This paper deals with the inversive congruential method with power of two modulus m for generating uniform pseudorandom numbers. Statistical independence properties of the generated sequences are studied based on the distribution of triples of successive pseudorandom numbers. It is shown that, on the average over the parameters in the inversive congruential method, the discrepancy of the corresponding point sets in the unit cube is of an order of magnitude between m \Gamma1=2 and m \Gamma1=2 (log m)³. The method of proof relies on a detailed discussion of the properties of certain exponential sums.

If x,)...) X, are independent identically distributed Rd-valued random vectors with probability measure p and empirical probability measure p,, and if QZ is a subset of the Bore1 sets on Rd, then we show that P{sup,, ~

This paper investigates the stability of optimal-solution sets to stochastic programs with complete recourse, where the underlying probability measure is understood as a parameter varying in some space of probability measures. In [29] Shapiro has proved Lipschitz upper semicontinuity of the solution set mapping. Inspired by this result we introduce a subgradient distance for probability distributions and establish the persistence of optimal solutions. For a subclass of recourse models we show that the solution set mapping is (Hausdorff) Lipschitz continuous with respect to the subgradient distance. Moreover, the subgradient distance is estimated above by the Kolmogorov-Smirnov distance of certain distribution functions related to the recourse model. The Lipschitz continuity result is illustrated by verifiable sufficient conditions for stochastic programs to belong to the mentioned subclass and by examples showing its validity and limitations. Finally, the Lipschitz continuity result ...

We study bounds on the classical ∗-discrepancy and on its inverse. Let n ∗ ∞(d, ε) be the inverse of the ∗-discrepancy, i.e., the minimal number of points in dimension d with the ∗-discrepancy at most ε. We prove that n ∗ ∞(d, ε) depends linearly on d and at most quadratically on ε −1. We present three upper bounds on n ∗ ∞(d, ε), all of them are based on probabilistic arguments and therefore they are non-constructive. The linear in d upper bound directly follows from deep results of the theory of empirical processes but it contains an unknown multiplicative factor. Two other upper bounds are without unknown factors but do not yield the linear (in d) upper bound. One upper bound is based on an average case analysis for the Lp-star discrepancy and our numerical results seem to indicate that it gives the best estimates for specific values of d and ε. We also present two lower bounds on n ∗ ∞(d, ε). For lower bounds, we allow arbitrary coefficients in the discrepancy formula. We prove that n ∗ ∞(d, ε) must be of order d log ε −1 and, roughly, of order d λ ε −(1−λ) for any λ ∈ (0, 1).

A sequence of s-dimensional quasi random numbers fills the unit cube evenly at a much faster rate than a sequence of pseudo uniform deviates does. It has been successfully used in many Monte Carlo problems to speed up the convergence. Direct use of a sequence of quasi random numbers, however, does not work in Gibbs samplers because the successive draws are now dependent. We develop a quasi random Gibbs algorithm in which a randomly permuted quasi random sequence is used in place of a sequence of pseudo deviates. One layer of unnecessary variation in the Gibbs sample is eliminated. A simulation study with three examples shows that the proposed quasi random algorithm provides much tighter estimates of the quantiles of the stationary distribution and is about 4–25 times as efficient as the pseudo algorithm. No rigorous theoretical justification for the quasi random algorithm, however, is available at this point.

Abstract. Recent developments in modeling stream of variation in multistage manufacturing system along with the urgent need for yield enhancement in the semiconductor industry has led to complex large scale simulation problems in design and performance prediction, thus challenging current Monte Carlo (MC) based simulation techniques. MC method prevails in statistical simulation approaches for multi-dimensional cases with general (i.e., non-Gaussian) distributions and/or complex response functions. A method is proposed based on number theory (NT-net) to reduce computing effort and the variability of MC’s results in tolerance design and circuit performance simulation. The sampling strategy is improved by introducing NT-net that can provide better convergent rate over MC. The new method is presented and verified using several case studies, including analytical and industrial cases of a filter design and analyses of a four-bar mechanism. Results indicate a 90–95 % reduction of computation effort with significant improvement in accuracy that can be achieved by the proposed technique. Key Words: tolerance analysis, Monte Carlo simulation

deviations for Ko1mogorov-Smirnov statistics, obtained earlier by Gnedenko, Karo1uk and Skorokhod, and by Kiefer and Wo1fowitz, among others. probability of moderate deviations and reverse sub-martingales. permitted for any purpose of the U.

In the context of sequential (point as well as interval) estimation, a general formulation of permutation-invariant stopping rules is considered. These stopping rules lead to savings in the ASN at the cost of some elevation of the associated risk--a phenomenon which may be attributed to the violation of sufficiency principle. For the (point and interval) sequential estimation of the mean of a normal distribution, it is shown.e that such permutation-invariant stopping rules may lead to a substantial sav1ng 1n the ASN with only a small increase in the associated risk •.e

For a general class of statistics expressible as extrema of certain sample functions, an almost sure invariance principle, particularly useful in the context of the law of iterated logarithm and the probabilities of moderate deviations, is established, and its applications are stressed.

The classical jackknifing based on a resampling scheme with deletion of of one observation at a time serves the dual purpose of bias reduction and variance estimation. Delete-k jackknifing, for some k ~ 1, is a variant of this scheme. In the light of second order asymptotic distributional representations, it is shown that for second order (compact) differentiable statistical functionals, for any k = o(n), delete-k jackknifing behaves very much similar to the classical one. This raises the question: To what degree delete-k jackknifing is preferable in practice?!.!~E:~~~~E~~~. Let Xl' · • •,X n be n independent and identically distributed random variables (i.i.d.r.v.) with a distribution function (d.f.) F, defined on the real line R. Let F be the ¢ample (emp~eatJ d. n. For a general statistical n functional e T(F), a natural estimator is T = T(F). In general, T may not be n n n