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Exact Bounds on the Inverse Mills Ratio and Its Derivatives

COMPLEX ANALYSIS AND OPERATOR THEORY 2019年第4期13卷 1643-1651页

作者： Pinelis, Iosif Michigan Technol Univ Dept Math Sci Houghton MI 49931 USA

The inverse Mills ratio is R := phi/Psi, where phi and are Psi respectively, the probability density function and the tail function of the standard normal distribution. Exact bounds on R(z) for complex z with Rz >= 0 are obtained, which then yield logarithmically exact upper bounds on high-order derivatives of R. These results complement the many known bounds on the (inverse) Mills ratio of the real argument. The main idea of the proof is a non-asymptotic version of the so-called stationary-phase method. This study was prompted by a recently discovered alternative to the Euler-Maclaurin formula.

关键词： Inverse Mills ratio complementary error function functions of a complex variable Rate of growth of functions Inequalities for integrals Monotonic functions Entire and meromorphic functions Boundary behavior Inequalities in approximation Best constants Approximations to distributions

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Analytical and numerical aspects of a generalization of the complementary error function

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APPLIED MATHEMATICS AND COMPUTATION 2010年第12期216卷 3680-3693页

作者： Deano, Alfredo Temme, Nico M. Ctr Wiskunde & Informat NL-1098 XG Amsterdam Netherlands Univ Carlos III Madrid Dept Matemat Madrid Spain

In this paper we discuss analytical and numerical properties of the function V-nu,V-mu(alpha,beta,z) = integral(infinity)(0)e(-zt) (t + alpha)(nu) (t + beta)(mu)dt, with alpha, beta;Rz > 0, which can be viewed as a generalization of the complementary error function, and in fact also as a generalization of the Kummer U-function. The function V-nu,V-mu(alpha, beta, z) is used for certain values of the parameters as an approximate in a singular perturbation problem. We consider the relation with other special functions and give asymptotic expansions as well as recurrence relations. Several methods for its numerical evaluation and examples are given. (C) 2010 Elsevier Inc. All rights reserved.

关键词： complementary error function Singular perturbation problem Incomplete gamma function Confluent hypergeometric function Asymptotic expansions Recurrence relations

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A SUBADDITIVE PROPERTY OF THE error function

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2014年第8期142卷 2697-2704页

作者： Alzer, Horst Kwong, Man Kam Hong Kong Polytech Univ Dept Appl Math Hunghom Hong Kong Peoples R China

We prove the following subadditive property of the error function: erf (x) = 2/root pi integral(x)(0) e(-t2) dt (x is an element of R). Let a and b be real numbers. The inequality erf ((x + y)(a))(b) < erf (y(a))(b... 详细信息

关键词： error function complementary error function sub-and superadditive inequalities convex concave

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The zeros of the complementary error function

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NUMERICAL ALGORITHMS 2008年第1-4期49卷 153-157页

作者： Elbert, Arpad Laforgia, Andrea Rome Tre Univ Dept Math I-00146 Rome Italy

We show that the complementary error function, , has no zeros in .

关键词： Zeros complementary error function Primary 33B20

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The functions erf and erfc computed with arbitrary precision and explicit error bounds

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INFORMATION AND COMPUTATION 2012年 216卷 72-95页

作者： Chevillard, S. INRIA Sophia Antipolis Mediterranee Apics Project Team F-06902 Sophia Antipolis France

The error function erf is a special function. It is widely used in statistical computations for instance, where it is also known as the standard normal cumulative probability. The complementary error function is defined as erfc(x) = 1 - erf(x). In this paper, the computation of erf(x) and erfc(x) in arbitrary precision is detailed: our algorithms take as input a target precision t' and deliver approximate values of erf(x) or erfc(x) with a relative error guaranteed to be bounded by 2(-t'). We study three different algorithms for evaluating erf and erfc. These algorithms are completely detailed. In particular, the determination of the order of truncation, the analysis of roundoff errors and the way of choosing the working precision are presented. The scheme used for implementing erf and erfc and the proofs are expressed in a general setting, so they can directly be reused for the implementation of other functions. We have implemented the three algorithms and studied experimentally what is the best algorithm to use in function of the point x and the target precision t'. (C) 2012 Elsevier Inc. All rights reserved.

关键词： error function complementary error function erf erfc Floating-point arithmetic Arbitrary precision Multiple precision

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Bounds for the generalized Marcum Q-function

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APPLIED MATHEMATICS AND COMPUTATION 2010年第5期217卷 2238-2250页

作者： Baricz, Arpad Sun, Yin Univ Babes Bolyai Dept Econ Cluj Napoca 400591 Romania Tsinghua Univ Tsinghua Natl Lab Informat Sci & Technol State Key Lab Microwave & Digital Commun Beijing Peoples R China Tsinghua Univ Dept Elect Engn Beijing Peoples R China

In this paper we consider the generalized Marcum Q-function of order m > 0 real, defined by Q(v)(a,b) = 1/a(v-1)integral(infinity)(b) t(v)e(-t2+a2/2)I(v-1)(at)dt, where a > 0, b >= 0 and I-v stands for the modified Bessel function of the first kind. Our aim is to extend some results on the ( first order) Marcum Q-function to the generalized Marcum Q-function in order to deduce some new lower and upper bounds. Moreover, we show that the proposed bounds are very tight for the generalized Marcum Q-function of integer order, and we deduce some new inequalities for the more general case of real order. The chief tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind, which are based on a criterion for the monotonicity of the quotient of two Maclaurin series. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Generalized Marcum Q-function Modified Bessel functions complementary error function Incomplete gamma function Bounds

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A Simple Approximation for the Symbol error Rate of Triangular Quadrature Amplitude Modulation

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IEICE TRANSACTIONS ON COMMUNICATIONS 2010年第3期E93B卷 753-756页

作者： Duy, Tran Trung Kong, Hyung Yun Univ Ulsan Ulsan South Korea

In this paper. we consider the error performance of the regular triangular quadrature amplitude modulation (TQAM) In particular, using an accurate exponential bound of the complemental y error function. we derive a simple approximation for the average symbol error rate (SER) of TQAM Over Additive White Gaussian Noise (AWGN) and fading channels The accuracy of our approach is verified by some simulation results

关键词： Triangular QAM complementary error function AWGN channel fading channels symbol error rate

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Performance Analysis of Square M-QAM with Dual-Hop Relay Link in Nakagami-m Fading Channel

Performance Analysis of Square M-QAM with Dual-Hop Relay Lin...

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International Conference on Signal Processing and Communications

作者： Datta, Soumendra Nath Chakrabarti, Saswat Roy, Rajarshi Indian Inst Technol GSSST Kharagpur 721302 W Bengal India Indian Inst Technol Dept Elect & Elect Commun Engn Kharagpur 721302 W Bengal India

ISBN: (纸本)9781424471379

In this paper, we analyze the symbol error probability (SEP) perfonnance of square M-ary quadrature amplitude modulation (M-QAM) with dual-hop relay transmission considering independent and non-identical flat Nakagami-m fading channels. The relay is operating under amplify-and-forward (AF) mode with average power scaling (APS) and instantaneous power scaling (IPS) constraints. Simplified closed-form expressions for the SEP are presented following the cumulative distribution function (cdf) based approach. Moreover, using a series representation of complementary error function er f c (.), we derive exact SEP expression under sufficiently large signal-to-noise ratio (SNR) for the source-relay link and compared the error perfonnance with the simplified SEP solution. Numerical and computer simulations results are presented to verify the accuracy of the proposed mathematical analysis.

关键词： Nakagami channels Nakagami-m fading channel amplify-and-forward mode average power scaling closed-form expressions complementary error function cumulative distribution function dual-hop relay link dual-hop relay transmission exact SEP expression instantaneous power scaling performance analysis quadrature amplitude modulation radio links signal-to-noise ratio source-relay link square M-QAM square M-ary quadrature amplitude modulation symbol error probability

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Tight bounds for the generalized Marcum Q-function

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2009年第1期360卷 265-277页

作者： Baricz, Arpad Univ Babes Bolyai Dept Econ Cluj Napoca 400591 Romania

In this paper we study the generalized Marcum Q-function of order nu > 0 real, defined by Q(nu)(a, b) = 1/a(nu-1) integral(infinity)(b) t(nu)e(-)t(2)+a(2)/2 I nu-1 (at) dt, where a > 0, b >= 0 and I-nu stands for the modified Bessel function of the first kind. Our aim is to improve and extend some recent results of Wang to the generalized Marcum Q-function in order to deduce some sharp lower and upper bounds. In both cases b >= a and b < a we give the best possible upper bound for Q(nu) (a, b). The key tools in our proofs are some monotonicity properties of certain functions involving the modified Bessel function of the first kind. These monotonicity properties are deduced from some results on modified Bessel functions, which have been used in wave mechanics and finite elasticity. (C) 2009 Elsevier Inc. All rights reserved.

关键词： Marcum Q-function Generalized Marcum Q-function Modified Bessel functions Lower and upper bounds complementary error function Sharp bounds

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Correction to "On the linear combination of normal and Laplace random variables" (vol 21, pg 63, 2006)

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COMPUTATIONAL STATISTICS 2008年第4期23卷 661-666页

作者： Diaz-Frances, Eloisa Montoya, Jose A. Ctr Invest Matemat Guanajuato 36000 Mexico

In the above mentioned paper, some errors were found in the expressions given for the distribution of a linear combination of Normal and Laplace random variables, Z, given in formulae (3, Theorem 1), (6), and (7) that can lead to obtaining negative values for the mentioned distribution. The corrected versions for these expressions are presented here. In addition, the density function of Z is also provided.

关键词： complementary error function density function of linear combinations of Laplace and Normal variables Laplace distribution normal distribution

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