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Source code for the paper "Inference for Continuous Time Random Maxima with Heavy-Tailed Waiting Times"

Home Page: https://strakaps.github.io/bursty-POT/

TeX 90.15% R 9.85%

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bursty-pot's Issues

Hill plots

bursty-POT/article_springer/article_springer.Rmd

Line 356 in 0b9b4a4

```{r Hillplots, echo=FALSE, fig.align="center", fig.cap="\\label{fig:Hillplots} Hillplots for simulated Mittag-Leffler distributed rv's with tail parameter 0.5 (two plots in the right column) and 0.8 (two plots in the left column) and sample sizes 200 (two plots in the lower row) and 1000 (two plots in the upper row). ", out.width = '90%'}

Hillplots.pdf hat nur 2 plots, aber du schreibst da wären 4. @KHees Wolltest du nicht die 2 verschiedenen sample sizes reinmachen?

power as beta to 1

bursty-POT/article_springer/article_springer.Rmd

Line 361 in 0b9b4a4

Since the exponential distribution is nested in the Mittag-Leffler family of distributions, an appropriate way to choose between a model with exponential and Mittag-Leffler inter-exceedance times seems to be a Likelihood ratio test. Although the two models are nested, the assymptotic distribution is not $\chi^2_1$-distributed, the classical Theorem of Wilk doesn't apply since under $H_0$ the parameter $\beta$ of the Mittag-Leffler distribution is equal to one and hence lies under $H_0$ on the boundary of the parameter space $(0,1]$. Nonetheless, one can perform a bootstrapped log-likelihood ratio test. In Figure \ref{Fig:LRT} one can see the power for the bootstrapped LRT for Mittag-Leffler distributions with different tail parameters. As expected the power gets smaller for tail parameters close to one, since the Mittag-Leffler distribution converges for $\beta \rightarrow 1$ to an exponential distribution. Consequently, for tail parameters close to one it is hard to differentiate a Mittag-Lefller distribution from an exponential.

@KHees, wolltest du diesen Plot noch für verschiedene betas machen? (Im text steht, "In Figure 4 one can see the power for the bootstrapped LRT for Mittag-Leffler distributions with different tail parameters.")

Bilder nicht lesbar

Die Schriftgröße auf den Bildern ist zu klein, kannst du mal probieren die fig.width und fig.height Parameter kleiner zu machen, so dass die Schrift größer aussieht?

Section "Simulation Study"

bursty-POT/article_springer/article_springer.Rmd

Line 507 in 0b9b4a4

# Simulation study

In dieser Sektion passieren zwei Sachen:

  1. Schätzung von \beta
  2. Schätzung von \sigma_0 (dem Skalenparameter)

Mit "have exact analytical values available" habe ich gemeint dass wir den exakten Skalenparameter verfügbar haben. Das geht dann aber nur im stable case. Den habe ich gemacht, und gezeigt dass die Konvergenz gegen \sigma_0 stattfindet.

Jetzt hast du aber noch die zwei anderen Fälle in den Absatz mit reingetan... für die wissen wir \sigma_0 nicht (kann man aber berechnen). Von daher ist der ganze Absatz jetzt verwirrend.

Wollen wir die Konvergenz für den Skalenparameter auch drin haben? Ich denke das wäre gut zu wissen dass der auch konvergiert... was meinst du @KHees ?

Quelle

bursty-POT/article_springer/article_springer.Rmd

Line 350 in 0b9b4a4

Since the Mittag-Leffler distribution is heavy-tailed, many researchers would intuitively give the highest importance to the tail behaviour of the distribution. Of course, one can also use established tail exponent estimators for the estimation of the parameter $\beta$ such as the Hill estimator

Falls nicht schon vorhanden, bitte Quelle angeben @KHees

3 Deckungsgleiche Bilder

Im Appendix sind die 3 Bilder deckungsgleich (identisch).
Sollten die nicht in Section "Simulation Study", siehe Issue #7?

opacity parameter in plots

@KHees wenn du alpha=0.3 oder so setzt, in den ggplots von den Simulationen, dann würde doch der Plot um den Mittelwert dunkler aussehen und auf dem Rand heller, oder? Und dann würde man doch mehr sehen... nur so ne Idee.

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