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Exact Likelihood Inference and Robust Filtering for Gauss-Cauchy Convolution Models
{Keywords:}{ Voigt profile, Convolutions, Heavy Tails, Robust Filtering, Kalman Filter.}
{JEL Classification:}{ C13, C16, C22, C58, G17} \setstretch{1.5}
The convolution of a Gaussian and a Cauchy distribution, known as the Voigt distribution, arises naturally when finite-variance background noise is combined with heavy-tailed disturbances. The Voigt profile is widely used in spectroscopy, where Gaussian broadening is associated with Doppler effects and Cauchy broadening with pressure or lifetime effects. It is also a natural statistical model for measurement errors that combine ordinary noise with occasional extreme observations.
Although an expression for the Voigt density has been known since Kendall_D:1938, its use as a likelihood-based statistical model has been limited by the perception that exact inference is computationally difficult. Because the density is not an elementary function, applications often rely on numerical convolution, pseudo-Voigt approximations, nonlinear least squares, or simulation-based methods. We show that these compromises are unnecessary for likelihood inference. The scaled complementary error function provides a stable representation of the Voigt density, and its differential identities yield closed-form expressions for the likelihood derivatives.
This paper develops likelihood inference and filtering methods for the Gauss-Cauchy convolution model. We first derive analytical expressions for the density, score, Hessian, and conditional moments of the Voigt distribution. These expressions reduce likelihood evaluation and differentiation to evaluations of a standard special function and algebraic operations. We then establish standard large-sample properties of the maximum likelihood estimator and document its numerical performance in Monte Carlo simulations.
The same analytical structure is useful for robust signal extraction. If an observation is the sum of a latent Gaussian signal and Cauchy measurement noise, the conditional expectation of the Gaussian component is nonlinear in the observation. The associated location score is approximately linear near the center of the distribution but redescends in the tails. Hence, extreme observations are automatically discounted rather than propagated into the latent signal. This differs from the Kalman update, which is linear in the prediction error, and from Student-$t$ or Huber-type procedures, where redescending or bounded influence is obtained by specifying a robust observation density or loss function directly. In the Gauss-Cauchy convolution, the redescending update is instead an implication of signal extraction: it is the exact conditional mean of the latent Gaussian component when the observation is contaminated by an additive Cauchy component. Thus, robustness is not imposed as an influence function but derived from the convolution structure itself.
We use this result to construct the Gauss-Cauchy Convolution (GCC) filter for linear state-space models with Gaussian latent dynamics and Voigt measurement errors. The key step is that the Masreliez Gaussian prediction approximation makes Tweedie's formula applicable to the state-prediction error, while the Gaussian-Cauchy convolution structure keeps the prediction-error density inside the Voigt family. Specifically, if the conditional state-prediction error is approximated as Gaussian with variance $h_{t|t-1}$, then $$ N(0,h_{t|t-1})+\mathcal V(0,\sigma,\gamma) = \mathcal V(0,\delta_t,\gamma), \qquad \delta_t^2=h_{t|t-1}+\sigma^2. $$ Thus the state update is the Tweedie conditional-mean correction associated with the Voigt prediction-error score. The filter nests the Kalman filter as the special case $\gamma=0$ and the pure Cauchy measurement-error filter as the special case $\sigma=0$. We assess the Masreliez approximation directly by comparing the GCC recursion with an exact benchmark filter based on numerical density propagation. The approximation error is small at both the predictive-density level and the one-step-correction level, especially in the empirically relevant region where the estimated Cauchy component is small relative to the Gaussian component.
We illustrate the method using daily log realized volatility for the Technology Select Sector SPDR Fund (XLK). The realized-volatility series contains both persistent volatility movements and transient extreme observations associated with market stress, liquidity disruptions, and microstructure effects. The GCC filter separates these components by retaining a Gaussian core for ordinary measurement variation while using the Cauchy component to absorb occasional large deviations. In the empirical application, the GCC specification attains the largest pseudo-log likelihood among the competing filters considered, including Gaussian, Cauchy, Normal-Laplace, Student-$t$, and Huber alternatives.
The paper is related to several strands of literature. It contributes to work on convolution-based distributions and Voigt-profile estimation by showing that the special-function representation yields a tractable likelihood theory. It is also related to robust filtering and score-driven dynamics CrealKoopmanLucas:2013, Harvey:2013, where heavy-tailed observation densities are used to reduce the influence of outliers. Recent contributions include robust score-driven filters based on Student-$t$ scores, including the multivariate filter of DInnocenzoLuatiMazzocchi:2023, and approximate filtering approaches connected to dynamic adaptive mixture models CataniaDInnocenzoLuati:2026. The GCC filter differs by deriving the redescending update as the conditional mean correction implied by the Gaussian-Cauchy convolution structure, rather than by specifying a heavy-tailed observation density or mixture structure directly.
The redescending conditional expectation also connects the GCC model to recent theoretical work on information aggregation with thin- and heavy-tailed signals. In global games, MorrisYildiz:2019 show that large shocks can trigger equilibrium shifts when agents rationally attribute extreme observations to a common heavy-tailed component rather than to idiosyncratic noise. A related “too good to be true” logic appears in WeibullMattssonVoorneveld:2007, where sufficiently extreme signals are discounted because they are more likely to reflect noise than fundamentals. HautschHessMuller:2012 extend this exact mechanism to financial price discovery, demonstrating that fat-tailed noise causes rational agents to optimally discount extreme signals. The Gauss-Cauchy convolution provides a tractable parametric realization of this behavior: the conditional expectation of the latent Gaussian component is explicitly redescending, and the parameters $(\sigma,\gamma)$ dictate precisely when extreme observations are treated as valid signals and when they are discarded as noise.
The remainder of the paper is organized as follows. Section (ref) defines the Gauss-Cauchy convolution and derives its conditional moment structure. Section (ref) develops likelihood inference for the Voigt distribution. Section (ref) introduces the GCC filter. Section (ref) evaluates the Masreliez approximation using exact benchmark filtering. Section (ref) applies the filter to log realized volatility. Section (ref) concludes. Technical proofs are collected in the Appendix.
Let $Z\sim \mathcal{N}(0,\sigma^{2})$ and $X\sim\operatorname{Cauchy}(0,\gamma)$ be independent where $\sigma,\gamma>0$, and consider the convolution $$ Y = \mu + Z + X. $$ We denote the resulting distribution by $\mathcal{V}(\mu,\sigma,\gamma)$ because it is known as the Voigt profile in the field of spectroscopy, and we let $f_Y(y;\theta)$ denote its density, where $\theta=(\mu,\sigma,\gamma)^\prime$. This density can be expressed in terms of the scaled complementary error function, $\mathsf{e}(w) \equiv \operatorname{erfcx}(w)$, where $\mathsf{e}(-iw)$ is the Faddeeva function. For $\operatorname{Re}(w)>0$, we have $$ \mathsf{e}(w) = \frac{1}{\pi} \int_{-\infty}^{\infty} \frac{\exp(-t^2)}{w+it}dt, $$ and for $Z\sim \mathcal{N}(0,1)$ we have $\Pr(|Z|>r)=\mathsf{e}(\tfrac{r}{\sqrt{2}})e^{-r^2/2}$ for $r>0$.
We are particularly interested in evaluating $\mathsf{e}(w)$ on the vertical Voigt line in $\mathbb{C}$ traced out by $y\mapsto w_{y,\theta} = \tfrac{\gamma+i(y-\mu)}{\sigma\sqrt{2}}$, and we introduce the notation for the real and imaginary parts of $\mathsf{e}(w_{y,\theta})$ $$ \mathsf{u}(y;\theta) \equiv \operatorname{Re}[\mathsf{e}(w_{y,\theta})], \qquad \mathsf{v}(y;\theta) \equiv \operatorname{Im}[\mathsf{e}(w_{y,\theta})], $$ so that $\mathsf{e}(w_{y,\theta})= \mathsf{u}(y;\theta)+i\mathsf{v}(y;\theta)$.
The expression for this density was first obtained by Kendall_D:1938. We include a proof of Proposition (ref) in the appendix based on the characteristic function $\varphi_{Z+X}(s)=e^{-s^{2}/2-|s|}$. Figure (ref) shows the density of $Y$ for $\mu=0$ and $\sigma=\gamma=1$, together with the best approximating Student-$t$ distribution, as defined by the Kullback-Leibler discrepancy. The visible differences highlight the specification error from using Student-$t$ distributions as proxies.\footnote{The Voigt density can be viewed as a limiting case of the convolution of two Student-$t$ distributions, a class of convolution problems known to be analytically difficult. For recent progress in this area, see Nason:2006, Forchini:2008, BergVignat:2010, and HansenTong:2026.} A second common approximation is the pseudo-Voigt profile, a two-component mixture of a Gaussian density and a Cauchy density with the same location.\footnote{The pseudo-Voigt density takes the form $(1-\eta_p)\phi(y;\mu,\sigma_p)+\eta_p c(y;\mu,\gamma_p)$, with $0\leq\eta_p\leq1$, where $\phi$ is the Gaussian density and $c$ is the Cauchy density.} It provides a closer density approximation in this example, but it is not a convolution and therefore does not preserve the additive-noise interpretation used below. The Voigt profile uniquely captures the superposition of finite-variance noise and infinite-variance jumps, a geometry that generic heavy-tailed approximations fail to reproduce.
Interestingly, the density ((ref)) can be expressed from the Mills ratio for a standard normal random variable, $m(t)=\Phi(-t)/\phi(t)$, where $\phi$ and $\Phi$ are the density and cumulative distribution function for $\mathcal{N}(0,1)$, respectively.\footnote{We thank George Tauchen for pointing out the connection between the Voigt representation and the Mills ratio.} This representation highlights the connection between the Voigt profile and the hazard rate of the normal distribution, providing another avenue for numerical evaluation.
Lemma (ref) collects the special-function identities that make the Voigt likelihood tractable. Part (i) states that $\mathsf{e}=\operatorname{erfcx}$ is closed under differentiation in the sense that every derivative is an algebraic function of $w$ and $\mathsf{e}(w)$. Part (ii) gives the positivity and symmetry properties needed for the real and imaginary components along the Voigt line.
The positivity of $\mathsf{u}(y;\theta)$ ensures that the density, log-density, score, Hessian, and conditional moment formulas involving ratios such as $$ \frac{\mathsf{v}(y;\theta)}{\mathsf{u}(y;\theta)} $$ are well defined for all real $y$. The symmetry relations explain the corresponding even-odd structure of the density and the conditional moments derived below.
The main likelihood implication of Lemma (ref) is the following algebraic closure property.
Corollary (ref) is the central tractability result for likelihood inference. Although the Voigt density is defined as a Gaussian-Cauchy convolution, the scaled complementary error function absorbs the otherwise unstable exponential term and turns differentiation of the density into algebraic operations on a single complex-valued function. Thus the likelihood, score, Hessian, and higher-order derivatives can all be evaluated from the same pair of real-valued quantities, $\mathsf{u}(y;\theta)$ and $\mathsf{v}(y;\theta)$, without numerical convolution, quadrature, finite-difference approximation, or pseudo-Voigt approximations. This is why exact maximum likelihood estimation is more direct than the convolution representation might suggest.
We now consider the signal-extraction problem implied by the convolution $$ Y=\mu+Z+X. $$ The objective is to infer the latent Gaussian component $Z$ from the observed realization of $Y$, treating the Cauchy component $X$ as measurement noise. This formulation is directly relevant for robust filtering, where $Z$ represents the latent signal of interest and $X$ captures transient heavy-tailed contamination. The following theorem gives the closed form conditional density of $Z$ given $Y=y$.
This conditional density is shown in Figure (ref) for selected values of $y$ with $\sigma=\gamma=1$. The vertical lines along the $x$-axis indicate the conditional expected value for each conditional density.
For the special case, $y=\mu$, the conditional density is $$ f_{Z|Y}(z|y=\mu)=\frac{1}{\gamma\pi}\frac{1}{\Pr(|Z|\geq\tfrac{\gamma}{\sigma})}\frac{\exp(-\frac{1}{2}\frac{z^{2}+\gamma^{2}}{\sigma^{2}})}{1+(\frac{z}{\gamma})^{2}}. $$
The conditional moments of $Z$ given $Y$ can be obtained from a Gaussian-convolution version of Tweedie's formula. The key point is that the conditional mean of an additive Gaussian component is determined by the score of the marginal density, and the conditional variance by its derivative. See Efron:2011 for a modern treatment of Tweedie's formula and Robbins:1956 for the original published attribution.
In the Gauss-Cauchy convolution, Proposition (ref) gives the marginal density $f_Y$ explicitly in terms of $\mathsf{u}(y;\theta)$. Therefore Proposition (ref) reduces the conditional-moment calculation to differentiating the Voigt log-density. Using the score and Hessian identities in Lemma (ref) gives the following closed-form expressions.
The conditional expectation of $Z$ given $Y$ is antisymmetric about $\mu$ and, conversely, the conditional variance is symmetric about $\mu$, driven by the fact that it depends on the prediction error strictly through the even function $\mathsf{u}(y)$ and the squared odd function $\mathsf{v}(y)^2$. The conditional expectation and variance are plotted for $(\mu,\gamma,\sigma)=(0,1,1)$ in Figure (ref). For this case, the maximum conditional expectation is $\max_{y}\mathbb{E}\left[Z|Y=y\right]\approx0.7486$ for $y\approx2.4637$ and the conditional variance ranges between $\mathbb{V}(Z|Y=0)\approx 0.5251$ and $\mathbb{V}(Z|Y=y)\approx 1.1603$ for $y^\ast \approx \pm 3.6621$.\footnote{That $\operatorname{var}(Z|Y=y)$ can exceed $\operatorname{var}(Z)$ is not inconsistent with the law of total variance, because the latter implies only that $\mathbb{E}[\operatorname{var}(Z|Y)]\leq\operatorname{var}(Z)$, not that $\operatorname{var}(Z|Y=y)\leq\operatorname{var}(Z)$ for every $y$. In the standardized case, $\sigma=\gamma=1$ and $\mu=0$, the thresholds for maximum signal extraction ($y\approx2.4637$) and maximum signal confusion ($y\approx3.6621$) are the positive roots of the second and third derivatives of the Voigt log-likelihood, respectively.} As $y\rightarrow\pm\infty$ the conditional moments converge to the unconditional moments, $\mathbb{E}\left[Z|Y=y\right]\rightarrow 0$ and $\mathbb{V}\left[Z|Y=y\right]\rightarrow 1$. The reason is that the conditional distribution converges to the unconditional distribution as $y$ increases in absolute value, i.e. $f_{Z|Y}(z|y)\rightarrow f_{Z}(z)$ (almost everywhere) as $y\rightarrow\pm\infty$. The intuition for this is simply that the tail of a Gaussian vanishes much faster than that of a Cauchy, such that a very extreme realization of $y$ is almost fully attributed to the Cauchy component.
Figure (ref) provides deep insight into the filter we propose later. The shape of ((ref)) is approximately linear near the origin similar to a standard Kalman filter. However, for large values of $|y|$ the filter enters a nonlinear regime, and the expectation does not merely saturate but redescends toward zero. This non-monotonicity is key and implies that the filter identifies observations that are “too large to be true signals” and isolates them as pure noise, protecting the latent trend estimate from contamination. Since $Y=\mu + Z+X$ we may define $\mathbb{E}[X|Y]= -\gamma \frac{\mathsf{v}(Y)}{\mathsf{u}(Y)}$, despite $X$ being Cauchy distributed, with $\mathbb{E}|X|=\infty$.
The conditional mean in Figure (ref) bears a striking resemblance to the score function of the Student-$t$ distribution shown in HarveyLuati:2014. This is noteworthy because the two objects arise from different constructions. In the score-driven literature, the redescending shape is induced by specifying a heavy-tailed observation density, such as the Student-$t$, whose score limits the influence of large observations. In contrast, the redescending shape in Figure (ref) emerges endogenously as the exact conditional mean of a latent Gaussian signal observed with Cauchy noise. Thus, the Gauss-Cauchy convolution provides a probabilistic micro-foundation for robust update functions of the type used in score-driven models.
The non-monotonicity of the conditional variance is inextricably linked to the redescending nature of the conditional expectation. By differentiating Tweedie's formula for the conditional mean, $\mathbb{E}[Z | Y=y] = -\sigma^2 \frac{\partial \log f_Y(y)}{\partial y}$, with respect to $y$, we obtain $\frac{\partial}{\partial y}\mathbb{E}[Z| Y=y] = -\sigma^2 \frac{\partial^2 \log f_Y(y)}{\partial y^2}$. Substituting this into the conditional variance equation yields the exact identity:
This establishes three distinct filtering regimes based on the slope of the conditional expectation. Near zero, where $\mathbb{E}(Z|Y=y)$ is increasing in $|Y|$, the observation is deemed reliable, and the conditional variance is strictly less than the unconditional variance $\sigma^2$, reflecting a reduction in uncertainty. Then, at the exact local extrema of the conditional expectation ($y \approx \pm 2.4637$ for the standard case), the slope is zero, and the conditional variance exactly equals $\sigma^2$. This represents the exact threshold where the filter ceases to trust the magnitude of the innovation. As the observation moves further into the tails, the expectation redescends and the conditional variance strictly exceeds $\sigma^2$. The increased uncertainty occurs because the filter cannot confidently attribute the moderate outlier to either the Gaussian signal or the Cauchy noise. As $|y| \to \infty$, $\frac{\partial}{\partial y}\mathbb{E}(Z|Y=y) \to 0$ and the conditional variance smoothly asymptotes back to the unconditional variance as the observation is entirely discounted.
Higher-order conditional cumulants can be obtained in the same way. The first step is not specific to the Voigt model, but holds for any Gaussian convolution. Let $$ K(s|y)=\log\mathbb{E}\left[\exp(sZ)|Y=y\right] $$ denote the conditional cumulant generating function of $Z|Y=y$. If $Y=\mu+Z+X$, where $Z\sim\mathcal{N}(0,\sigma^2)$ is independent of an arbitrary proper random variable $X$, then completing the square gives $$ K(s|y)=\frac{\sigma^2s^2}{2}+\log f_Y(y-\sigma^2s)-\log f_Y(y). $$ The conditional cumulants are defined by $$ \kappa_r(y)=\left.\frac{\partial^r}{\partial s^r}K(s|y)\right|_{s=0}. $$ Therefore $$ \kappa_1(y)=-\sigma^2\frac{\partial}{\partial y}\log f_Y(y), \qquad \kappa_2(y)=\sigma^2+\sigma^4\frac{\partial^2}{\partial y^2}\log f_Y(y). $$ These are the conditional mean and variance identities in Proposition (ref). For $r\geq3$, $$ \kappa_r(y)=(-\sigma^2)^r\frac{\partial^r}{\partial y^r}\log f_Y(y), $$ and equivalently, $ \kappa_{r+1}(y)=-\sigma^2\frac{\partial}{\partial y}\kappa_r(y)$, for all $r\geq2$.
For the Voigt Gauss-Cauchy convolution, Proposition (ref) gives $f_Y$ explicitly in terms of $\mathsf{u}(y;\theta)$. Hence all higher-order conditional cumulants are obtained by differentiating the closed-form variance expression in Corollary (ref). By Lemma (ref), these derivatives are again algebraic functions of $\mathsf{u}(y;\theta)$, $\mathsf{v}(y;\theta)$, and the parameters.
The tractability is therefore not limited to the conditional mean and variance. Although the Voigt random variable $Y$ has Cauchy tails and hence does not have finite positive integer moments when $\gamma>0$, the conditional distribution of the latent Gaussian component $Z|Y=y$ has moments of all orders. Thus conditional skewness, kurtosis, and higher conditional moments of the latent Gaussian component can be evaluated without numerical integration.
Having established the exact density and conditional properties of the Voigt distribution, we now turn to the problem of parameter estimation. While the Voigt distribution is fundamental in spectroscopy and physics, rigorous statistical inference for its parameters has historically been underdeveloped. The closed-form expression in ((ref)) allows us to proceed directly with Maximum Likelihood Estimation (MLE), avoiding the approximations commonly used in the existing literature. These results are obtained by leveraging the differential properties of the scaled complementary error function, which leads to exact analytical expressions for the score vector and Hessian matrix.
One significance of Lemma (ref) is computational. The perceived intractability of the Voigt integral has led researchers to rely on computationally expensive methods or approximation-based substitutes.\footnote{For example, CannasPiras:2025 note that direct numerical integration is time consuming and discuss the use of pseudo-Voigt approximations and Bayesian methods. The Voigt profile also has a substantial analytic literature on special-function representations, including Kummer-function expansions DiRoccoAguirreTellez:2004 and Mellin-Barnes, Fox $H$-, and Meijer $G$-function representations PagniniSaxena:2008. Related computational bottlenecks have also motivated workarounds in physics and spectroscopy, including deep learning approaches StemockChurchillLeeHassanDoughtyOchoa:2024, AliVanZijlPrasuhnWirestamKnutssonYadav:2025. Lemma (ref) shows that these complications are not intrinsic to likelihood-based inference for the Voigt model.} However, once the density is represented through the scaled complementary error function, the likelihood, score, and Hessian are all computed from the same quantities $\mathsf{u}(y;\theta)$ and $\mathsf{v}(y;\theta)$. Exact maximum likelihood estimation therefore reduces to repeated evaluation of a standard special function and algebraic operations, without numerical convolution, finite-difference derivatives, or pseudo-Voigt approximations.
A natural concern is whether standard likelihood-based inference remains valid for a model whose distribution has Cauchy tails. Since the Cauchy component dominates the tails of $Y$, the distribution has no finite positive integer moments; in particular, even the mean is undefined. One might therefore expect conventional asymptotic normality to fail. The relevant issue for maximum likelihood estimation, however, is not whether $Y$ has finite moments, but whether the score, Hessian, and Fisher information are well defined. In the Gauss-Cauchy convolution model they are, so exact maximum likelihood remains a regular estimation problem despite the absence of moments for $Y$. This is analogous to cases that arise in GARCH models, where asymptotic normality can hold even when the observed process has limited or no finite moments; see JensenRahbek:2004. Theorem (ref) establishes consistency and asymptotic normality of the MLE for the Voigt model.
Assumption (ref) restricts attention to the regular Voigt case in which both scale parameters are strictly positive and bounded away from zero. This excludes the boundary cases $\sigma=0$ and $\gamma=0$, corresponding to the pure Cauchy and pure Gaussian limits, respectively. These limiting cases are mathematically meaningful, but they require separate treatment because the analytical expressions below involve ratios and derivatives evaluated under $\sigma>0$ and $\gamma>0$.
The lower bound on $\gamma$ is also useful for the uniform integrability condition used in the consistency proof. If the true distribution has Cauchy tails but the parameter space includes the pure Gaussian boundary $\gamma=0$, the Gaussian tail can make $|\log f_Y(Y;\theta)|$ grow quadratically in $Y$ for some admissible parameter values, while $Y$ itself has Cauchy tails. Bounding $\gamma$ away from zero avoids this difficulty and yields a common logarithmic envelope for the likelihood. The lower bound on $\sigma$ similarly keeps the analysis within the interior of the regular Voigt family, where the score and Hessian formulas are ordinary derivatives with respect to $(\mu,\sigma,\gamma)$.
Despite the lack of moments, $\mathbb{E}|Y|=\infty$, Theorem (ref) shows that likelihood-based inference is quite standard in the Gauss-Cauchy convolution model. Convergence in distribution is at the conventional root-$n$ rate and the limit distribution is Gaussian, such that standard errors and $t$-statistics have their usual interpretation. The underlying reason is that the likelihood function and its derivatives are well-behaved. Moreover, the information-matrix equality holds, and the simplest way to estimate $\mathcal{I}_{\theta_{0}}$ appears to be to evaluate $\mathbb{E}_{\theta_{0}}[s_{\theta_{0}}s_{\theta_{0}}^{\prime}]$ by numerical integration. The asymptotic independence of $\hat{\mu}$ and $(\hat{\sigma},\hat{\gamma})$ is a consequence of $s_\mu$ being an odd function of the $\tilde{y}=y-\mu$, whereas $s_\gamma$ and $s_\sigma$ are both even functions. Consequently, ($\sigma, \gamma$) will have the same asymptotic variances in the simpler model where $\mu$ is known.
Moreover, the asymptotic covariance matrix for $(\sigma,\gamma)$ is homogeneous of degree two. Hence the ratio of their asymptotic standard deviations depends only on $\lambda=\gamma/\sigma$. We define this ratio as\footnote{The ratio $\lambda$ is proportional to the Voigt damping parameter used in astrophysics, $a=\lambda/\sqrt{2}$.} $$ R(\lambda)\equiv \frac{\operatorname{aStd}(\sigma;\mu,\sigma,\lambda\sigma)} {\operatorname{aStd}(\gamma;\mu,\sigma,\lambda\sigma)}. $$ Figure (ref) reports the resulting asymptotic standard deviations and their ratio. The left panel shows the asymptotic standard deviations for $\sigma$ and $\gamma$ as functions of $\lambda=\gamma/\sigma$, under the normalization $(\sigma,\gamma)=(1,\lambda)$. Both axes are log transformed. The relationship is close to log-linear for $\gamma$, corresponding to an approximately constant elasticity, whereas the relationship for $\sigma$ is convex.
The right panel plots $R(\lambda)$ against $\log_{10}\lambda$. The ratio is larger than one throughout the range considered, so the Cauchy scale parameter $\gamma$ is asymptotically more precisely estimated than the Gaussian scale parameter $\sigma$. This may seem counterintuitive because $\gamma$ governs the heavy-tailed component. The reason is that the Cauchy scale is strongly identified by the frequency and magnitude of extreme observations, where the Gaussian component contributes little. By contrast, the Gaussian scale is identified primarily from the central part of the distribution, where changes in $\sigma$ and $\gamma$ can partially offset one another. The U-shape of $R(\lambda)$ shows that the relative precision is especially favorable to $\gamma$ when the Cauchy scale is either small or large relative to the Gaussian scale.
We examine the finite-sample behavior of the Voigt maximum likelihood estimator through a Monte Carlo study. The purpose is to assess how accurately the asymptotic distribution in Theorem (ref) approximates the sampling distribution of the MLE. We generate $N=100,000$ independent datasets for sample sizes ranging from $n=100$ to $n=10,000$. For each replication, we compute the MLE and compare the empirical distribution of the estimates with the corresponding asymptotic approximation.
The simulations cover three values of the Cauchy-to-Gaussian scale ratio, $$ \lambda=\frac{\gamma_0}{\sigma_0}, $$ corresponding to $\lambda=0.01$, $\lambda=0.1$, and $\lambda=1$. Thus the designs range from a nearly Gaussian case to a balanced Voigt case in which the Gaussian and Cauchy scales are equal.
Table (ref) summarizes the results. We report the mean of the parameter estimates, their empirical standard deviation (Std), and the asymptotic standard deviation (aStd) derived from the Fisher information matrix. For each parameter component, we also report the empirical left and right tail probabilities
where $j$ indexes the components of $\theta=(\mu,\sigma,\gamma)^\prime$. Under asymptotic normality, both probabilities should approach $0.025$ as $n\rightarrow\infty$.
The simulation results support the asymptotic approximation. The estimator of $\mu$ is essentially unbiased across all designs and sample sizes, and its empirical standard deviation is close to the asymptotic standard deviation even for small samples. The scale parameters also behave well once the sample size is moderate. The main finite-sample distortions occur in the smaller samples and involve the scale parameters. In the near-Gaussian design, $\lambda=0.01$, the Cauchy scale is close to the lower boundary of the parameter space, so the finite-sample distribution of $\hat\gamma$ is visibly skewed even though the magnitude of its standard deviation is well approximated by the Fisher information. In the balanced design, $\lambda=1$, the estimator of $\sigma$ exhibits some small-sample skewness, reflecting the difficulty of separating the Gaussian core from the Cauchy tails with limited data. These distortions diminish rapidly as $n$ increases.
Overall, the table indicates that the Voigt MLE is well centered and that the Fisher-information approximation provides an accurate description of sampling variability over a wide range of designs. The remaining discrepancies are finite-sample effects associated with weak identification of one of the two scale components, either because $\gamma_0$ is close to zero or because the sample is too small to clearly distinguish the Gaussian and Cauchy contributions.
We now introduce the Gauss-Cauchy Convolution (GCC) filter, a robust recursive method for extracting a latent state when measurement errors contain both Gaussian background noise and Cauchy outliers. The filter is built from the Voigt likelihood derived above and uses its score, or equivalently the conditional mean of the latent Gaussian component, to update the state recursively.
The standard Kalman filter is optimal in linear Gaussian systems, but it is not robust to heavy-tailed measurement noise: a single large outlier can move the state estimate in proportion to its magnitude. The GCC filter replaces the linear Kalman update with a nonlinear update implied by the Gauss-Cauchy convolution. For moderate prediction errors, the update is approximately linear, resembling the Kalman filter. For large prediction errors, however, the conditional mean redescends toward zero, so the filter automatically discounts observations that are more plausibly attributed to the Cauchy component than to the latent Gaussian signal.
A key advantage of the GCC filter is analytical tractability. Conditional on a Gaussian prediction density for the latent state, the prediction-error density is Voigt and therefore available in closed form. This contrasts with many robust alternatives, such as filters based on Student-$t$ measurement noise, where the convolution of a Gaussian prediction density and a Student-$t$ error does not generally yield a closed-form prediction-error density. The pure Cauchy measurement-error filter of McCabeGualdoni:2024 is an important exception and is nested as a limiting case of the present framework. The GCC filter extends this case by allowing measurement errors to contain both Gaussian and Cauchy components, a distinction that is empirically useful when ordinary background noise is approximately Gaussian but occasional observations are contaminated by heavy-tailed outliers.
Finally, the convolution parameters are estimated by quasi maximum likelihood using the analytical density, score, and Hessian derived above. Thus the filter inputs are obtained from likelihood-based estimation rather than ad-hoc tuning.
Let $x_t$ denote a latent state variable that follows a Gaussian AR(1) process, and $y_t$ denote the observed variable, measured with additive error:
The state innovations, Gaussian measurement-error components, and Cauchy measurement-error components are mutually independent and independent over time. The initial state is independent of all subsequent innovations. Thus the latent state is Gaussian, while the measurement error follows a Voigt distribution. Equivalently, the measurement error contains a Gaussian component with scale $\sigma$ and a Cauchy component with scale $\gamma$.
We use standard state-space notation Harvey:1989, DurbinKoopman:2012, with lowercase letters for the state and observation variables to avoid conflict with the convolution variables used above. Let $\mathcal{F}_t=\sigma(y_t,y_{t-1},\ldots)$ denote the natural filtration, and define $$ x_{t|t-1} = \mathbb{E}[x_t|\mathcal{F}_{t-1}], \quad\text{and}\quad h_{t|t-1} = \operatorname{var}(x_t|\mathcal{F}_{t-1}). $$ Here $h_{t|t-1}$ denotes the one-step-ahead prediction variance of the latent state. The linear Gaussian state transition gives the usual prediction equations: $$ x_{t|t-1} = (1-\phi)\mu+\phi x_{t-1|t-1}\quad\text{and}\quad h_{t|t-1} = \phi^2 h_{t-1|t-1}+\tau^2. $$
The nonlinear part of the GCC filter enters through the update step. Following the approximation introduced by Masreliez:1975, we approximate the one-step-ahead prediction density of the state by a Gaussian distribution.
This approximation was originally introduced by Masreliez:1975 for robust radar tracking in the presence of so-called glint noise, a form of sporadic heavy-tailed measurement contamination. In the present setting, the approximation is useful for a more specific reason: it turns the prediction error into a Gaussian convolution. Let $$ \xi_t=x_t-x_{t|t-1}, \qquad e_t=\xi_t+\eta_t. $$ Under Assumption (ref), conditionally on $\mathcal F_{t-1}$, $\xi_t$ is Gaussian and independent of $\eta_t$. Proposition (ref) can therefore be applied conditionally to obtain the state correction from the score of the one-step-ahead prediction-error density: $$ \mathbb{E}[\xi_t|e_t,\mathcal F_{t-1}] = -h_{t|t-1}\frac{\partial}{\partial e_t}\log f_{t-1}(e_t), $$ where $f_{t-1}$ denotes the conditional density of $e_t$ given $\mathcal F_{t-1}$.
The second ingredient is specific to the GCC model. Since $\eta_t\sim\mathcal V(0,\sigma,\gamma)$, we may write $\eta_t=Z_t+C_t$, where $Z_t\sim\mathcal N(0,\sigma^2)$ and $C_t\sim\operatorname{Cauchy}(0,\gamma)$. Hence $$ e_t=(\xi_t+Z_t)+C_t, \qquad \xi_t+Z_t|\mathcal F_{t-1}\sim\mathcal N(0,h_{t|t-1}+\sigma^2), $$ so the prediction-error density remains Voigt, specifically $e_t|\mathcal F_{t-1}\sim\mathcal V(0,\delta_t,\gamma)$ with $\delta_t^2=h_{t|t-1}+\sigma^2$. Thus Assumption (ref) and the Gaussian-Cauchy convolution structure work together: Tweedie's formula gives the moment update, and the Voigt closure property makes the required prediction-error score available in closed form.
The update has the same structure as the Kalman filter, except that the linear Gaussian score $e_t/\delta_t^2$ is replaced by the Voigt location score $\psi_t$. For moderate prediction errors, the update is approximately linear and resembles the Kalman update. For large prediction errors, however, the Voigt score redescends toward zero, so the filter discounts observations that are more plausibly attributed to the Cauchy component than to the latent Gaussian state.
Figure (ref) illustrates this mechanism. The left panel shows how the Voigt score changes with the Cauchy-to-Gaussian ratio $\lambda=\gamma/\sigma$. Larger values of $\lambda$ generate stronger redescending behavior. The right panel compares the GCC score with Gaussian, Huber, and Student-$t$ scores. The Gaussian score is linear, the Huber score is bounded, and the Student-$t$ score is redescending. The GCC score is also redescending, but its shape is tied directly to the Gaussian-Cauchy convolution rather than imposed as a robustification device.
Under the Gaussian prediction approximation in Assumption (ref), the recursions in Theorem (ref) coincide with the approximate conditional mean filter of Masreliez:1975. The update is therefore optimal within the class of filters that represent the one-step-ahead state density by a Gaussian approximation and update its first two conditional moments using the prediction-error density.
The next lemma records that the variance recursion remains positive under the Gaussian prediction approximation.
Next, we introduce the smoothing algorithm.
The GCC filter produces filtered estimates $x_{t|t}$ and $h_{t|t}$. For applications in which a full-sample estimate of the latent state is desired, we construct smoothed estimates using a fixed-interval recursion (the Rauch-Tung-Striebel smoother) based on the Gaussian state transition and the moment approximation that underlies the GCC filter.
Starting from $x_{T|T}$ and $h_{T|T}$, the smoothed estimates are obtained backward by $$ x_{t|T} = x_{t|t} + c_t \left( x_{t+1|T}-x_{t+1|t} \right), \qquad c_t = \frac{\phi h_{t|t}}{h_{t+1|t}}. $$ for $t=T-1,\ldots,1$. The smoothed variance is $$ h_{t|T} = h_{t|t} + c_t^2 \left( h_{t+1|T}-h_{t+1|t} \right). $$ Here $x_{t|t}$, $h_{t|t}$, $x_{t+1|t}$, and $h_{t+1|t}$ are produced by the forward GCC filter.
Given the system model $y_t = x_t + Z_t + C_t$, where $x_t$ is the latent state, $Z_t \sim \mathcal{N}(0, \sigma^2)$ represents Gaussian measurement noise, and $C_t \sim \operatorname{Cauchy}(0, \gamma)$ captures the heavy-tailed measurement, we can characterize the optimal extraction of the Cauchy component as follows.
Let $\xi_t = x_t - x_{t|t-1}$ denote the state prediction error, and define the total Gaussian unobserved component as $G_t = \xi_t + Z_t$. The prediction error can then be written as $e_t = y_t - x_{t|t-1} = G_t + C_t$.
Under the Masreliez approximation, $G_t \mid \mathcal{F}_{t-1} \sim \mathcal{N}(0, \delta_t^2)$, where the total Gaussian scale is $\delta_t^2 = h_{t|t-1} + \sigma^2$. Applying the exact Voigt identity (Corollary (ref)), the expected total Gaussian component given the concurrent information is:
The filtered Cauchy component is isolated residually:
The total Gaussian component $G_t$ is subsequently allocated between the state update and the measurement noise update in proportion to their prior variances:
The parameters of the GCC state-space model are estimated by maximizing the quasi log-likelihood implied by the Masreliez prediction approximation. Let $\vartheta=(\mu,\sigma,\gamma,\phi,\tau)^\prime$ and define, recursively, $e_t(\vartheta)=y_t-x_{t|t-1}(\vartheta)$ and $\delta_t^2(\vartheta)=h_{t|t-1}(\vartheta)+\sigma^2$. The one-step quasi log-likelihood contribution is $$ \ell_t(\vartheta) = -\tfrac{1}{2}\log(2\pi) -\log\delta_t(\vartheta) + \log\mathsf{u}(e_t(\vartheta);0,\delta_t(\vartheta),\gamma), $$ and the full-sample objective is $$ \ell_T(\vartheta)=\sum_{t=1}^T\ell_t(\vartheta). $$ The QMLE is $$ \hat\vartheta_T = \arg\max_{\vartheta\in\Theta}\ell_T(\vartheta). $$ The recursion is initialized at the stationary moments, $x_{1|0}=\mu$ and $h_{1|0}=\tau^2/(1-\phi^2)$.
The objective is a quasi-likelihood because the prediction-error density is evaluated under the Gaussian prediction approximation in Assumption (ref). Conditional on the recursively generated prediction moments, however, each likelihood contribution is the exact Voigt density of the implied prediction error. The analytical derivatives in Lemma (ref) and the algebraic closure property in Corollary (ref) make it possible to compute the likelihood and its derivatives without numerical convolution or finite differences.
Under the idealized model in which the Gaussian prediction approximation is exact (Assumption (ref)), the quasi-likelihood is correctly specified, so $\vartheta_\star=\vartheta_0$, $\mathcal{I}_{\vartheta_0}=\mathcal{J}_{\vartheta_0}$, and $\sqrt{T}\left(\hat{\vartheta}_T-\vartheta_0\right) \xrightarrow{d} \mathcal{N} \left(0,\mathcal{I}_{\vartheta_0}^{-1} \right)$. For the exact GCC state-space model, we treat this as a conjecture and evaluate it numerically below.
Table (ref) reports Monte Carlo evidence for the finite-sample behavior of the GCC QMLE. The simulations cover two measurement-error designs: a small Cauchy component, $\gamma_0/\sigma_0=0.1$, and a balanced design, $\gamma_0/\sigma_0=1$. In both cases the estimators are well centered, and the empirical standard deviations are close to the inverse-information standard deviations even for moderate sample sizes.
The small-Cauchy design in Panel A is especially regular. The estimates of $\sigma$, $\gamma$, $\mu$, and the state innovation scale are nearly unbiased throughout the table, and the empirical standard deviations track the asymptotic standard deviations closely. The main finite-sample distortion is in the persistence parameter $\phi$, which is biased downward in small samples and has right-tail rejection probabilities above the nominal benchmark. This distortion shrinks steadily as $T$ increases, although at sample sizes typical of financial applications the remaining bias in $\hat\phi$ can still imply some understatement of persistence. This is most naturally interpreted as a finite-sample dynamic-parameter issue rather than a Masreliez approximation effect.
Panel B shows the more challenging case in which the Gaussian and Cauchy measurement-error scales are equal. The estimator of $\gamma$ remains well centered even in small samples, whereas $\hat\sigma$ displays a noticeable downward bias for small $T$. This reflects the difficulty of separating the Gaussian core from the Cauchy component when the two scales are of comparable magnitude and the sample is short. The bias decreases monotonically with $T$, and by $T=8{,}000$ or $T=16{,}000$ the empirical standard deviations are very close to their asymptotic counterparts.
The tail probabilities provide a useful diagnostic for the normal approximation. For $\mu$, the empirical left and right probabilities are close to the nominal $0.025$ benchmark across both panels, even for small samples. The scale and persistence parameters show more finite-sample skewness, especially $\sigma$ in Panel B and $\phi$ in both panels. These deviations diminish with the sample size, consistent with the regular inverse-information approximation becoming accurate away from boundary cases.
The Monte Carlo evidence is consistent with the conjecture that the pseudo-true parameter, $\vartheta_{\star}$, coincides with the true data-generating parameter, $\vartheta_{0}$. Across the designs considered, the QMLE is well centered, and the inverse-information standard errors provide a good approximation to the sampling variability. This suggests that the information loss induced by the Masreliez approximation is small in these settings. The remaining discrepancies are concentrated in the small-sample behavior of the scale decomposition and the persistence parameter.
The GCC filter rests on two linked implications of Assumption (ref). First, the Gaussian approximation to the predictive state density makes the update a Tweedie moment correction for a Gaussian convolution. Second, conditional on this approximation, the GCC measurement-error structure keeps the prediction-error density Voigt, so the required score is available in closed form. The second step is exact given the approximation; the simulations below assess the numerical cost of the first step.
The purpose of this section is therefore not to evaluate finite-sample maximum likelihood performance. Instead, we hold the model parameters fixed at their true values and assess the numerical consequences of Assumption (ref) for the filtering recursion itself. The central question is how much is lost by replacing the exact one-step-ahead predictive distribution of the latent state by the Gaussian approximation used by the GCC filter.
The benchmark is an exact filter computed by numerical density propagation, in the spirit of the non-Gaussian filtering and smoothing approach of Kitagawa:1987. At each date, Bayes' rule is used to update the full predictive density of the latent state, and the resulting filtering density is propagated forward through the Gaussian state transition. This benchmark does not impose Gaussianity on the predictive state density. It is therefore useful for assessing the accuracy of the Masreliez approximation, although it is not the operational filter we propose. Details on the numerical benchmark and the diagnostic definitions are provided in Appendix (ref).
We compare the exact benchmark with two Gaussian approximations. The first is a moment-matched Gaussian approximation that uses the exact predictive mean and variance. This isolates the shape error from Gaussianizing the exact predictive density. The second is the operational GCC approximation, which uses the predictive mean and variance generated by the GCC recursion. This measures the full approximation error of the implemented filter.
We evaluate the approximation in two ways. First, we compute Kullback-Leibler discrepancies for both the latent-state predictive density and the implied observation predictive density. Second, we compare the one-step correction generated by the exact benchmark with the correction generated by the GCC filter, since this correction is the object that drives the filtered state.
Because the model is equivariant to common changes in scale, the simulation design is indexed by the dimensionless ratios $\lambda=\gamma/\sigma$ and $\tau/\sigma$, together with the persistence parameter $\phi$. The grid includes $\lambda=0.10$, which is close to the value estimated in the empirical application.
The density-level diagnostics in Table (ref) show that the Gaussian prediction approximation is highly accurate across the simulation designs. For empirically relevant values $\lambda\leq0.10$, the average KL discrepancies for the latent-state density are on the order of $10^{-4}$ or smaller, and the discrepancies for the observation density are an order of magnitude smaller. This attenuation reflects the smoothing effect of convolution with the Voigt measurement-error density.
Table (ref) shows that these density-level differences have little effect on the filter update. For $\lambda\leq0.10$, the mean absolute operational distortion is between $10^{-3}$ and $5\times10^{-3}$, with root mean square errors below $1.5\times10^{-2}$. The operational diagnostics are only modestly larger than the shape diagnostics, indicating that the GCC recursion does not materially amplify the approximation error.
The largest discrepancies occur in stress designs with $\lambda=0.50$ or $\lambda=1.00$, but even there the distortions remain moderate. Overall, the evidence suggests that violations of Assumption (ref) have limited practical consequences for the GCC filter, especially in the empirically relevant region where the Cauchy component is small relative to the Gaussian measurement-error component.
We now apply the GCC filter to realized volatility data. The objective is to evaluate whether the Gauss-Cauchy measurement-error specification provides a useful empirical decomposition of realized volatility into a persistent latent component and transitory measurement noise. We compare the GCC filter with several alternative filters that use the same Gaussian AR(1) state equation but differ in their measurement-error distributions.
We apply the GCC filter to the daily logarithm of realized volatility for the Technology Select Sector SPDR Fund (XLK). The sample runs from December 22, 1998 to November 7, 2025, and the realized-volatility series is obtained from the Dacheng Xiu Risk Lab. Realized measures are constructed from high-frequency intraday returns and are therefore informative about daily variation in volatility, but they may also contain extreme observations associated with market stress, liquidity disruptions, and market microstructure effects.
We work with log realized volatility. This transformation reduces the right-skewness of realized volatility and makes a symmetric measurement-error specification more plausible. In the XLK sample, the skewness of the log realized-volatility series is 0.084.
We compare the GCC filter with several filters that retain the same Gaussian AR(1) state equation but impose different specifications on the measurement error $\eta_t$. The comparison is designed to separate two issues. The first is whether the measurement-error distribution is sufficiently flexible to account for outliers in realized volatility. The second is whether the implied prediction-error density is available in closed form after convolution with the Gaussian state prediction error.
Table (ref) summarizes the competing specifications. The Gaussian filter corresponds to the Kalman benchmark. The Cauchy filter is the opposite limiting case in which all measurement error is heavy-tailed. The Normal-Laplace filter provides a light-tailed robust alternative with a closed-form Gaussian convolution. The Student-$t$ and Huber filters are included as common robust benchmarks, but their prediction-error densities are not closed under convolution with Gaussian state uncertainty. Implementations of these filters therefore require either numerical integration or an approximation to the prediction-error density.
Table (ref) reports estimation results for the competing filters. The GCC specification attains the largest value of the implemented prediction-error criterion, $\ell=-1,306$, substantially improving upon the Gaussian benchmark, $\ell=-2,465$. This large difference indicates that a purely Gaussian measurement-error specification is too restrictive for log realized volatility.
Allowing for heavy tails improves the criterion across all alternatives. The Student-$t$ filter increases the implemented criterion to $\ell=-1,349$, with an estimated degrees-of-freedom parameter $\nu\approx 5.25$, while the pure Cauchy specification yields $\ell=-1,536$. The comparison with the Student-$t$ and Huber filters should be interpreted as a comparison of the operational filtering rules, since their reported values are pseudo-log likelihoods based on same-family approximations to the prediction-error density. With this qualification, the empirical ranking favors the GCC filter.
The Normal-Laplace estimate is also informative: its Gaussian scale is essentially zero, so the fitted specification is close to a pure-Laplace measurement-error model. Thus, within the Normal-Laplace family, the data do not support a separately identified Gaussian measurement-error component once the Laplace component is included.
The GCC estimates imply a measurement-error decomposition with a dominant Gaussian component and a small but non-negligible Cauchy component. The estimated parameters are $\sigma=0.1817$ and $\gamma=0.0199$, corresponding to $\lambda=\gamma/\sigma\approx 0.11$. This estimate lies in the range for which the simulation results in Section (ref) show that the Gaussian prediction approximation is essentially exact. Thus, the improvement is not driven by replacing Gaussian noise with a purely heavy-tailed specification, but by allowing for occasional large deviations while retaining a Gaussian core. In contrast, the Gaussian filter absorbs these extreme observations by inflating the measurement variance, $\sigma=0.2980$, which leads to a noisier extraction of the latent state.
Beyond empirical fit, the GCC specification also has an analytical advantage. As indicated in Table (ref), the prediction-error density is available in closed form for the GCC, Gaussian, Cauchy, and Normal-Laplace models under the Gaussian prediction approximation. This allows the likelihood contribution and score to be evaluated without numerical convolution. By contrast, the Student-$t$ and Huber filters are not closed under convolution with Gaussian state uncertainty, so their implementations rely on approximations to the prediction-error density.
Finally, all models imply highly persistent latent volatility dynamics, with estimates of $\phi$ close to $0.97$. The exception is the pure Cauchy specification, which yields a somewhat lower persistence estimate ($\phi\approx 0.936$), reflecting its greater reliance on the measurement-error component to absorb large fluctuations.
Figure (ref) plots the observed log realized-volatility series and the filtered GCC estimate of the latent volatility component. The observed series contains several large spikes associated with periods of market stress, including the bursting of the Dot-Com bubble, the period following September 11, 2001, the Flash Crash on May 6, 2010, and the market disruption on August 24, 2015. The sample also contains unusually low realized-volatility measurements in the early part of the sample, which are plausibly related to market microstructure effects and thin trading.
The filtered state is substantially smoother than the observed series and is not pulled strongly toward isolated extreme observations. This behavior is a direct consequence of the redescending location score of the Voigt prediction-error density, for which $\psi_t\to0$ as $|e_t|\to\infty$. Large prediction errors are therefore downweighted in the state update and are instead attributed primarily to the heavy-tailed component of the measurement error.
The right panel of Figure (ref), labeled “Measurement errors,” illustrates the filtered measurement-error decomposition. The Gaussian component captures regular, small-to-moderate measurement variation around the latent state, while the Cauchy component absorbs the most extreme deviations. Thus, the GCC filter preserves sensitivity to ordinary daily variation in realized volatility while limiting the influence of observations that are more plausibly interpreted as transitory measurement disturbances.
This paper develops likelihood-based inference and filtering methods for the Gauss-Cauchy convolution model. The resulting Voigt density has long been known in spectroscopy, but its statistical use has been limited by the perception that likelihood evaluation, differentiation, and filtering are computationally burdensome. We show that this perception is misleading.
The key observation is that the scaled complementary error function provides an algebraically tractable representation of the Voigt density. Its differential properties yield closed-form expressions for the likelihood, score, Hessian, and conditional moments. As a result, maximum likelihood estimation for the Gauss-Cauchy convolution can be implemented without numerical convolution, quadrature, finite-difference derivatives, or pseudo-Voigt approximations.
The same structure also yields a natural robust filtering method. In the GCC state-space model, the prediction error has a Voigt density under the Masreliez Gaussian prediction approximation. The state update is therefore driven by the location score of this density. This score is approximately linear near the center but redescends in the tails, so extreme observations are discounted rather than treated as persistent movements in the latent state. In this sense, the GCC filter provides a probabilistic alternative to ad hoc robust filtering rules: robustness arises from the conditional expectation implied by the convolution model itself.
We also assess the numerical consequences of the Masreliez approximation directly. Using an exact benchmark filter based on density propagation, we show that the Gaussian prediction approximation is highly accurate over a broad range of parameter configurations. The approximation error is small both at the level of predictive densities and, more importantly, at the level of the one-step correction that drives the filter. In the empirically relevant region where the estimated Cauchy-to-Gaussian ratio is small, the approximation error is negligible.
The empirical application to log realized volatility for XLK illustrates the practical value of the approach. The GCC filter improves substantially on the Gaussian benchmark and outperforms common robust alternatives. The estimates indicate a dominant Gaussian measurement-error component together with a small Cauchy component that absorbs occasional extreme observations. This decomposition allows the filtered volatility state to remain smooth without being unduly influenced by transient spikes or microstructure-related disturbances.
The likelihood theory developed here may also be useful for modern machine-learning approaches to Voigt fitting. Recent work in MRI spectroscopy and quasar absorption-line analysis uses machine learning to accelerate Voigt-profile fitting by training algorithms to reproduce least-squares or traditional profile-fitting outputs. Our results complement this approach by characterizing the likelihood-based statistical target itself. In particular, the score and Fisher information can be used to assess local parameter identification, construct uncertainty measures, and provide one-step likelihood corrections to fast machine-learning estimates.
Overall, the paper shows that the Gauss-Cauchy convolution is not merely a useful heavy-tailed density, but a tractable framework for likelihood inference, robust signal extraction, and state-space filtering. Natural extensions include multivariate GCC models, richer latent-state dynamics, hybrid machine-learning and likelihood-based estimation, and applications in other settings where persistent Gaussian variation is observed through intermittent heavy-tailed measurement noise.