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19
NonUniform Random Variate Generation
, 1986
"... Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various ..."
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Cited by 623 (21 self)
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Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.
The equivalence of weak, strong and complete convergence in L1 for kernel density estimates
 ANNALS OF STATISTICS
, 1983
"... Let f be a density on R", and let f, be the kernel estimate off, fn(x) _ ( nh d) ' ~ = 1 K((x X1)lh) where h = h n is a sequence of positive numbers, and K is an absolutely integrable function with f K(x) dx =1. Let J, = f l f,(x) f (x) ( dx. We show that when limnh = 0 and limn nh d = o ..."
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Cited by 23 (11 self)
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Let f be a density on R", and let f, be the kernel estimate off, fn(x) _ ( nh d) ' ~ = 1 K((x X1)lh) where h = h n is a sequence of positive numbers, and K is an absolutely integrable function with f K(x) dx =1. Let J, = f l f,(x) f (x) ( dx. We show that when limnh = 0 and limn nh d = oo, then for every e> 0 there exist constants r, no> 0 such that P(Jn> e) < _ exp(rn), n? no. Also, when J,p 0 in probability as np oo and K is a density, then lim nh = 0 and limnnh d = oo. 1. Introduction. The
Convergence rates of evolutionary algorithms for a class of convex objective functions
 Control and Cybernetics
, 1997
"... Probabilistic optimization algorithms that mimic the process of biological evolution are usually subsumed under the term `evolutionary algorithms. ' This work extends the convergence theory of evolutionary algorithms by presenting a su cient convergence condition for those evolutionary algorithms th ..."
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Cited by 21 (1 self)
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Probabilistic optimization algorithms that mimic the process of biological evolution are usually subsumed under the term `evolutionary algorithms. ' This work extends the convergence theory of evolutionary algorithms by presenting a su cient convergence condition for those evolutionary algorithms that do not necessarily generate a sequence of feasible points such that the associated objective function values decrease monotonically to the global minimum. Moreover, it is investigated how fast the sequence of objective function values generated by anevolutionary algorithm approaches the minimum of strongly convex functions in a probabilistic sense. The theoretical analysis presented here distinguishes from related studies in three points: First, it does not require advanced calculus. Second, only the rst partial derivatives of the objective function are assumed to exist. Third, one obtains sharp bounds on the convergence rates for a class of functions being a superset of the class of quadratic functions with positive de nite Hessian matrix. 1
Local Convergence Rates of Simple Evolutionary Algorithms with Cauchy Mutations
 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 1998
"... The standard choice for mutating an individual of an evolutionary algorithm with continuous variables is the normal distribution; however other distributions, especially some versions of the multivariate Cauchy distribution, have recently gained increased popularity in practical applications. Here t ..."
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Cited by 18 (1 self)
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The standard choice for mutating an individual of an evolutionary algorithm with continuous variables is the normal distribution; however other distributions, especially some versions of the multivariate Cauchy distribution, have recently gained increased popularity in practical applications. Here the extent to which Cauchy mutation distributions may affect the local convergence behavior of evolutionary algorithms is analyzed. The results show that the order of local convergence is identical for Gaussian and spherical Cauchy distributions, whereas nonspherical Cauchy mutations lead to slower local convergence. As a byproduct of the analysis some recommendations for the parametrization of the selfadaptive step size control mechanism can be derived.
Mean Value Analysis for Computer Systems with Variabilities in Workload
, 1996
"... When evaluating the performance of computer systems, often uncertainties or variabilities in service demands may be observed. Applying well known mean value analysis (MVA) for single or multiclass queueing network models of such systems is inappropriate and ineffective, because these models fail to ..."
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Cited by 11 (8 self)
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When evaluating the performance of computer systems, often uncertainties or variabilities in service demands may be observed. Applying well known mean value analysis (MVA) for single or multiclass queueing network models of such systems is inappropriate and ineffective, because these models fail to represent variations within a class. This paper proposes to use histograms for characterizing model parameters that are associated with uncertainty or variability and presents an adaptation of the single class MVA algorithm, which traditionally accepts single (mean) values for service demands, so that one or more input parameters can be specified as a histogram. The adapted algorithm generates a histogram output for the performance measures, thus providing a more detailed information (e.g. percentile values) than the mean values obtained from conventional MVA. The proposed technique is demonstrated on selected examples in different problem domains. It is shown, that the computational comple...
A NOTE ON THE L1 CONSISTENCY OF VARIABLE KERNEL ESTIMATES
, 1985
"... A sample X,, •.., X, ~ of i.i.d. R dvalued random vectors with common density f is used to construct the density estimate fn(x) _ ( 1/n) ~n, HndK((x Xi)/Hm), where K is a given density on Rd, and the H's are positive functions of n, i and X,, • • • , X „ (but not of x). The H, 's can be thou ..."
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Cited by 8 (7 self)
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A sample X,, •.., X, ~ of i.i.d. R dvalued random vectors with common density f is used to construct the density estimate fn(x) _ ( 1/n) ~n, HndK((x Xi)/Hm), where K is a given density on Rd, and the H's are positive functions of n, i and X,, • • • , X „ (but not of x). The H, 's can be thought of as locally adapted smoothing parameters. We give sufficient conditons for the weak convergence to 0 of f I fn f I for all f. This is illustrated for the estimate of Breiman, Meisel and Purcell (1977).
HistogramBased Performance Analysis for Computer Systems with Variabilities or Uncertainties in Workload
 in Workload. Res. Rep. SCE9522, Carleton Univ
, 1995
"... A conventional analytic model used for evaluating the performance of computer and communication systems accepts single values as model inputs and computes a single value for each performance measure of interest. However uncertainties regarding parameter values exist in different situations such as d ..."
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Cited by 7 (4 self)
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A conventional analytic model used for evaluating the performance of computer and communication systems accepts single values as model inputs and computes a single value for each performance measure of interest. However uncertainties regarding parameter values exist in different situations such as during early stages of system design. Although the clients in a system are statistically identical factors such as the time of the day and the current size of the data files can introduce variabilities in service demands for the server devices. Existence of uncertainties or variabilities in service demands makes the use of a single mean value for each model parameter inappropriate causing the conventional modelling approach to become ineffective. This paper proposes to use histograms for characterizing one or more model parameters that are associated with such uncertainty or variability and demonstrates its application with separable queueing network models. A histogram consists of a number o...
A Universal Lower Bound For The Kernel Estimate
, 1989
"... Let f, be the kernel density estimate with arbitrary smoothing factor h and arbitrary (absolutely integrable) kernel K, based upon an i.i.d. sample of size n drawn from a density f. It is shown that ) inf E([If,fl > , h,K,f " 52V and that liminf// inf E([If.fl)>. ..."
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Cited by 6 (4 self)
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Let f, be the kernel density estimate with arbitrary smoothing factor h and arbitrary (absolutely integrable) kernel K, based upon an i.i.d. sample of size n drawn from a density f. It is shown that ) inf E([If,fl > , h,K,f " 52V and that liminf// inf E([If.fl)>.
empirical probability measure can converge in the total variation sense for all distributions
 Annals of Statistics
, 1990
"... For any sequence of empirical probability measures {p) on the Borel sets of the real line and any 6> 0, there exists a singular continuous probability measure µ such that inf sup µ n (A) µ (A) I> 2 6 almost surely. n A We consider a probability measure µ on the Borel sets of the real line, from wh ..."
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Cited by 6 (1 self)
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For any sequence of empirical probability measures {p) on the Borel sets of the real line and any 6> 0, there exists a singular continuous probability measure µ such that inf sup µ n (A) µ (A) I> 2 6 almost surely. n A We consider a probability measure µ on the Borel sets of the real line, from which we draw an i.i.d. sample X 1,..., X,. An empirical probability measure µ n is a probability measure on the same Borel sets and for a fixed set A, µ(A) is a measurable function of the data X 1,..., X n. In particular, we are interested in the total variation Tn 4 suPllt n (A) A(A)I, where the supremum is over all the Borel sets. By considering suprema over left infinite intervals only, it is easy to see that Tn> sup x IFn(x) F(x)I, the GlivenkoCantelli norm, where Fn and F are the distribution functions corresponding to µ and µ, respectively. The standard empirical measure, defined by 0 1 µn ( A) ~ I[Xl E AJ, is atomic in nature. Hence, whenever µ is continuous, we have Tn = 1 almost surely for all n. This is in stark contrast with the GlivenkoCantelli norm, which is known to converge to zero almost surely as n ~ oo (by the GlivenkoCantelli theorem). If µ is atomic, it is quite obvious that Tn ~ 0 almost surely as n ~ 0. In order for Tn to be small when µ is nonatomic, we should not use the standard empirical measure. For example, for absolutely continuous µ (with density f), SchefF 's lemma [SchefF (1947) states that Tn n n L=i 2 fIfn f l,
Density estimation in linear time
"... We consider the problem of choosing a density estimate from a set of densities F, minimizing the L1distance to an unknown distribution. Devroye and Lugosi [DL01] analyze two algorithms for the problem: Scheffé tournament winner and minimum distance estimate. The Scheffé tournament estimate requires ..."
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Cited by 2 (0 self)
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We consider the problem of choosing a density estimate from a set of densities F, minimizing the L1distance to an unknown distribution. Devroye and Lugosi [DL01] analyze two algorithms for the problem: Scheffé tournament winner and minimum distance estimate. The Scheffé tournament estimate requires fewer computations than the minimum distance estimate, but has strictly weaker guarantees than the latter. We focus on the computational aspect of density estimation. We present two algorithms, both with the same guarantee as the minimum distance estimate. The first one, a modification of the minimum distance estimate, uses the same number (quadratic in F) of computations as the Scheffé tournament. The second one, called “efficient minimum lossweight estimate,” uses only a linear number of computations, assuming that F is preprocessed. We then apply our algorithms to bandwidth selection for kernel estimates and binwidth selection for histogram estimates, yielding efficient procedures for these problems. We also give examples showing that the guarantees of the algorithms cannot be improved and explore randomized algorithms for density estimation. 1