for given values of p₁ and p ₂ the necessary sample size ( n ) and the allowable number of defectives in the sample ( c ).
The nomogram may reversedly be used to determine the producers and consumers 1 in 20 risk points for a given single sampling scheme.
The curves in this nomogram were computed from a table of percentage points of the χ² distribution. For v > 30 Wilson and Hilferty's approximation to the χ² distribution was used.     

Asymptotic properties of estimators for the volume fractions of jointly stationary random sets

In the present paper, we show how a consistent estimator can be derived for the asymptotic covariance matrix of stationary 0–1-valued vector fields in R ^d , whose supports are jointly stationary random closed sets. As an example, which is of particular interest for statistical applications, we consider jointly stationary random closed sets associated with the Boolean model in R ^d such that the components indicate the frequency of coverage by the single grains of the Boolean model. For this model, a representation formula for the entries of the covariance matrix is obtained.     

Discrete Spacings

Consider a string of n positions, i.e. a discrete string of length n . Units of length k are placed at random on this string in such a way that they do not overlap, and as often as possible, i.e. until all spacings between neighboring units have length less than k . When centered and scaled by n ^−1/2 the resulting numbers of spacings of length 1, 2,…,  k −1 have simultaneously a limiting normal distribution as n →∞. This is proved by the classical method of moments.     

Large deviation principles and loglog laws for observation processes

Let ( X_n, n ≥ 1) be an i.i.d. sequence of positive random variables with distribution function H . Let φ _H := {(n, X_n ), n ≥ 1) be the associated observation process. We view φ h as a measure on E := [0, ∞) ∞ (0, φ] where φ_H (A) is the number of points of φ _H which lie in A . A family ( V_s, s> 0) of transformations is defined on E in such a way that for suitable H the distributions of ( V_sφ_H, S > 0) satisfy a large deviation principle and that a related Strassen-type law of the iterated logarithm also holds. Some consequent large deviation principles and loglog laws are derived for extreme values. Similar results are proved for φ _H replaced by certain planar Poisson processes.     

Rendering Alternative Offerings Less Profitable with Resale Price Maintenance

When it was legal, resale price maintenance (RPM) was commonly observed on items such as aspirin, pens, pencils, toothpaste, soap, shaving cream, and milk. In providing a theory that is based on compensating retailers for their opportunity cost of shelf space, and that does not hinge on the existence of externalities in nonprice competition, this article explains why a manufacturer might impose RPM on these and many other products. By contrast, the use of RPM on food, grocery, and drug store items is not easily explained by standard theories such as free riding on presale services and quality certification by high-priced retailers.

The typical supermarket has room for fewer than 25,000 products. Yet there are some 100,000 available, and between 10,000 and 25,000 items aye introduced each year. ¹