scholarly journals Asymptotic Properties of Longitudinal Weighted Averages for Strongly Mixing Data

2017 ◽  
Vol 9 (2) ◽  
pp. 65
Author(s):  
Brahima Soro ◽  
Ouagnina Hili ◽  
Sophie Dabo- Niang
Keyword(s):  
Covariance Function ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Weighted Averages ◽  
Strongly Mixing

We present general results of consistency and normality of a real-valued longitudinal random variable. We suppose that this random variable is some formed weighted averages of alpha-mixing data. The results can be applied to within-subject covariance function.

scholarly journals Asymptotic Properties for Weighted Averages of Longitudinal Dependent Data

2018 ◽  
Vol 7 (6) ◽  
pp. 100
Author(s):  
Brahima Soro ◽  
Ouagnina Hili ◽  
Youssouf Diagana
Keyword(s):  
Longitudinal Data ◽  
Covariance Function ◽  
Kernel Method ◽  
Asymptotic Properties ◽  
Dependent Data ◽  
Independent Data ◽  
Function Estimation ◽  
Mixing Conditions ◽  
Weighted Averages ◽  
Nonparametric Kernel

This paper presents a set of normality general results for kernel weighted averages. We extend existing literature for independent data (Yao, 2007) to stationary dependent longitudinal data. The asymptotic properties of proposed weighted averages are investigate under α-mixing conditions. These results are useful for covariance function estimation based on nonparametric kernel method.

Asymptotic properties of the number of replications of a paired comparison

1974 ◽  
Vol 11 (1) ◽  
pp. 43-52 ◽  
Author(s):  
V. R. R. Uppuluri ◽  
W. J. Blot
Keyword(s):  
Paired Comparison ◽  
Beta Function ◽  
Asymptotic Properties ◽  
Random Variable ◽  
Incomplete Beta Function ◽  
Discrete Random Variable ◽  
World Series ◽  
Mean And Variance ◽  
The Mean ◽  
Made In

A discrete random variable describing the number of comparisons made in a sequence of comparisons between two opponents which terminates as soon as one opponent wins m comparisons is studied. By equating two different expressions for the mean of the variable, a closed form for the incomplete beta function with equal arguments is obtained. This expression is used in deriving asymptotic (m-large) expressions for the mean and variance. The standardized variate is shown to converge to the Gaussian distribution as m→ ∞. A result corresponding to the DeMoivre-Laplace limit theorem is proved. Finally applications are made to the genetic code problem, to Banach's Match Box Problem, and to the World Series of baseball.

scholarly journals Double Kernel Estimation of Sensitivities

2009 ◽  
Vol 46 (03) ◽  
pp. 791-811
Author(s):  
Romuald Elie
Keyword(s):  
Asymptotic Properties ◽  
Kernel Estimation ◽  
Score Function ◽  
Random Variable ◽  
Kernel Functions ◽  
Contingent Claim ◽  
Main Interest ◽  
Stringent Condition ◽  
Technical Interest ◽  
The One

In this paper we address the general issue of estimating the sensitivity of the expectation of a random variable with respect to a parameter characterizing its evolution. In finance, for example, the sensitivities of the price of a contingent claim are called the Greeks. A new way of estimating the Greeks has recently been introduced in Elie, Fermanian and Touzi (2007) through a randomization of the parameter of interest combined with nonparametric estimation techniques. In this paper we study another type of estimator that turns out to be closely related to the score function, which is well known to be the optimal Greek weight. This estimator relies on the use of two distinct kernel functions and the main interest of this paper is to provide its asymptotic properties. Under a slightly more stringent condition, its rate of convergence is the same as the one of the estimator introduced in Elie, Fermanian and Touzi (2007) and outperforms the finite differences estimator. In addition to the technical interest of the proofs, this result is very encouraging in the dynamic of creating new types of estimator for the sensitivities.

scholarly journals A new urn model

2005 ◽  
Vol 42 (4) ◽  
pp. 964-976 ◽  
Author(s):  
May-Ru Chen ◽  
Ching-Zong Wei
Keyword(s):  
Asymptotic Properties ◽  
Random Variable ◽  
Computational Studies ◽  
Urn Model ◽  
Absolutely Continuous

In this paper, we propose a new urn model. A single urn contains b black balls and w white balls. For each observation, we randomly draw m balls and note their colors, say k black balls and m − k white balls. We return the drawn balls to the urn with an additional ck black balls and c(m − k) white balls. We repeat this procedure n times and denote by Xn the fraction of black balls after the nth draw. To investigate the asymptotic properties of Xn, we first perform some computational studies. We then show that {Xn} forms a martingale, which converges almost surely to a random variable X. The distribution of X is then shown to be absolutely continuous.

On the asymptotic properties of a supercritical bisexual branching process

1986 ◽  
Vol 23 (03) ◽  
pp. 820-826 ◽  
Author(s):  
J. H. Bagley
Keyword(s):  
Population Size ◽  
General Class ◽  
Population Model ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Convergence Result ◽  
Bisexual Population

An almost sure convergence result for the normed population size of a bisexual population model is proved. Properties of the limit random variable are deduced. The derivation of similar results for a general class of such processes is discussed.

scholarly journals A new urn model

2005 ◽  
Vol 42 (04) ◽  
pp. 964-976 ◽  
Author(s):  
May-Ru Chen ◽  
Ching-Zong Wei
Keyword(s):  
Asymptotic Properties ◽  
Random Variable ◽  
Computational Studies ◽  
Urn Model ◽  
Absolutely Continuous

In this paper, we propose a new urn model. A single urn contains b black balls and w white balls. For each observation, we randomly draw m balls and note their colors, say k black balls and m − k white balls. We return the drawn balls to the urn with an additional ck black balls and c(m − k) white balls. We repeat this procedure n times and denote by X n the fraction of black balls after the nth draw. To investigate the asymptotic properties of X n , we first perform some computational studies. We then show that {X n } forms a martingale, which converges almost surely to a random variable X. The distribution of X is then shown to be absolutely continuous.

scholarly journals Exact Lower Bounds on the Exponential Moments of Truncated Random Variables

2011 ◽  
Vol 48 (02) ◽  
pp. 547-560 ◽  
Author(s):  
Iosif Pinelis
Keyword(s):  
Option Pricing ◽  
Lower Bounds ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Random Variable ◽  
Exponential Moments ◽  
Comparative Advantages ◽  
Large Deviation Probabilities

Exact lower bounds on the exponential moments of min(y,X) andX1{X<y} are provided given the first two moments of a random variableX. These bounds are useful in work on large deviation probabilities and nonuniform Berry-Esseen bounds, when the Cramér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the so-called Winsorization min(y,X) over the truncationX1{X<y} are demonstrated. An application to option pricing is given.

Asymptotic Properties of a Recursive Procedure for Simultaneous Estimation

1990 ◽  
Vol 4 (4) ◽  
pp. 461-475
Author(s):  
Martin I. Reiman
Keyword(s):  
Central Limit ◽  
Heavy Traffic ◽  
Asymptotic Properties ◽  
Queueing Network ◽  
Random Variable ◽  
Simultaneous Estimation ◽  
Recursive Procedure ◽  
Heavy Traffic Limit ◽  
Sojourn Time Distribution ◽  
Pass Through

In this paper we consider a problem that arises in estimating the heavy traffic limit of a sojourn time distribution in a queueing network during the course of a medium traffic simulation. We need to estimate α = E[f(γ, M)], where γ is an unknown constant and M a random variable. More specifically, we are given an iid sequence of random vectors {(Xi, Mi), 1 ≤ i ≤ n}, with γ = E[Xi] and Mi having the same distribution as M.For known γ, we have a standard estimation problem, which we describe here. The standard estimate is unbiased and asymptotically (as n → 8 ) consistent. There is also a central limit theorem for this estimator. For unknown γ, we provide two estimation procedures, one that requires two passes through the data (as well as storage of {Mi, 1 ≤ i ≤ n}), and another one, which is recursive, requiring only one pass through and bounded storage. The estimators obtained from these two procedures are shown to be strongly consistent, and central limit theorems are also proven for them.

On the asymptotic properties of a supercritical bisexual branching process

1986 ◽  
Vol 23 (3) ◽  
pp. 820-826 ◽  
Author(s):  
J. H. Bagley
Keyword(s):  
Population Size ◽  
General Class ◽  
Population Model ◽  
Branching Process ◽  
Asymptotic Properties ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Convergence Result ◽  
Bisexual Population

An almost sure convergence result for the normed population size of a bisexual population model is proved. Properties of the limit random variable are deduced. The derivation of similar results for a general class of such processes is discussed.

Asymptotic properties for the loglog laws under positive association

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
Xiao-Rong Yang ◽  
Ke-Ang Fu
Keyword(s):  
Gamma Function ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Positive Association ◽  
Random Variable ◽  
Stationary Sequence ◽  
Normal Random Variable ◽  
Finite Variance ◽  
Associated Random Variables ◽  
Standard Normal

AbstractLet {X n: n ≥ 1} be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set $$S_n = \sum\limits_{k = 1}^n {X_k }$$, $$Mn = \mathop {\max }\limits_{k \leqslant n} \left| {S_k } \right|$$, n ≥ 1. Suppose that $$0 < \sigma ^2 = EX_1^2 + 2\sum\limits_{k = 2}^\infty {EX_1 X_k < \infty }$$. In this paper, we prove that if E|X 1|2+δ < for some δ ∈ (0, 1], and $$\sum\limits_{j = n + 1}^\infty {Cov\left( {X_1 ,X_j } \right) = O\left( {n^{ - \alpha } } \right)}$$ for some α > 1, then for any b > −1/2 $$\mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2b + 1} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^{b - 1/2} }} {{n^{3/2} \log n}}} E\left\{ {M_n - \sigma \varepsilon \sqrt {2n\log \log n} } \right\}_ + = \frac{{2^{ - 1/2 - b} E\left| N \right|^{2(b + 1)} }} {{(b + 1)(2b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2(b + 1)} }}}$$ and $$\mathop {\lim }\limits_{\varepsilon \nearrow \infty } \varepsilon ^{ - 2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(\log \log n)^b }} {{n^{3/2} \log n}}E\left\{ {\sigma \varepsilon \sqrt {\frac{{\pi ^2 n}} {{8\log \log n}}} - M_n } \right\}} _ + = \frac{{\Gamma (b + 1/2)}} {{\sqrt 2 (b + 1)}}\sum\limits_{k = 0}^\infty {\frac{{( - 1)^k }} {{(2k + 1)^{2b + 2} }}} ,$$ where x + = max{x, 0}, N is a standard normal random variable, and Γ(·) is a Gamma function.

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