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Likelihood inference for exponential-trawl processes

Neil Shephard, Justin J. Yang

Abstract

Integer-valued trawl processes are a class of serially correlated, stationary and infinitely divisible processes that Ole E. Barndorff-Nielsen has been working on in recent years. In this Chapter, we provide the first analysis of likelihood inference for trawl processes by focusing on the so-called exponential-trawl process, which is also a continuous time hidden Markov process with countable state space. The core ideas include prediction decomposition, filtering and smoothing, complete-data analysis and EM algorithm. These can be easily scaled up to adapt to more general trawl processes but with increasing computation efforts.

Likelihood inference for exponential-trawl processes

Abstract

Integer-valued trawl processes are a class of serially correlated, stationary and infinitely divisible processes that Ole E. Barndorff-Nielsen has been working on in recent years. In this Chapter, we provide the first analysis of likelihood inference for trawl processes by focusing on the so-called exponential-trawl process, which is also a continuous time hidden Markov process with countable state space. The core ideas include prediction decomposition, filtering and smoothing, complete-data analysis and EM algorithm. These can be easily scaled up to adapt to more general trawl processes but with increasing computation efforts.

Paper Structure

This paper contains 26 sections, 6 theorems, 85 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Exponential-Trawl Processes
  3. Definition
  4. Markovian Counting Process
  5. Conditional Intensities and Log-likelihood
  6. Exact Filter and Smoother for Exponential-Trawl Processes
  7. Filtering
  8. Smoothing
  9. Likelihood Inference for General Exponential-Trawl Processes
  10. MLE Calculation based on Filtering
  11. Complete-Data Likelihood Inference
  12. MLE Calculation based on EM Algorithm
  13. Likelihood Inference without the Initial Information
  14. Likelihood Inference for Non-negative Exponential-Trawl Processes
  15. Partial Likelihood Inference
  16. ...and 11 more sections

Key Result

theorem \oldthetheorem

Let $X_{t}$ be any integer-valued stochastic process and $\left\{ \mathcal{F}_{t}^{X} \right\} _{t \geq 0}$ be its associated natural filtration. Assume that, under both $\mathbb{P}$ and $\mathbb{Q}$, (i) it has finite expected number of jumps during $(0,T]$, and (ii) the conditional intensities $\l

Figures (6)

  • Figure 1: A moving trawl $A_{t}$ is joined by the Skellam Lévy basis $L(\mathrm{d}x,\mathrm{d}s)$, where the horizontal axis $s$ is time and the vertical axis $x$ is height. The shaded area is an example of the exponential trawl $A$, while we also show the outlines of $A_{t}$ when $t=1/2$ and $t=1$. Also shown below is the implied trawl process $Y_{t}=L(A_{t})$. Code: EPTprocess_Illurstration.R
  • Figure 2: Top left: A simulated path for the Skellam exponential-trawl process $Y_{t}$. Top right, Bottom left, Bottom right: Paths of the true hidden counting processes $C_{t}^{\left( +\right) }$, $C_{t}^{\left( -\right) }$ and $D_{t}=C_{t}^{\left( +\right) }+C_{t}^{\left( -\right) }$ of surviving events in the trawl along with their filtering estimations. Code: EPTprocess_FilteringSmoothing_Illustration.R
  • Figure 3: Top left: A simulated path for the Skellam exponential-trawl process $Y_{t}$. Top right, Bottom left, Bottom right: Paths of the true hidden counting processes $C_{t}^{\left( +\right) }$, $C_{t}^{\left( -\right) }$ and $D_{t}=C_{t}^{\left( +\right) }+C_{t}^{\left( -\right) }$ of surviving events in the trawl along with their smoothing estimations. Code: EPTprocess_FilteringSmoothing_Illustration.R
  • Figure 4: Log-likelihood plots over $\phi$ (with $\nu ^{+}$ and $\nu ^{-}$ fixed at the truth) using different $\delta _{\mathrm{inactivity}}$ and a simulated $10$-day-long ($T=756,000$ (sec.)) Skellam exponential-trawl process. The one-day-long data is the first tenth of the simulated data. The dashed lines indicate the true value of $\phi$, while the solid lines indicate the optimal value of $\phi$ in each plot. The $p$-values using the likelihood ratio test are $0.104\%$ (Top left), $21.0\%$ (Bottom left), $8.82\times 10^{-13}$ (Top right) and $46.1\%$ (Bottom right). Code: EPTprocess_MLE_Inference_Simulation_Small_vs_Large.R
  • Figure 5: Log-likelihood plots over either $\nu ^{+}$ or $\nu ^{-}$ for one simulated Skellam exponential-trawl process. The dashed lines indicate the true value, while the solid lines indicate the optimal value of $\nu ^{+}$ or $\nu ^{-}$ in the individual plot. The $p$-values using the likelihood ratio test are $40.5\%$ (Left) and $33.4\%$ (Right). Code: EPTprocess_MLE_Inference_Simulation_Small_vs_Large.R
  • ...and 1 more figures

Theorems & Definitions (8)

  • remark 1
  • remark 2
  • theorem \oldthetheorem
  • corollary 1
  • theorem \oldthetheorem: Forward Filtering
  • theorem \oldthetheorem: Backward Smoothing
  • theorem \oldthetheorem
  • proposition 1