Exponential {hawkesbow}

Reproduction kernels for the Hawkes processes

Description

These classes are derived from the class Model, each implementing a different reproduction kernel for the Hawkes process. They inherit all fields from Model.

Details

• The kernel Exponential has density function

  h^\ast(t) = \beta \exp(-\beta t) 1_{\{t \ge 0\}}.

  Its vector of parameters must be of the form (\eta, \mu, \beta). Both loglik, its derivatives, and whittle can be used with this reproduction kernel.

• The kernel SymmetricExponential has density function

  h^\ast(t) = 0.5 \beta \exp(-\beta |t|).

  Its vector of parameters must be of the form (\eta, \mu, \beta). Only whittle can be used with this reproduction kernel.

• The kernel Gaussian has density function

  h^\ast(t) = \frac{1}{\sigma \sqrt{2\pi}}\exp\left(-\frac{(t-\nu)^2}{2\sigma^2}\right).

  Its vector of parameters must be of the form (\eta, \mu, \nu, \sigma^2). Only whittle is available with this reproduction kernel.

• The kernel PowerLaw has density function

  h^\ast(t) = \theta a^\theta (t+a)^{-\theta-1} 1_{\{\theta > 0 \}}.

  Its vector of parameters must be of the form (\eta, \mu, \theta, a). Both loglik, its derivatives, and whittle can be used with this reproduction kernel.

• The kernels Pareto3, Pareto2 and Pareto1 have density function

  h_\theta^\ast(t) = \theta a^\theta t^{-\theta - 1} 1_{\{t > a\}},

  with \theta = 3, 2 and 1 respectively. Their vectors of parameters must be of the form (\eta, \mu, a). Only whittle is available with this reproduction kernel.

See Also

Model