The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance

By Samuel Kotz, Tomasz J. Kozubowski, Krzysztof Podgórski. 2001

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Citations in the corpus

Private Counting from Anonymous Messages: Near-Optimal Accuracy with Vanishing Communication Overhead (2020, background): “This is because the Discrete Laplace distribution4 is∞-div as DLap(ε) = NB(1, e−ε)− NB(1, e−ε) (see, e.g., (Kotz et al., 2001)), and NB(1, e−ε) is∞-div, and is in fact the geometric distribution.”
A Comprehensive Comparison of Multiparty Secure Additions with Differential Privacy (2015, methodology): “Infinite divisibility of the Laplace distribution is stable [37].”
A Tutorial on Fisher Information (2017, background): “It can be shown that the1 MLE for this model is the sample median, θ̂ =M , and the unit Fisher information is IX(θ) = b−2.2 The exact sampling distribution of the MLE is unwieldy (Kotz et al., 2001) and not presented3 here.”
A General Closed-Form Spread Option Pricing Formula (2013, unknown): “For a detailed description of the asymmetric Laplace distribution and its properties, see Kotz et al. (2001).”
Model-Based Clustering (2016, background): “Crucially, the random variableX can be generated via X = μ+Wα+ √ WV, (13) where W Exp(1) and V N(0,Σ) is independent of W (Kotz, Kozubowski, and Podgorski 2001; Franczak et al. 2014).”
Nonparametric Estimation for Lévy Models Based on Discrete-Sampling (2009, unknown): “[21].”
Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood (2016, methodology): “By Kotz et al. (2001), i can be represented by a scale mixture of normals, i D 1vi C 2 ́i p vi ; (2.5) where 1 D .1 2 /=1 .1 /o, 2 2 D 2=1 .1 /o, ́i N.0; 1/, vi follows the standard exponential distribution, and ́i and vi are independent. By using the representation (2.5) and assuming a proper Gau”
Modeling asymptotically independent spatial extremes based on Laplace random fields (2015, methodology): “This definition and basic properties can be found in the monograph of Kotz et al. (2001).”
Bayesian Tail Risk Interdependence Using Quantile Regression (2015, methodology): “Following Kozumi and Kobayashi (2011), we instead adopt the well-known representation (see e.”
A Skew Laplace Distribution on Integers (2006, background): “…form Y d= IX, (16) where I takes on the values 0 and ±1, X has a discrete distribution on non-negative integers, and the two variables are independent (if Y has the skew Laplace distribution (2) then a similar representation holds with an exponentially distributed X, see, e.g., Kotz et al. 2001).”
Fractional Laplace model for hydraulic conductivity (2004, methodology): “…new data being carefully measured in two well-defined sandstone facies [Castle et al., 2004], we report that the increment PDFs, including the full tails, appear to follow the double exponential (Laplace) PDF or stretched exponential PDF, exemplified by the so-called Bessel PDF [Kotz et al., 2001].”
A mixture of generalized hyperbolic distributions (2013, background): “se of the distribution known as the generalized, Bessel function or variance-gamma distribution (Barndor-Nielsen, 1978). If = 1, = 2 and ˜!0, then we obtain the asymmetric Laplace distribution (cf. Kotz et al., 2001) and if = 0, we have the symmetric generalized hyperbolic distribution (Barndor-Niel”
I Have a DREAM! (DiffeRentially privatE smArt Metering) (2011, background): “Lemma 1 (Divisibility of Laplace distribution [17]) Let L(λ) denote a random variable which has a Laplace distribution with PDF f(x, λ) = 1 2λe |x| λ .”
Asymmetric power distribution: Theory and applications to risk measurement (2007, methodology): “t=1 [ |xt − θ|λ αλ 1(xt 6 θ) + |xt − θ|λ (1− α)λt > θ)], (14)”