Theory: GPD tails + DPM bulk + splicing

This page summarizes the two ingredients that the package “splices” together:

1. an extreme value theory (EVT) tail model based on the Generalized Pareto Distribution (GPD) for threshold exceedances;
2. a Dirichlet Process Mixture (DPM) regression model for the bulk (where data are more abundant).

Threshold exceedances and the GPD

Let \(Y\) be a continuous outcome and \(X\) a (possibly vector-valued) covariate. For a high threshold \(u(x)\), define the exceedance variable

\[ Z = Y - u(x), \]

and consider the conditional excess distribution (given that \(Y>u(x)\)):

\[ F_u(z\mid x) = \Pr\bigl(Y-u(x)\le z \mid Y>u(x), X=x\bigr), \qquad z\ge 0. \]

Under EVT regularity conditions, when \(u(x)\) is large, the exceedances are well-approximated by a GPD. Write the upper-tail CDF (for \(y>u(x)\)) as

\[ F_{\mathrm{GPD}}\bigl(y\mid u(x),\sigma(x),\xi\bigr) = \begin{cases} 1-\left(1+\xi\dfrac{y-u(x)}{\sigma(x)}\right)^{-1/\xi}, & \xi\neq 0,\\[6pt] 1-\exp\!\left(-\dfrac{y-u(x)}{\sigma(x)}\right), & \xi=0. \end{cases} \]

The corresponding density is

\[ f_{\mathrm{GPD}}(y\mid u(x),\sigma(x),\xi) = \begin{cases} \dfrac{1}{\sigma(x)}\left(1+\xi\dfrac{y-u(x)}{\sigma(x)}\right)^{-1/\xi-1}, & \xi\neq 0,\\[8pt] \dfrac{1}{\sigma(x)}\exp\!\left(-\dfrac{y-u(x)}{\sigma(x)}\right), & \xi=0. \end{cases} \]

and the (upper-tail) quantile for \(0<\tau<1\) can be written as

\[ Q_{\mathrm{GPD}}(\tau\mid u(x),\sigma(x),\xi) = \begin{cases} u(x)-\sigma(x)\log(1-\tau), & \xi=0,\\[6pt] u(x)+\dfrac{\sigma(x)}{\xi}\Bigl((1-\tau)^{-\xi}-1\Bigr), & \xi\neq 0. \end{cases} \]

Interpretation:

• \(\xi>0\) corresponds to heavier-than-exponential tails (infinite support);
• \(\xi=0\) corresponds to the exponential limit;
• \(\xi<0\) corresponds to a lighter tail with finite upper endpoint.

DPM regression model for conditional densities (bulk)

For the bulk region (below \(u(x)\)), the package uses a DPM regression model. Let \(k(\cdot\mid x;\theta)\) denote a user-chosen kernel family. The DPM hierarchical structure can be summarized as

\[ Y\mid X=x,\theta \sim k(\cdot\mid x;\theta), \quad \theta \mid H \sim H, \quad H \mid \kappa, H_0 \sim \mathrm{DP}(\kappa, H_0), \]

which marginalizes to an infinite mixture representation

\[ f_{\mathrm{DP}}(y\mid x) =\int k(y\mid x;\theta)\,dH(\theta) =\sum_{j=1}^{\infty} w_j\,k(y\mid x;\theta_j), \quad w_j\ge 0,\ \sum_{j=1}^{\infty} w_j=1. \]

A common way to represent the weights is stick-breaking:

\[ w_1 = V_1,\qquad w_j = V_j\prod_{\ell<j}(1-V_\ell), \qquad V_j \stackrel{iid}{\sim} \mathrm{Beta}(1,\kappa), \qquad \theta_j \stackrel{iid}{\sim} H_0. \]

An equivalent clustering representation is the P'olya urn / CRP view, which induces a random partition of observations through latent component ties.

Spliced DPM–GPD conditional distribution (bulk + tail)

The core package construction is a spliced (two-part) conditional distribution: below a threshold, use the DPM fit; above the threshold, replace the tail with a GPD component.

Let \(p_u(x)=F_{\mathrm{DP}}(u(x)\mid x)\) be the bulk probability mass below the threshold under the DPM model. The spliced conditional CDF is

\[ F(y\mid x;\Theta,\Phi)= \begin{cases} F_{\mathrm{DP}}(y\mid x;\Theta), & y\le u(x),\\[6pt] p_u(x;\Theta) + \{1-p_u(x;\Theta)\}\,F_{\mathrm{GPD}}(y\mid x;\Phi), & y>u(x), \end{cases} \]

which is continuous at \(u(x)\). Differentiating yields the spliced density:

\[ f(y\mid x;\Theta,\Phi)= \begin{cases} f_{\mathrm{DP}}(y\mid x;\Theta), & y\le u(x),\\[8pt] \{1-p_u(x;\Theta)\}\,f_{\mathrm{GPD}}(y\mid x;\Phi), & y>u(x). \end{cases} \]

For quantiles, the index must be rescaled on the tail part. Define the tail-exceedance quantile index

\[ \tilde{\tau}(x;\Theta)=\dfrac{\tau-p_u(x;\Theta)}{1-p_u(x;\Theta)}\in(0,1). \]

Then the spliced quantile is

\[ Q(\tau\mid x;\Theta,\Phi)= \begin{cases} Q_{\mathrm{DP}}(\tau\mid x;\Theta), & \tau \le p_u(x;\Theta),\\[8pt] Q_{\mathrm{GPD}}\!\bigl(\tilde{\tau}(x;\Theta)\mid x;\Phi\bigr), & \tau>p_u(x;\Theta). \end{cases} \]

In the package’s notation, the bulk parameters (DPM regression coefficients and related truncation quantities) are collected into \(\Theta\), and the tail parameters governing \(u(x)\), \(\sigma(x)\), and \(\xi\) are collected into \(\Phi\).

What this means for the package

• dpmix() corresponds to the bulk-only DPM (GPD splicing disabled).
• dpmgpd() corresponds to the spliced DPM–GPD model (bulk DPM + GPD tail beyond the threshold).
• Posterior prediction and quantile estimation use the spliced formulas above, not separate post-processing.

References (key)

• Antoniak (1974), Mixtures of Dirichlet Processes… — doi:10.1214/aos/1176342871
• Ishwaran & James (2001), Gibbs Sampling Methods for Stick-Breaking Priors — doi:10.1198/016214501750332758
• Neal (2000), Markov Chain Sampling Methods for Dirichlet Process Mixture Models — doi:10.1080/10618600.2000.10474879
• Balkema & de Haan (1974), Residual Life Time at Great Age — doi:10.1214/aop/1176996548
• Pickands (1975), Statistical Inference Using Extreme Order Statistics — doi:10.1214/aos/1176343003
• Davison & Smith (1990), Models for Exceedances over High Thresholds — doi:10.1111/j.2517-6161.1990.tb01796.x

Prereqs

• Required packages and data for this page are listed in the setup chunks above.

Outputs

• This page renders model fits, diagnostics, and summary artifacts generated by package APIs.

Interpretation

• Canonical concept page: Index
• Treat this page as an application/example view and use the canonical page for core definitions.

Next

• Continue to the linked canonical concept page, then return for implementation-specific details.