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# Nonparametric Estimation of the Laplace Transform from Empirical Data Using Symmetric Kernels Method

Raogo Frank Emile Er Jumeau Kabore, Sobom Matthieu Some, Abel Zongo, S Pierre Clovis Nitiema

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## To cite this version:

Raogo Frank Emile Er Jumeau Kabore, Sobom Matthieu Some, Abel Zongo, S Pierre Clovis Nitiema. Nonparametric Estimation of the Laplace Transform from Empirical Data Using Symmetric Kernels Method. Journal of Advances in Mathematics and Computer Science, 2025, 40, pp.23 - 32.
10.9734/jamcs/2025/v40i62008. hal-05094202

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# Nonparametric Estimation of the Laplace Transform from Empirical Data Using Symmetric Kernels Method

Raogo Frank Emile 1er Jumeau KABORE a*, Sobom Matthieu SOME b,
Abel ZONGO a and S. Pierre Clovis NITIEMA b

## Authors' contributions

This work was carried out in collaboration among all authors. All authors read and approved the final manuscript.

## Article Information

DOI: https://doi.org/10.9734/jamcs/2025/v40i62008

## Open Peer Review History:

This journal follows the Advanced Open Peer Review policy. Identity of the Reviewers, Editor(s) and additional Reviewers, peer review comments, different versions of the manuscript, comments of the editors, etc are available here: https://pr.sdiarticle5.com/review-history/136473

Received: 10/03/2025

Accepted: 15/05/2025

Published: 30/05/2025

## Original Research Article

### Abstract

Focusing on the use of a Gaussian kernel, the paper constructs a nonparametric Laplace transform estimator via symmetric kernel methodology. Convergence rates and an asymptotic mean integrated squared error (AMISE) formula are used to establish theoretical properties of the estimator and guide optimal bandwidth choice. Finite-sample performance is evaluated through comprehensive simulations. Real and synthetic data applications illustrate practical utility. Estimator robustness and accuracy are showed in outcomes, interpretation the implement valuable across statistical and practical fields.

<footnote>a Département de Mathématiques, Université Joseph KI-ZERBO, 03 BP 7021, Ouagadougou, Burkina Faso.</footnote>

<footnote>b Département de Mathématiques de Décision, Université Thomas SANKARA, 12 BP 417, Ouagadougou, Burkina Faso.</footnote>

<footnote>*Corresponding author: E-mail: rkefrank@gmail.com;</footnote>
Keywords: Laplace transform; gaussian kernel; integral estimator; bandwidth.

2020 Mathematics Subject Classification: 60E10, 62G05, 62G07.

# 1 Introduction

The analytical computation of the Laplace transform of certain probability densities often encounters difficulties due to the complexity of the functions involved.

In many situations for example, when the Laplace transform of the claim size distribution, the claim size distribution itself, or the inter-claim time distribution are not explicit or are unknown several authors have proposed estimation and approximation methods (see notably (Chauveau et al., 1994; Libengué Dobélé Kopoka, 2013; Albrecher and Boxma, 2004; Lima et al., 2002; Usabel, 1999; Goffard et al., 2015; Nitiema et al., 2016; Thakur et al., 2024; Zhang and Yang, 2013; Harfouche and Bareche, 2021; Yasutaka; Touazi et al., 2014; Dussap, 2022)). Mnatsakanov et al. (2008) studied a nonparametric estimation of the Laplace transform of the survival probability in classical compound Poisson risk model (see Ross (2003); Geurts (2019); Jarman (2002) for ruin theory), where the claim size distribution is unknown and is estimated using a discrete Laplace transform. Furthermore, to model situations in which claims can occur in large numbers over a short period, Zhang et al. proposed a nonparametric estimation of the unknown density $f_X$ using continuous kernel methods, relying on the Pollaczek-Khinchine formula to express the ruin probability. More recently, Goffard et al. Jammalamadaka and Meintanis (2022) introduced a kernel estimator of the Laplace transform of a density, with applications to goodness-of-fit tests for exponential and gamma distributions. Mnatsakanov et al. Elmagbri and Mnatsakanov (2018) also proposed a nonparametric procedure based on an approximation of the Laplace transform inversion, allowing the estimation of the density of a positive random variable using a moment-reconstruction approach. They show that the mean squared error (MSE) of the estimated density is bounded, with a convergence rate of order $o(\sqrt{\alpha/n})$, where $\alpha \sim n^{2/5}$. In this paper, we propose a nonparametric estimation of the Laplace transform $\varphi(s)$, considered as a function of a complex variable $s$, of an unknown density $f_X(x)$, based on random observations and using the method of symmetric continuous kernels.

Our contribution lies in estimating the Laplace transform as a continuous function of the complex variable $s$, along with an analysis of the bias, variance, smoothing parameter, and the asymptotic mean integrated squared error (AMISE), all expressed in terms of the Laplace transform. Assuming we are given a sample of independent and identically distributed (i.i.d.) random variables $X_1, X_2, \dots, X_n$, with mean $m = E(X)$, and following an unknown density $f_X$, and under certain technical assumptions on the complex-variable function $\varphi(s)$, we define an empirical estimator of the Laplace transform, denoted $\hat{\varphi}_n(s)$. The paper is organized as follows:

Section 2 introduces the definition of the estimator $\hat{\varphi}_n(s)$ of the Laplace transform of the unknown density, and provides an expression for it, leading to the derivation of the optimal smoothing parameter as a function of the Laplace transform.

Section 3 is devoted to simulations, where the MISE is computed from the theoretical Laplace transform and the estimated one. For real datasets, the optimal smoothing parameter is selected using cross-validation based on the Laplace transform of the Gaussian kernel.

Section 4 concludes the paper.

Let us define: $\mathbb{R}^+$: the set of positive real numbers; $\mathbb{E}$: the expectation operator; $\mathbb{Var}$: the variance operator; $\mathbb{C}$: the set of complex numbers; $\mathbb{T}_{\mathbb{C}}$: a domain in $\mathbb{C}$ such that $\text{Re}(s) > 0$ (strictly positive real part); $L$: the set of integrable functions; $L^2$: the set of square-integrable functions. The theoretical results are based on the following assumptions:

$A_1$: $f_X$ is Laplace-transformable;

$A_2$: $\varphi(s) \in L(\mathbb{T}_{\mathbb{C}})$, where $\varphi(s)$ denotes the Laplace transform of an unknown probability density $f_X$;
A3: $\varphi(s) \in L^2(\mathbb{T}_{\mathbb{C}_+})$, with $\mathbb{T}_{\mathbb{C}_+} \subset \mathbb{C}$;

A4: $\int_{\mathbb{T}_{\mathbb{C}_+}} \varphi(s) ds \le 1$;

A5: The estimators $\hat{\varphi}_1(s), \hat{\varphi}_2(s), \dots, \hat{\varphi}_n(s)$ are assumed to be independent and identically distributed (i.i.d.).

## 2 Estimation of the Unknown Density via Laplace Transform

Inspired by the work of Zhang et al., particularly their formulation of the Fourier transform applied to an unknown density, we propose an estimation of the Laplace transform of an unknown probability density using a continuous symmetric kernel.

**Definition 2.1.** Let $h > 0$ be the smoothing bandwidth and $K$ a continuous symmetric kernel function with support $\mathcal{S}$. The pointwise continuous symmetric kernel estimator of the Laplace transform $\varphi(s)$, defined on a complex domain $\mathbb{T}_{\mathbb{C}_+}$, for an unknown density $f_X$ with support $\mathbb{T} \subseteq \mathbb{R}^+$, is given at a point $x \in \mathbb{T}$ by:

$$ \hat{\varphi}_n(x, s) := \frac{1}{nh} \sum_{i=1}^{n} e^{-xs} K \left( \frac{x - X_i}{h} \right), \quad (1) $$

where $x$ is the target point, $X_i$ the $i$-th random variable, and $s \in \mathbb{T}_{\mathbb{C}_+}$ such that $\mathrm{Re}(s) > 0$.

We then give the definition of Laplace transform integral estimator for the unknown density at $x \in \mathbb{T}$.

**Definition 2.2.** The continuous symmetric kernel integral estimator of the Laplace transform $\varphi(s)$ of the unknown density $f_X$ with support $\mathbb{T}$ is defined for all $x \in \mathbb{T} \subseteq \mathbb{R}^+$ at a fixed point $s \in \mathbb{T}_{\mathbb{C}_+}$ by:

$$ \hat{\varphi}_n(s) := \int_{\mathbb{T}} \hat{\varphi}_n(x, s) dx, \quad (2) $$

where $\hat{\varphi}_n(x, s)$ is the pointwise estimator at a given point $x$ defined by: (1).

The following lemma provides the bias and variance of the integral estimator $\hat{\varphi}_n(s)$ with continuous symmetric kernel $K$.

**Lemma 2.3.** Let $h > 0$ a real number, $K$ the continuous symmetric kernel with support $\mathcal{S}$, $f_X$ the unknown density with support $\mathbb{T}$. Let $\varphi(x, s)$ a function integrable with respect to variables $x \in \mathbb{T}$, $s \in \mathbb{T}_{\mathbb{C}_+}$ with first and second derivatives bounded in $x \in \mathbb{T}$. Let $\hat{\varphi}_n(s)$ with support $\mathbb{T}_{\mathbb{C}_+}$ the integral estimator of Laplace transform of $f_X$,

$$ \mathrm{Bias} \{ \hat{\varphi}_n(s) \} = \frac{h^2}{2} \sigma_K^2 \int_{\mathbb{T}} \varphi''(x, s) dx + o(h^2), $$

$$ \mathrm{Var} \{ \hat{\varphi}_n(s) \} = \frac{\theta_K}{nh} \left\{ \int_{\mathbb{T}} e^{-xs} \varphi(x, s) dx \right\} + o \left( \frac{1}{nh} \right), $$

where: $\sigma_K^2 = \int_{\mathbb{T}} w^2 K(w) dw < \infty$, $\theta_K = \int_{\mathbb{T}} K^2(w) dw < \infty$, $\varphi(x, s) = e^{-xs} f_X(x)$ and $\varphi''(x, s) = \frac{\partial^2}{\partial x^2} \varphi(x, s)$.

**Proof 2.4.** For a fixed point $x \in \mathbb{T}$ and fixed complex point $s \in \mathbb{T}_{\mathbb{C}}$, starting from the pointwise estimator $\hat{\varphi}_n(x, s)$ given by Definition 2.1, we have:

$$ \begin{align} \mathbb{E} \{ \hat{\varphi}_n(x, s) \} &= \mathbb{E} \left\{ \frac{1}{nh} \sum_{i=1}^{n} e^{-xs} K \left( \frac{x - X_i}{h} \right) \right\} \nonumber \\ &= \frac{1}{h} \mathbb{E} \left\{ e^{-xs} K \left( \frac{x - X_1}{h} \right) \right\} \nonumber \\ &= \frac{1}{h} \int_{\mathbb{T}} e^{-xs} K \left( \frac{x - z}{h} \right) f_X(z) dz, \tag{3} \end{align} $$
by setting $\varphi(z,s) = e^{-xs}f_X(z)$, $-w = \frac{x-z}{h}$, $K(-w) = K(w)$ thus $z = wh+x$ and (3) becomes:

$$ \mathbb{E}\{\hat{\phi}_n(x,s)\} = \int_{\mathbb{T}} K(w) \varphi(x+wh,s) dw $$

and the second-order Taylor expansion of $\varphi(x+wh,s)$ yields:

$$
\begin{aligned}
\mathbb{E}\{\hat{\phi}_n(x,s)\} &= \int_{\mathbb{T}} K(w) \varphi(x+wh,s) dw \\
&= \int_{\mathbb{T}} K(w) \{\varphi(x,s) + wh\varphi'(x,s) + \frac{1}{2}(wh)^2 \varphi''(x,s) + o(w^2h^2)\} dw \\
&= \int_{\mathbb{T}} K(w) \varphi(x,s) dw + h\varphi'(x,s) \int_{\mathbb{T}} wK(w) dw \\
&\quad + \frac{1}{2}h^2 \varphi''(x,s) \int_{\mathbb{T}} w^2 K(w) dw + o(h^2) \\
&= \varphi(x,s) \int_{\mathbb{T}} K(w) dw + h\varphi'(x,s) \int_{\mathbb{T}} wK(w) dw \\
&\quad + \frac{1}{2}h^2 \varphi''(x,s) \int_{\mathbb{T}} w^2 K(w) dw + o(h^2);
\end{aligned}
$$

where $\varphi'(x,s) = \frac{\partial}{\partial x}\varphi(x,s)$ and $\varphi''(x,s) = \frac{\partial^2}{\partial x^2}\varphi(x,s)$.

Let $\sigma_K^2 = \int_{\mathbb{T}} w^2 K(w) dw$ and $\theta_K = \int_{\mathbb{T}} K^2(w) dw$ the kernel constants. From the properties of the continuous symmetric kernel, we obtain:

$$
\begin{aligned}
\text{Bias}\{\hat{\phi}_n(x,s)\} &= \mathbb{E}\{\hat{\phi}_n(x,s)\} - \varphi(x,s) \\
&= \frac{h^2}{2} \sigma_K^2 \varphi''(x,s) + o(h^2),
\end{aligned}
\qquad (4)
$$

by integrating both sides of equality (4), we obtain the bias of the integral estimator $\hat{\phi}_n(s)$:

$$
\begin{aligned}
\text{Bias}\{\hat{\phi}_n(s)\} &= \text{Bias}\left\{\int_{\mathbb{T}} \hat{\phi}_n(x,s) dx\right\} \\
(\text{using independence}) &= \int_{\mathbb{T}} \text{Bias}\{\hat{\phi}_n(x,s)\} dx \\
&= \frac{h^2}{2} \sigma_K^2 \int_{\mathbb{T}} \varphi''(x,s) dx + o(h^2)
\end{aligned}
$$

where $\int_{\mathbb{T}} o(h^2) dx = o(h^2)$,

by indépendance, the covariance is zero, and the variance is then calculated by

$$
\begin{aligned}
\mathbb{V}\text{ar}\{\hat{\phi}_n(x,s)\} &= \mathbb{V}\text{ar}\left\{\frac{1}{nh}\sum_{i=1}^n e^{-xs}K\left(\frac{x-X_i}{h}\right)\right\} \\
&= \frac{1}{n^2h^2}\sum_{i=1}^n \mathbb{V}\text{ar}\left\{e^{-xs}K\left(\frac{x-X_i}{h}\right)\right\} \\
&= \frac{1}{nh^2}\mathbb{E}\left\{e^{-xs}K\left(\frac{x-X_1}{h}\right)\right\}^2 - \frac{1}{n^2h^2}\left\{\mathbb{E}\left\{e^{-xs}K\left(\frac{x-X_1}{h}\right)\right\}\right\}^2 \\
&= \frac{1}{nh^2}\int_{\mathbb{T}}\left\{e^{-xs}K\left(\frac{x-z}{h}\right)\right\}^2 f_X(z) dz \\
&\quad - \frac{1}{n}\left\{\frac{1}{h}\int_{\mathbb{T}}e^{-xs}K\left(\frac{x-z}{h}\right) f_X(z) dz\right\}^2,
\end{aligned}
\qquad (5)
$$
Setting $p(z,s) = e^{-xs}\varphi(z,s)$, $-w = \frac{x-z}{h}$, $K(-w) = K(w)$ thus $z = wh + x$, equation (6) becomes:

$$ 
\begin{aligned}
\mathrm{Var}\{\hat{\Phi}_n(x,s)\} &= \frac{1}{nh^2} \int_{\mathbb{T}} \left\{ K\left(\frac{x-z}{h}\right) \right\}^2 p(z,s) dz - \frac{1}{n} \left\{ \frac{1}{h} \int_{\mathbb{T}} K\left(\frac{x-z}{h}\right) \varphi(z,s) dz \right\}^2 \\
&= \frac{1}{nh} \int_{\mathbb{T}} \{K(w)\}^2 p(x+wh,s) dw - \frac{1}{n} \{E\{\hat{\Phi}_n(x,s)\}\}^2.
\end{aligned} 
\quad (6) $$

A second-order Taylor expansion of $p(x+wh,s)$ yields:

$$ p(x+wh,s) = p(x,s) + whp'(x,s) + \frac{1}{2}(wh)^2 p''(x,s) + o(h^2), $$

$$ \text{where: } p'(x,s) = \frac{\partial}{\partial x} p(x,s) \text{ and } p''(x,s) = \frac{\partial^2}{\partial x^2} p(x,s), $$

by substituting the Taylor expansion of $p(x+wh,s)$ into equation (6), we obtain:

$$ \mathrm{Var}\{\hat{\Phi}_n(x,s)\} = \frac{1}{nh} p(x,s) \int_{\mathbb{T}} K^2(w) dw + \frac{1}{n} p'(x,s) \int_{\mathbb{T}} wK^2(w) dw \\ - \frac{1}{n} [p(x,s) + \mathrm{Bias}\{\hat{\Phi}_n(x,s)\}]^2 + o\left(\frac{1}{nh}\right). $$

As $n \to \infty$ then:

$$ \frac{1}{n} p'(x,s) \int_{\mathbb{T}} w K^2(w) dw \to 0, \text{ and } \frac{1}{n} [p(x,s) + \mathrm{Bias}\{\hat{\Phi}_n(x,s)\}]^2 \to 0, $$

and

$$ 
\begin{aligned}
\mathrm{Var}\{\hat{\Phi}_n(x,s)\} &= \frac{1}{nh} p(x,s) \int_{\mathbb{T}} K^2(w) dw + o\left(\frac{1}{nh}\right) \\
&= \frac{1}{nh} e^{-xs} \varphi(x,s) \int_{\mathbb{T}} K^2(w) dw + o\left(\frac{1}{nh}\right).
\end{aligned} 
\quad (7) $$

Finally, by integrating both sides of (7) with respect to $x$ over $\mathbb{T}$, we obtain the variance of integral estimator $\hat{\Phi}_n(s)$:

$$ \mathrm{Var}\{\hat{\Phi}_n(s)\} = \mathrm{Var}\left\{\int_{\mathbb{T}} \Phi_n(x,s) dx\right\} $$

$$ 
\begin{aligned}
(\text{using independence}) &= \int_{\mathbb{T}} \mathrm{Var}\{\hat{\Phi}_n(x,s)\} dx \\
&= \int_{\mathbb{T}} \left\{ \frac{\theta_K}{nh} p(x,s) + o\left(\frac{1}{nh}\right) \right\} dx \\
&= \frac{\theta_K}{nh} \int_{\mathbb{T}} e^{-xs} \varphi(x,s) dx + o\left(\frac{1}{nh}\right);
\end{aligned} $$

$$ \text{where } \int_{\mathbb{T}} o\left(\frac{1}{nh}\right) dx = o\left(\frac{1}{nh}\right). $$

**Remark 2.5.** Unlike the bias and variance of estimator for the unknown density (which are expressed in terms of the real target $x$), the bias and variance of Laplace transform estimator for unknown density are expressed in terms of Laplace variable $s$.

Through the following theorem, we establish the quadratic approximation of integrated mean squared error (integral) for the Laplace transform estimator of the unknown density and its convergence rate $h$ in case of a continuous symmetric kernel estimator.
**Theorem 2.6.** Let $\varphi(s)$ with complex domain $\mathbb{T}_{C_+}$ be the Laplace transform of the unknown density $f_X$, and K a continuous symmetric kernel. Let $\varphi(x,s)$ be a function integrable with respect to variables $x \in \mathbb{T}$, $s \in \mathbb{T}_{C_+}$ with first and second derivatives bounded in $x \in \mathbb{T}$. Then AMISE and the optimal bandwidth minimizing AMISE are:

$$ AMISE(h) = \frac{\theta_K}{nh} \int_{\mathbb{T}_{C_+}} \left( \int_{\mathbb{T}} e^{-xs} \varphi(x,s) dx \right) ds + \frac{\sigma_K^4 h^4}{4} \int_{\mathbb{T}_{C_+}} \left( \int_{\mathbb{T}} \varphi''(x,s) dx \right)^2 ds, \\ h_{opt} = n^{-\frac{1}{3}} \left\{ \frac{\theta_K \int_{\mathbb{T}_{C_+}} \left( \int_{\mathbb{T}} e^{-xs} \varphi(x,s) dx \right) ds}{\sigma_K^4 \int_{\mathbb{T}_{C_+}} \left( \int_{\mathbb{T}} \varphi''(x,s) dx \right)^2 ds} \right\}^{\frac{1}{3}}, \quad (8) $$

where $\theta_K$, $\sigma_K$, $\varphi(x,s)$ and $\varphi''(x,s)$ are given in Lemma 2.3.

**Proof 2.7.** Using the results of the Lemma 2.3, we have:

$$ AMSE(h,s) = \mathbb{E} \left[ \int_0^\infty (\hat{\phi}_n(x,s) - \varphi(x,s))^2 dx \right] \\ = \mathrm{Var} \{ \hat{\phi}_n(s) \} + \{ \mathrm{Bias} \hat{\phi}_n(s) \}^2 \\ = \frac{\theta_K}{nh} \int_{\mathbb{T}} e^{-xs} \varphi(x,s) dx + \frac{h^4}{4} \sigma_K^4 \left( \int_{\mathbb{T}} \varphi''(x,s) dx \right)^2, \quad (9) $$

after integrating with respect to complex variable $s$ in (9), we obtain the asymptotic approximation of integrated mean squared error, denoted by AMISE(h):

$$ AMISE(h) = \int_{\mathbb{T}_{C_+}} AMSE(h,s) ds \\ = \frac{\theta_K}{nh} \int_{\mathbb{T}_{C_+}} \left( \int_{\mathbb{T}} e^{-xs} \varphi(x,s) dx \right) ds + \frac{h^4}{4} \sigma_K^4 \int_{\mathbb{T}_{C_+}} \left( \int_{\mathbb{T}} \varphi''(x,s) dx \right)^2 ds, \quad (10) $$

by partially differentiating both sides of the equation (10) with respect to $h$ and setting the result to zero and after some calculations and simplifications, we obtain the result (8).

## 3 Simulations

To facilitate numerical simulations and interpretation of results, the complex variable $s$ of Laplace transform is restricted to positive real values, as has also been adopted in several recent works such as Todorov and Tauchen (2012), Curato et al. (2018), Elmagbri and Mnatsakanov (2018) and Gadallah et al. (2022).

We consider a known theoretical density, compute the optimal smoothing parameter $h_{opt}$ given in (8), which will be used to evaluate the MISE after $N$ replications, and compute the variance of the integral estimator $\hat{\phi}_n$ given in equation (2). For each known distribution, we generate random variables following this distribution, which are then estimated using the Gaussian kernel. The Laplace variable $s$ is discretized over a given interval to allow for the numerical evaluation of the estimated Laplace transform. All integrals over $x$ and $s$ are approximated using Euler's method (constant step discretization) implemented in MATLAB R2022b.

Table 1 presents the performance of Laplace transform estimator $\hat{\phi}_n(s)$ for different distributions and sample sizes. We observe that the mean integrated squared error (MISE) and variance systematically decrease with $n$, indicating better accuracy for larger samples. The estimation is significantly more stable for the Gaussian and Gamma distributions compared to Weibull distribution, which initially shows higher variance.
<table><caption>Table 1. Estimation of $\phi_n(s)$ for different probability densities, with $N = 50$ replications</caption><thead><tr><th rowspan="2">n</th><th colspan="3">X ~ Gauss (0, 1)</th><th colspan="3">X ~ Gamma (2, 1/2)</th><th colspan="3">X ~ Weibull (1, 3)</th></tr><tr><th>h<sub>opt</sub></th><th>MISE</th><th>Var</th><th>h<sub>opt</sub></th><th>MISE</th><th>Var</th><th>h<sub>opt</sub></th><th>MISE</th><th>Var</th></tr></thead><tbody><tr><td>50</td><td>0.0654</td><td>0.0101</td><td>1.8860e-04</td><td>0.0265</td><td>0.0072</td><td>6.1256e-05</td><td>0.3875</td><td>0.9878</td><td>0.1218</td></tr><tr><td>100</td><td>0.0570</td><td>0.0031</td><td>1.3833e-05</td><td>0.0231</td><td>0.0055</td><td>3.8449e-05</td><td>0.3373</td><td>0.7111</td><td>0.0565</td></tr><tr><td>500</td><td>0.0413</td><td>0.0011</td><td>2.4427e-06</td><td>0.0167</td><td>0.0032</td><td>7.7349e-06</td><td>0.2445</td><td>0.4413</td><td>0.0032</td></tr><tr><td>1000</td><td>0.0359</td><td>0.0006</td><td>5.8952e-07</td><td>0.0146</td><td>0.0016</td><td>2.2051e-06</td><td>0.2128</td><td>0.4169</td><td>0.0012</td></tr><tr><td>3000</td><td>0.0288</td><td>0.0002</td><td>6.6949e-08</td><td>0.0117</td><td>0.0004</td><td>1.4986e-07</td><td>0.1709</td><td>0.4117</td><td>0.0002</td></tr><tr><td>5000</td><td>0.0260</td><td>0.0001</td><td>1.5428e-08</td><td>0.0106</td><td>0.0004</td><td>9.4548e-08</td><td>0.1543</td><td>0.4229</td><td>0.0001</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>Fig. 1. Estimation of the Laplace transform for the Weibull(1, 3) density</figcaption></figure>

Fig. 1 illustrates the convergence of Laplace transform estimator towards its theoretical value as the sample size *n* increases. *hop* is the optimal smoothing parameter computed by applying the formula (8). The gradual decrease of the smoothing parameter
$h_{opt}$ improves the estimation accuracy while controlling variance. For $n \ge 1000$, estimator shows excellent stability and becomes particularly reliable.

<figure><img src="image_6.png" /><figcaption>Fig. 1. Evolution of $\hat{\phi}_n(s)$ for the random claim severity function AutoClaims</figcaption></figure>

Fig. 2 shows evolution of Laplace transform estimator $\hat{\phi}_n(s)$ as a function of variable $s$, for a sample of size $n = 6773$ and an optimal smoothing parameter $h_{CV}$ calculated by cross-validation in terms of the Laplace transform. In this case, MISE cannot be computed since the true density is both unknown and random. *AutoClaims* dataset is available in *CASdatasets* package of R software.

# 4 Conclusion

We have proposed a nonparametric estimation method for Laplace transform of an unknown density based on a sample of independent and identically distributed random variables. This approach, founded on the use of continuous symmetric kernels, enables analytical expressions for the bias, variance, and asymptotic mean integrated squared error (AMISE) in terms of Laplace transform. It thus provides a flexible and robust framework for analyzing situations where explicit form of density, or its Laplace transform, is difficult or even impossible to obtain.

# Disclaimer (Artificial Intelligence)

Author(s) hereby declare that generative AI technologies such as Large Language Models, etc have been used during writing or editing of manuscripts. This explanation will include the name, version, model, and source of the generative AI technology and as well as all input prompts provided to the generative AI technology.

Details of the AI usage are given below:

1. ChatGPT (GPT-4-turbo, OpenAI, April 2024 version), used via the ChatGPT Plus platform was used to rephrase certain sentences in the introduction and conclusion for improved clarity and flow.

2. ChatGPT (GPT-4-turbo, OpenAI, April 2024 version), used via the ChatGPT Plus platform and DeepSeek Chat (Version DeepSeek-V3 2024) was used to check grammar, punctuation, and sentence structure throughout the manuscript.
3. No AI-generated content was accepted without human review and critical editing. All scientific ideas, data analysis, and interpretations are entirely the authors' own.

**Competing Interests**

Author has declared that no competing interests exist.

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