Etheta_imaginary

Calculates the value of
$$-ix e^{ix} E_\theta(ix) = -ix e{ix} \int_1^\infty t^{-\theta} e^{-ixt} \mathrm d t$$
for $\theta &gt; 0$.
This is achieved using recursive integrations by parts until $0 &lt; \theta \le 1$,
then using either the exponential integral <code>E1_imaginary</code> if $\theta = 1$,
or the incomplete gamma function <code>inc_gamma_imag</code> if $0 &lt; \theta &lt; 1$.

Implements an estimation method for Hawkes processes when count data are only observed in discrete time, using a spectral approach derived from the Bartlett spectrum, see Cheysson and Lang (2020) <arXiv:2003.04314>. Some general use functions for Hawkes processes are also included: simulation of (in)homogeneous Hawkes process, maximum likelihood estimation, residual analysis, etc.

Felix Cheysson

hawkesbow

Estimation of Hawkes Processes from Binned Observations

Etheta_imaginary function

<dl><dt>theta</dt>
<dd>A strictly positive number</dd>
<dt>x</dt>
<dd>A vector of non-negative numbers</dd></dl>

Arguments

Incomplete gamma function of imaginary argument with arbitrary power — Etheta_imaginary

<dl>

<dt>theta</dt>
<dd>A strictly positive number</dd>


<dt>x</dt>
<dd>A vector of non-negative numbers</dd>

</dl>

Etheta_imaginary: Incomplete gamma function of imaginary argument with arbitrary power

Description

Usage

Value

Arguments

Examples