Normal approximations for binary lattice systems

1980 ◽  
Vol 17 (03) ◽  
pp. 674-685 ◽  
Author(s):  
Richard J. Kryscio ◽  
Roy Saunders ◽  
Gerald M. Funk
Keyword(s):  
Random Variables ◽  
Rectangular Lattice ◽  
Multivariate Normal ◽  
Lattice Systems ◽  
Binary Variables ◽  
Normal Approximations ◽  
Binary Random Variables ◽  
Binary Lattice ◽  
Parameter Values ◽  
Independent And Identically Distributed

Consider an array of binary random variables distributed over an m 1(n) by m 2(n) rectangular lattice and let Y 1(n) denote the number of pairs of variables d, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditions Y(n) = (Y 1(n), · ··, Yr (n)) is asymptotically multivariate normal for n large and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.

Normal approximations for binary lattice systems

1980 ◽  
Vol 17 (3) ◽  
pp. 674-685 ◽  
Author(s):  
Richard J. Kryscio ◽  
Roy Saunders ◽  
Gerald M. Funk
Keyword(s):  
Random Variables ◽  
Rectangular Lattice ◽  
Multivariate Normal ◽  
Lattice Systems ◽  
Binary Variables ◽  
Normal Approximations ◽  
Binary Random Variables ◽  
Binary Lattice ◽  
Parameter Values ◽  
Independent And Identically Distributed

Consider an array of binary random variables distributed over an m1(n) by m2(n) rectangular lattice and let Y1(n) denote the number of pairs of variables d, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditions Y(n) = (Y1(n), · ··, Yr(n)) is asymptotically multivariate normal for n large and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.

scholarly journals Bounds for the Distance Between the Distributions of Sums of Absolutely Continuous i.i.d. Convex-Ordered Random Variables with Applications

2009 ◽  
Vol 46 (1) ◽  
pp. 255-271 ◽  
Author(s):  
Tasos C. Christofides ◽  
Eutichia Vaggelatou
Keyword(s):  
Total Variation ◽  
Random Variables ◽  
Upper Bounds ◽  
Partial Sums ◽  
Convex Order ◽  
Absolutely Continuous ◽  
Normal Approximations ◽  
Independent And Identically Distributed

Let X1, X2,… and Y1, Y2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(Xi)=E(Yi), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1nXi and ∑i=1nYi. In the case where the distributions of the Xis and the Yis are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.

scholarly journals Bounds for the Distance Between the Distributions of Sums of Absolutely Continuous i.i.d. Convex-Ordered Random Variables with Applications

2009 ◽  
Vol 46 (01) ◽  
pp. 255-271
Author(s):  
Tasos C. Christofides ◽  
Eutichia Vaggelatou
Keyword(s):  
Total Variation ◽  
Random Variables ◽  
Upper Bounds ◽  
Partial Sums ◽  
Convex Order ◽  
Absolutely Continuous ◽  
Normal Approximations ◽  
Independent And Identically Distributed

Let X 1, X 2,… and Y 1, Y 2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(X i)=E(Y i), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1 n X i and ∑i=1 n Y i. In the case where the distributions of the X is and the Y is are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

scholarly journals The number of failed components in a coherent working system when the lifetimes are discretely distributed

Metrika ◽  
2021 ◽  
Author(s):  
Krzysztof Jasiński
Keyword(s):  
Random Variables ◽  
Continuous Case ◽  
Coherent System ◽  
Coherent Systems ◽  
Discrete Random Variables ◽  
Independent And Identically Distributed

AbstractIn this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ i = 0 , … , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.

scholarly journals Path Analysis for Binary Random Variables

2021 ◽  
pp. 004912412110312
Author(s):  
Martina Raggi ◽  
Elena Stanghellini ◽  
Marco Doretti
Keyword(s):  
Random Variables ◽  
Standard Errors ◽  
Marginal Effect ◽  
Direct And Indirect Effects ◽  
Direct Effects ◽  
P Values ◽  
Recursive System ◽  
Binary Random Variables ◽  
Log Odds ◽  
Research Questions

The decomposition of the overall effect of a treatment into direct and indirect effects is here investigated with reference to a recursive system of binary random variables. We show how, for the single mediator context, the marginal effect measured on the log odds scale can be written as the sum of the indirect and direct effects plus a residual term that vanishes under some specific conditions. We then extend our definitions to situations involving multiple mediators and address research questions concerning the decomposition of the total effect when some mediators on the pathway from the treatment to the outcome are marginalized over. Connections to the counterfactual definitions of the effects are also made. Data coming from an encouragement design on students’ attitude to visit museums in Florence, Italy, are reanalyzed. The estimates of the defined quantities are reported together with their standard errors to compute p values and form confidence intervals.

Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

2021 ◽  
Author(s):  
José Correa ◽  
Paul Dütting ◽  
Felix Fischer ◽  
Kevin Schewior
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Optimal Solution ◽  
Random Order ◽  
Secretary Problem ◽  
Maximum Value ◽  
Optimal Stopping Theory ◽  
Long Time ◽  
Object Of Study ◽  
Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

scholarly journals On the law of the iterated logarithm in the infinite variance case

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Infinite Variance ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

On the ordered partial sums of real random variables

1977 ◽  
Vol 14 (1) ◽  
pp. 75-88 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Markov Sequence ◽  
Stieltjes Transforms ◽  
Independent And Identically Distributed

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

Infinite dams with inputs forming a Markov chain

1968 ◽  
Vol 5 (1) ◽  
pp. 72-83 ◽  
Author(s):  
M. S. Ali Khan ◽  
J. Gani
Keyword(s):  
Markov Chain ◽  
Storage Systems ◽  
Random Variables ◽  
Time Intervals ◽  
Storage Theory ◽  
Annual Time ◽  
Theory Of Storage ◽  
Independent And Identically Distributed

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

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