Proof of the antithetic-variates theorem for unbounded functions

1979 ◽  
Vol 86 (3) ◽  
pp. 477-479 ◽  
Author(s):  
James R. Wilson
Keyword(s):  
Marginal Distribution ◽  
Functional Dependence ◽  
Finite Variance ◽  
Response Functions ◽  
Unbiased Estimators ◽  
Monte Carlo Techniques ◽  
Joint Distributions ◽  
Antithetic Variates ◽  
Borel Functions ◽  
Image Position

The method of antithetic variates introduced by Hammersley and Morton (2) is one of the most widely used Monte Carlo techniques for estimating an unknown parameter θ. The basis for this method was established by Hammersley and Mauldon(l).in the case of unbiased estimators with the formwhere each of the variates ξj is required to have a uniform marginal distribution over the unit interval [0,1]. By assuming that n = 2 and that the gj are bounded Borel functions, Hammersley and Mauldon showed that the greatest lower bound of var (t) over all admissible joint distributions for the variates ξj can be approached simply by arranging an appropriate strict functional dependence between the ξj. Handscomb(3) extended this result to the case of n > 2 bounded antithetic variates gj(ξj). In many experiments involving distribution sampling or the simulation of some stochastic process over time, the response functions gry are unbounded. This paper further extends the antithetic-variates theorem to include the case of n ≥ 2 unbounded antithetic variates gj(ξj) each with finite variance.

scholarly journals Unbiased estimation of the solution to Zakai’s equation

2020 ◽  
Vol 26 (2) ◽  
pp. 113-129
Author(s):  
Hamza M. Ruzayqat ◽  
Ajay Jasra
Keyword(s):  
Continuous Time ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Linear Filtering ◽  
Multilevel Monte Carlo ◽  
Zakai Equation ◽  
Unbiased Estimators ◽  
First Order ◽  
Multilevel Monte Carlo Method ◽  
Filtering Problem

AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.

On the estimation of a harmonic component in a time series with stationary dependent residuals

1973 ◽  
Vol 5 (02) ◽  
pp. 217-241 ◽  
Author(s):  
A. M. Walker
Keyword(s):  
Time Series ◽  
Least Squares ◽  
Least Squares Method ◽  
Harmonic Component ◽  
Asymptotic Distributions ◽  
Finite Variance ◽  
Rigorous Derivation ◽  
Image Position ◽  
Vector Valued ◽  
Special Case

Let observations (X 1, X 2, …, Xn ) be obtained from a time series {Xt } such that where the ɛt are independently and identically distributed random variables each having mean zero and finite variance, and the gu (θ) are specified functions of a vector-valued parameter θ. This paper presents a rigorous derivation of the asymptotic distributions of the estimators of A, B, ω and θ obtained by an approximate least-squares method due to Whittle (1952). It is a sequel to a previous paper (Walker (1971)) in which a similar derivation was given for the special case of independent residuals where gu (θ) = 0 for u > 0, the parameter θ thus being absent.

scholarly journals Exhaustive operators and vector measures

1975 ◽  
Vol 19 (3) ◽  
pp. 291-300 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Vector Measures ◽  
Borel Functions ◽  
Image Position ◽  
Real Topological Vector Space ◽  
Sequence Of Functions

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 andthen Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representationwhere μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

scholarly journals The general theory of canonical correlation and its relation to functional analysis

1961 ◽  
Vol 2 (2) ◽  
pp. 229-242 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Linear Combination ◽  
Canonical Correlation ◽  
Random Variables ◽  
Random Variable ◽  
Finite Variance ◽  
Finite Sample ◽  
Standard Description ◽  
Joint Distributions ◽  
Gaussian Case ◽  
The Relationship

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variablesxs, and any linear combination ofqrandom variablesytinsofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, forp=q= 1, to include a description of the correlation of any function of a random variablexand any function of a random variabley(both functions having finite variance) for a class of joint distributions ofxandywhich is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations wherepandqare not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

A new autoregressive time series model in exponential variables (NEAR(1))

1981 ◽  
Vol 13 (4) ◽  
pp. 826-845 ◽  
Author(s):  
A. J. Lawrance ◽  
P. A. W. Lewis
Keyword(s):  
Time Series ◽  
Autoregressive Model ◽  
Marginal Distribution ◽  
Sample Path ◽  
Cross Coupling ◽  
Time Series Model ◽  
Autoregressive Time Series ◽  
First Order ◽  
Joint Distributions ◽  
Two Parameter

A new time series model for exponential variables having first-order autoregressive structure is presented. Unlike the recently studied standard autoregressive model in exponential variables (ear(1)), runs of constantly scaled values are avoidable, and the two parameter structure allows some adjustment of directional effects in sample path behaviour. The model is further developed by the use of cross-coupling and antithetic ideas to allow negative dependency. Joint distributions and autocorrelations are investigated. A transformed version of the model has a uniform marginal distribution and its correlation and regression structures are also obtained. Estimation aspects of the models are briefly considered.

On the relative merits of correlated and importance sampling for Monte Carlo integration

1965 ◽  
Vol 61 (2) ◽  
pp. 497-498 ◽  
Author(s):  
John H. Halton
Keyword(s):  
Importance Sampling ◽  
Measure Space ◽  
Finite Measure ◽  
Monte Carlo Integration ◽  
Close Approximation ◽  
Unbiased Estimators ◽  
Finite Measure Space ◽  
Correlated Sampling ◽  
Approach Zero ◽  
Image Position

Given a totally finite measure space (S, S, μ) and two μ-integrable, non-negative functions f(x) and φ(x) defined in S, such that whenthenwe define correlated sampling as the technique of estimatingby sampling an estimator functionwhere ξ is uniformly distributed in S with respect to μ (i.e. for any T ∈ S, p(T) = μ(T)/μ(S) is the probability that ξ lies in T): and importance sampling as estimating L by sampling the estimator functionwhere η is distributed in S with probability density φ(x)/ΦThen, clearly,It follows that υ(ξ) and ν(η) are both unbiased estimators of L, and that their variances can both be made to approach zero arbitrarily closely by making φ(x) a sufficiently close approximation to f(x).

Stationary joint distributions arising in the analysis of the M/G/1 queue by the method of the imbedded markov chain

1966 ◽  
Vol 3 (2) ◽  
pp. 512-520 ◽  
Author(s):  
J. H. Jenkins
Keyword(s):  
Markov Chain ◽  
Generating Functions ◽  
Joint Distribution ◽  
Marginal Distribution ◽  
Probability Generating Function ◽  
Joint Distributions ◽  
Left Behind ◽  
Markov Chain Theory ◽  
Imbedded Markov Chain ◽  
Chain Theory

SummaryProbability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.

scholarly journals An ordering inequality for exchangeable random variables

1986 ◽  
Vol 18 (01) ◽  
pp. 274-276
Author(s):  
G. S. Watson
Keyword(s):  
Random Variables ◽  
Finite Variance ◽  
Exchangeable Random Variables ◽  
Image Position

Let X 1, · ··, Xn be exchangeable random variables with finite variance and two sequences of constants satisfying a 1≦···≦an , b 1≦···≦bn . Suppose that a ′ 1, ···, a ′ n is a rearrangement of a 1, ···, an and that g(x) is a non-decreasing function. Then

scholarly journals Random Fields with Pólya Correlation Structure

2014 ◽  
Vol 51 (4) ◽  
pp. 1037-1050 ◽  
Author(s):  
Richard Finlay ◽  
Eugene Seneta
Keyword(s):  
Random Fields ◽  
Cross Correlation ◽  
Marginal Distribution ◽  
Gaussian Random Fields ◽  
Correlation Structure ◽  
Finite Variance ◽  
Infinitely Divisible ◽  
New Class ◽  
Joint Characteristic ◽  
Non Gaussian

We construct random fields with Pólya-type autocorrelation function and dampened Pólya cross-correlation function. The marginal distribution of the random fields may be taken as any infinitely divisible distribution with finite variance, and the random fields are fully characterized in terms of their joint characteristic function. This makes available a new class of non-Gaussian random fields with flexible correlation structure for use in modeling and estimation.

scholarly journals An ordering inequality for exchangeable random variables

1986 ◽  
Vol 18 (1) ◽  
pp. 274-276
Author(s):  
G. S. Watson
Keyword(s):  
Random Variables ◽  
Finite Variance ◽  
Exchangeable Random Variables ◽  
Image Position

Let X1, · ··, Xn be exchangeable random variables with finite variance and two sequences of constants satisfying a1≦···≦an, b1≦···≦bn. Suppose that a′1, ···, a′n is a rearrangement of a1, ···, an and that g(x) is a non-decreasing function. Then

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