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Lanpeng Ji

Publications and source records attributed to Lanpeng Ji.

At least 19 recordsLinked to original sources
arXiv2025

Bayesian CART models for aggregate claim modeling

This paper proposes three types of Bayesian CART (or BCART) models for aggregate claim amount, namely, frequency-severity models, sequential models and joint models. We propose a general framework for the BCART models applicable to data with multivariate responses, which is particularly useful for the joint BCART models with a bivariate response: the number of claims and aggregate claim amount. To facilitate frequency-severity modeling, we investigate BCART models for the right-skewed and heavy-tailed claim severity data by using various distributions. We discover that the Weibull distribution is superior to gamma and lognormal distributions, due to its ability to capture different tail characteristics in tree models. Additionally, we find that sequential BCART models and joint BCART models, which incorporate dependence between the number of claims and average severity, are beneficial and thus preferable to the frequency-severity BCART models in which independence is assumed. The effectiveness of these models' performance is illustrated by carefully designed simulations and real insurance data.

stat.ME↗
arXiv2024

Probability of entering an orthant by correlated fractional Brownian motion with drift: Exact asymptotics

For $\{B_H(t)= (B_{H,1}(t), \ldots, B_{H,d}(t))^\top,t\ge0\}$, where $\{B_{H,i}(t),t\ge 0\}, 1\le i\le d$ are mutually independent fractional Brownian motions, we obtain the exact asymptotics of $$ \mathbb P (\exists t\ge 0: A B_{H}(t) - μt >νu), \ \ \ \ u\to\infty, $$ where $A$ is a non-singular $d\times d$ matrix and $μ=(μ_1,\ldots, μ_d)^\top\in R^d$, $ν=(ν_1, \ldots, ν_d)^\top \in R^d$ are such that there exists some $1\le i\le d$ such that $μ_i>0, ν_i>0.$

math.PR↗
arXiv2023

Bayesian CART models for insurance claims frequency

Accuracy and interpretability of a (non-life) insurance pricing model are essential qualities to ensure fair and transparent premiums for policy-holders, that reflect their risk. In recent years, the classification and regression trees (CARTs) and their ensembles have gained popularity in the actuarial literature, since they offer good prediction performance and are relatively easily interpretable. In this paper, we introduce Bayesian CART models for insurance pricing, with a particular focus on claims frequency modelling. Additionally to the common Poisson and negative binomial (NB) distributions used for claims frequency, we implement Bayesian CART for the zero-inflated Poisson (ZIP) distribution to address the difficulty arising from the imbalanced insurance claims data. To this end, we introduce a general MCMC algorithm using data augmentation methods for posterior tree exploration. We also introduce the deviance information criterion (DIC) for the tree model selection. The proposed models are able to identify trees which can better classify the policy-holders into risk groups. Some simulations and real insurance data will be discussed to illustrate the applicability of these models.

stat.ML↗
arXiv2022

Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend

We investigate extreme value theory of a class of random sequences defined by the all-time suprema of aggregated self-similar Gaussian processes with trend. This study is motivated by its potential applications in various areas and its theoretical interestingness. We consider both stationary sequences and non-stationary sequences obtained by considering whether the trend functions are identical or not. We show that a sequence of suitably normalised $k$th order statistics converges in distribution to a limiting random variable which can be a negative log transformed Erlang distributed random variable, a Normal random variable or a mixture of them, according to three conditions deduced through the model parameters. Remarkably, this phenomenon resembles that for the stationary Normal sequence. We also show that various moments of the normalised $k$th order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyze various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory.

math.PR↗
arXiv2022

On the maxima of suprema of dependent Gaussian models

In this paper, we study the asymptotic distribution of the maxima of suprema of dependent Gaussian processes with trend. For different scales of the time horizon we obtain different normalizing functions for the convergence of the maxima. The obtained results not only have potential applications in estimating the delay of certain Gaussian fork-join queueing systems but also provide interesting insights to the extreme value theory for triangular arrays of random variables with row-wise dependence.

math.PR↗
arXiv2020

Extrema of multi-dimensional Gaussian processes over random intervals

This paper studies the joint tail asymptotics of extrema of the multi-dimensional Gaussian process over random intervals defined as $$ P(u):=\mathbb{P}\left\{\cap_{i=1}^n \left(\sup_{t\in[0,\mathcal{T}_i]} ( X_{i}(t) +c_i t )>a_i u \right)\right\}, \ \ \ u\to\infty, $$ where $X_i(t), t\ge0$, $i=1,2,\cdots,n,$ are independent centered Gaussian processes with stationary increments, $\boldsymbol{\mathcal{T}}=(\mathcal{T}_1, \cdots, \mathcal{T}_n)$ is a regularly varying random vector with positive components, which is independent of the Gaussian processes, and $c_i\in \mathbb{R}$, $a_i>0$, $i=1,2,\cdots,n$. Our result shows that the structure of the asymptotics of $P(u)$ is determined by the signs of the drifts $c_i$'s. We also discuss a relevant multi-dimensional regenerative model and derive the corresponding ruin probability.

math.PR↗
arXiv2020

On the cumulative Parisian ruin of multi-dimensional Brownian motion models

Consider a multi-dimensional Brownian motion which models the surplus processes of multiple lines of business of an insurance company. Our main result gives exact asymptotics for the cumulative Parisian ruin probability as the initial capital tends to infinity. An asymptotic distribution for the conditional cumulative Parisian ruin time is also derived. The obtained results on the cumulative Parisian ruin can be seen as generalizations of some of the results derived in Debicki et al (2018, Stochastic Processes and Their Applications). As a particular interesting case, the two-dimensional Brownian motion risk model is discussed in detail.

math.PR↗
arXiv2020

Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

We derive the exact asymptotics of \[ P\left( \sup_{t\ge 0} \Bigl( X_1(t) - μ_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - μ_2 s\Bigr)> u \right), \ \ u\to\infty, \] where $(X_1(t),X_2(s))_{t,s\ge0}$ is a correlated two-dimensional Brownian motion with correlation $ρ\in[-1,1]$ and $μ_1,μ_2>0$. It appears that the play between $ρ$ and $μ_1,μ_2$ leads to several types of asymptotics. Although the exponent in the asymptotics as a function of $ρ$ is continuous, one can observe different types of prefactor functions depending on the range of $ρ$, which constitute a phase-type transition phenomena.

math.PR↗
arXiv2019

Logarithmic asymptotics for probability of component-wise ruin in a two-dimensional Brownian model

We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function $P(u)$ for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where $u$ is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of $-\ln P(u)/u$ as $u$ tends to infinity, which depends essentially on the correlation $ρ$ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem.

math.PR↗
arXiv2019

Tail Asymptotic Behavior of the supremum of a class of chi-square processes

In this paper, we analyze a multivariate counterpart of the generalized weighted Kolmogorov-Smirnov statistic, which is the supremum of weighted locally stationary chi-square process over non-compact interval. The boundedness and the exact tail asymptotic behavior of the statistics are derived. We illustrate our findings by two examples where the statistic is defined by Brownian bridge and fractional Brownian motion respectively.

math.PR↗
arXiv2017

Extremal behaviour of hitting a cone by correlated Brownian motion with drift

This paper derives an exact asymptotic expression for \[ \mathbb{P}_{\mathbf{x}_u}\{\exists_{t\ge0} \mathbf{X}(t)- \boldsymbolμt\in \mathcal{U} \}, \ \ {\rm as}\ \ u\to\infty, \] where $\mathbf{X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated $d$-dimensional Brownian motion starting at the point $\mathbf{x}_u=-\boldsymbolαu$ with $\boldsymbolα\in \mathbb{R}^d$, $\boldsymbolμ \in \mathbb{R}^d$ and $\mathcal{U}=\prod_{i=1}^d [0,\infty)$. The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some cases to dimension-reduction of the considered problem. Complementary, we study asymptotic distribution of the conditional first passage time to $\mathcal{U}$, which depends on the dimension-reduction phenomena.

math.PR↗
arXiv2016

Extremes of locally stationary chi-square processes with trend

Chi-square processes with trend appear naturally as limiting processes in various statistical models. In this paper we are concerned with the exact tail asymptotics of the supremum taken over (0; 1) of a class of locally stationary chi-square processes with particular admissible trends. An important tool for establishing our results is a weak version of Slepian's lemma for chi-square processes. Some special cases including squared Brownian bridge and Bessel process are discussed.

math.PR↗
arXiv2016

Extremes of a class of nonhomogeneous Gaussian random fields

This contribution establishes exact tail asymptotics of $\sup_{(s,t)\in\mathbf{E}}$ $X(s,t)$ for a large class of nonhomogeneous Gaussian random fields $X$ on a bounded convex set $\mathbf{E}\subset\mathbb{R}^2$, with variance function that attains its maximum on a segment on $\mathbf{E}$. These findings extend the classical results for homogeneous Gaussian random fields and Gaussian random fields with unique maximum point of the variance. Applications of our result include the derivation of the exact tail asymptotics of the Shepp statistics for stationary Gaussian processes, Brownian bridge and fractional Brownian motion as well as the exact tail asymptotic expansion for the maximum loss and span of stationary Gaussian processes.

math.PR↗
arXiv2015

Extremes of vector-valued Gaussian processes: exact asymptotics

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of $$ P\left(\exists_{t \in [0,T]} \forall_{i=1 ... n} X_i(t)> u \right) $$ as $u\to\infty$, for both locally stationary $X_i$'s and $X_i$'s with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants, that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands-Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.

math.PR↗
arXiv2015

On Parisian ruin over a finite-time horizon

For a risk process $R_u(t)=u+ct-X(t), t\ge 0$, where $u\ge 0$ is the initial capital, $c>0$ is the premium rate and $X(t),t\ge 0$ is an aggregate claim process, we investigate the probability of the Parisian ruin \[ \mathcal{P}_S(u,T_u)=\mathbb{P}\{\inf_{t\in[0,S]} \sup_{s\in[t,t+T_u]} R_u(s)<0\}, \] with a given positive constant $S$ and a positive measurable function $T_u$. We derive asymptotic expansion of $\mathcal{P}_S(u,T_u)$, as $u\to\infty$, for the aggregate claim process $X$ modeled by Gaussian processes. As a by-product, we derive the exact tail asymptotics of the infimum of a standard Brownian motion with drift over a finite-time interval.

math.PR↗
arXiv2015

Extremes of Chi-square Processes with Trend

This paper studies the supremum of a chi-square process with trend over a threshold-dependent-time horizon. Under the assumption that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square process with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed.

math.PR↗
arXiv2014

Extremes of Order Statistics of Stationary Processes

Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$ the $r$th largest order statistics of $X_1(t), \ldots , X_n(t), t\ge 0$. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions $C_{r,T,u}$ is not empty. Imposing the Albin's conditions on $X$, in this paper we obtain an exact asymptotic expansion of this probability as $u$ tends to infinity. Further, we establish the tail asymptotics of the supremum of a generalized skew-Gaussian process and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.

math.PR↗