Package 'trawl'

Autocorrelation function of the double exponential trawl function

Description

This function computes the autocorrelation function associated with the double exponential trawl function.

Usage

acf_DExp(x, w, lambda1, lambda2)

Arguments

x	The argument (lag) at which the autocorrelation function associated with the double exponential trawl function will be evaluated
w	parameter in the double exponential trawl
lambda1	parameter in the double exponential trawl
lambda2	parameter in the double exponential trawl

Details

The trawl function is parametrised by parameters $0\le w\le 1$ and $\lambda_1, \lambda_2 > 0$ as follows:

$g(x) = w e^{\lambda_1 x}+(1-w) e^{\lambda_2 x}, \mbox{ for } x \le 0.$

Its autocorrelation function is given by:

$r(x) = (w e^{-\lambda_1 x}/\lambda_1+(1-w) e^{-\lambda_2 x}/\lambda_2)/c, \mbox{ for } x \ge 0,$

where

$c = w/\lambda_1+(1-w)/\lambda_2.$

Value

The autocorrelation function of the double exponential trawl function evaluated at x

Examples

acf_DExp(1,0.3,0.1,2)

---

Autocorrelation function of the exponential trawl function

Description

This function computes the autocorrelation function associated with the exponential trawl function.

Usage

acf_Exp(x, lambda)

Arguments

x	The argument (lag) at which the autocorrelation function associated with the exponential trawl function will be evaluated
lambda	parameter in the exponential trawl

Details

The trawl function is parametrised by the parameter $\lambda > 0$ as follows:

$g(x) = e^{\lambda x}, \mbox{ for } x \le 0.$

Its autocorrelation function is given by:

$r(x) = e^{-\lambda x}, \mbox{ for } x \ge 0.$

Value

The autocorrelation function of the exponential trawl function evaluated at x

Examples

acf_Exp(1,0.1)

---

Autocorrelation function of the long memory trawl function

Description

This function computes the autocorrelation function associated with the long memory trawl function.

Usage

acf_LM(x, alpha, H)

Arguments

x	The argument (lag) at which the autocorrelation function associated with the long memory trawl function will be evaluated
alpha	parameter in the long memory trawl
H	parameter in the long memory trawl

Details

The trawl function is parametrised by the two parameters $H> 1$ and $\alpha > 0$ as follows:

$g(x) = (1-x/\alpha)^{-H}, \mbox{ for } x \le 0.$

Its autocorrelation function is given by

$r(x)=(1+x/\alpha)^{(1-H)}, \mbox{ for } x \ge 0.$

Value

The autocorrelation function of the long memory trawl function evaluated at x

Examples

acf_LM(1,0.3,1.5)

---

Autocorrelation function of the supIG trawl function

Description

This function computes the autocorrelation function associated with the supIG trawl function.

Usage

acf_supIG(x, delta, gamma)

Arguments

x	The argument (lag) at which the autocorrelation function associated with the supIG trawl function will be evaluated
delta	parameter in the supIG trawl
gamma	parameter in the supIG trawl

Details

The trawl function is parametrised by the two parameters $\delta \geq 0$ and $\gamma \geq 0$ as follows:

$g(x) = (1-2x\gamma^{-2})^{-1/2}\exp(\delta \gamma(1-(1-2x\gamma^{-2})^{1/2})), \mbox{ for } x \le 0.$

It is assumed that $\delta$ and $\gamma$ are not simultaneously equal to zero. Its autocorrelation function is given by:

$r(x) = \exp(\delta\gamma (1-\sqrt{1+2 x/\gamma^2})), \mbox{ for } x \ge 0.$

Value

The autocorrelation function of the supIG trawl function evaluated at x

Examples

acf_supIG(1,0.3,0.1)

---

Simulates from the bivariate logarithmic series distribution

Description

Simulates from the bivariate logarithmic series distribution

Usage

Bivariate_LSDsim(N, p1, p2)

Arguments

N	number of data points to be simulated
p1	parameter $p_1$ of the bivariate logarithmic series distribution
p2	parameter $p_2$ of the bivariate logarithmic series distribution

Details

The probability mass function of a random vector $X=(X_1,X_2)'$ following the bivariate logarithmic series distribution with parameters $0<p_1, p_2<1$ with $p:=p_1+p_2<1$ is given by

$P(X_1=x_1,X_2=x_2)=\frac{\Gamma(x_1+x_2)}{x_1!x_2!} \frac{p_1^{x_1}p_2^{x_2}}{(-\log(1-p))},$

for $x_1,x_2=0,1,2,\dots$ such that $x_1+x_2>0$. The simulation proceeds in two steps: First, $X_1$ is simulated from the modified logarithmic distribution with parameters $\tilde p_1=p_1/(1-p_2)$ and $\delta_1=\log(1-p_2)/\log(1-p)$. Then we simulate $X_2$ conditional on $X_1$. We note that $X_2|X_1=x_1$ follows the logarithmic series distribution with parameter $p_2$ when $x_1=0$, and the negative binomial distribution with parameters $(x_1,p_2)$ when $x_1>0$.

Value

An $N \times 2$ matrix with $N$ simulated values from the bivariate logarithmic series distribution

---

Simulates from the bivariate negative binomial distribution

Description

Simulates from the bivariate negative binomial distribution

Usage

Bivariate_NBsim(N, kappa, p1, p2)

Arguments

N	number of data points to be simulated
kappa	parameter $\kappa$ of the bivariate negative binomial distribution
p1	parameter $p_1$ of the bivariate negative binomial distribution
p2	parameter $p_2$ of the bivariate negative binomial distribution

Details

A random vector ${\bf X}=(X_1,X_2)'$ is said to follow the bivariate negative binomial distribution with parameters $\kappa, p_1, p_2$ if its probability mass function is given by

$P({\bf X}={\bf x})=\frac{\Gamma(x_1+x_2+\kappa)}{x_1!x_2! \Gamma(\kappa)}p_1^{x_1}p_2^{x_2}(1-p_1-p_2)^{\kappa},$

where, for $i=1,2$, $x_i\in\{0,1,\dots\}$, $0<p_i<1$ such that $p_1+p_2<1$ and $\kappa>0$.

Value

An $N\times 2$ matrix with $N$ simulated values from the bivariate negative binomial distribution

---

Computes the correlation of the components of a bivariate vector following the bivariate logarithmic series distribution

Description

Computes the correlation of the components of a bivariate vector following the bivariate logarithmic series distribution

Usage

BivLSD_Cor(p1, p2)

Arguments

p1	parameter $p_1$ of the bivariate logarithmic series distribution
p2	parameter $p_2$ of the bivariate logarithmic series distribution

Value

Correlation of the components of a bivariate vector following the bivariate logarithmic series distribution

---

Computes the covariance of the components of a bivariate vector following the bivariate logarithmic series distribution

Description

Computes the covariance of the components of a bivariate vector following the bivariate logarithmic series distribution

Usage

BivLSD_Cov(p1, p2)

Arguments

p1	parameter $p_1$ of the bivariate logarithmic series distribution
p2	parameter $p_2$ of the bivariate logarithmic series distribution

Value

Covariance of the components of a bivariate vector following the bivariate logarithmic series distribution

---

Computes the correlation of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Description

Computes the correlation of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Usage

BivModLSD_Cor(delta, p1, p2)

Arguments

delta	parameter $\delta$ of the bivariate modified logarithmic series distribution
p1	parameter $p_1$ of the bivariate modified logarithmic series distribution
p2	parameter $p_2$ of the bivariate modified logarithmic series distribution

Value

Covariance of the components of a bivariate vector following the bivariate modified logarithmic series distribution

---

Computes the covariance of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Description

Computes the covariance of the components of a bivariate vector following the bivariate modified logarithmic series distribution

Usage

BivModLSD_Cov(delta, p1, p2)

Arguments

delta	parameter $\delta$ of the bivariate modified logarithmic series distribution
p1	parameter $p_1$ of the bivariate modified logarithmic series distribution
p2	parameter $p_2$ of the bivariate modified logarithmic series distribution

Value

Covariance of the components of a bivariate vector following the bivariate modified logarithmic series distribution

---

Fits the trawl function consisting of the weighted sum of two exponential functions

Description

Fits the trawl function consisting of the weighted sum of two exponential functions

Usage

fit_DExptrawl(x, Delta = 1, GMMlag = 5, plotacf = FALSE, lags = 100)

Arguments

x	vector of equidistant time series data
Delta	interval length of the time grid used in the time series, the default is 1
GMMlag	lag length used in the GMM estimation, the default is 5
plotacf	binary variable specifying whether or not the empirical and fitted autocorrelation function should be plotted
lags	number of lags to be used in the plot of the autocorrelation function

Details

The trawl function is parametrised by the three parameters $0\leq w \leq 1$ and $\lambda_1,\lambda_2 > 0$ as follows:

$g(x) = we^{\lambda_1 x}+(1-w)e^{\lambda_2 x}, \mbox{ for } x \le 0.$

The Lebesgue measure of the corresponding trawl set is given by $w/\lambda_1+(1-w)/\lambda_2$.

Value

w: the weight parameter (restricted to be in [0,0.5] for identifiability reasons)

lambda1: the first memory parameter (denoted by $\lambda_1$ above)

lambda2: the second memory parameter (denoted by $\lambda_2$ above)

LM: The Lebesgue measure of the trawl set associated with the double exponential trawl

---

Fits an exponential trawl function to equidistant time series data

Description

Fits an exponential trawl function to equidistant time series data

Usage

fit_Exptrawl(x, Delta = 1, plotacf = FALSE, lags = 100)

Arguments

x	vector of equidistant time series data
Delta	interval length of the time grid used in the time series, the default is 1
plotacf	binary variable specifying whether or not the empirical and fitted autocorrelation function should be plotted
lags	number of lags to be used in the plot of the autocorrelation function

Details

The trawl function is parametrised by the parameter $\lambda > 0$ as follows:

$g(x) = e^{\lambda x}, \mbox{ for } x \le 0.$

The Lebesgue measure of the corresponding trawl set is given by $1/\lambda$.

Value

lambda: the memory parameter $\lambda$ in the exponential trawl

LM: the Lebesgue measure of the trawl set associated with the exponential trawl, i.e. $1/\lambda$.

---

Fits a long memory trawl function to equidistant univariate time series data

Description

Fits a long memory trawl function to equidistant univariate time series data

Usage

fit_LMtrawl(x, Delta = 1, GMMlag = 5, plotacf = FALSE, lags = 100)

Arguments

x	vector of equidistant time series data
Delta	interval length of the time grid used in the time series, the default is 1
GMMlag	lag length used in the GMM estimation, the default is 5
plotacf	binary variable specifying whether or not the empirical and fitted autocorrelation function should be plotted
lags	number of lags to be used in the plot of the autocorrelation function

Details

The trawl function is parametrised by the two parameters $H> 1$ and $\alpha > 0$ as follows:

$g(x) = (1-x/\alpha)^{-H},\mbox{ for } x \le 0.$

The Lebesgue measure of the corresponding trawl set is given by $\alpha/(1-H)$.

Value

alpha: parameter in the long memory trawl

H: parameter in the long memory trawl

LM: The Lebesgue measure of the trawl set associated with the long memory trawl

---

Fist a negative binomial distribution as marginal law

Description

Fist a negative binomial distribution as marginal law

Usage

fit_marginalNB(x, LM, plotdiag = FALSE)

Arguments

x	vector of equidistant time series data
LM	Lebesgue measure of the estimated trawl
plotdiag	binary variable specifying whether or not diagnostic plots should be provided

Details

The moment estimator for the parameters of the negative binomial distribution are given by

$\hat \theta = 1-\mbox{E}(X)/\mbox{Var}(X),$

and

$\hat m = \mbox{E}(X)(1-\hat \theta)/(\widehat{ \mbox{LM}} \hat \theta).$

Value

m: parameter in the negative binomial marginal distribution

theta: parameter in the negative binomial marginal distribution

a: Here $a=\theta/(1-\theta)$. This is given for an alternative parametrisation of the negative binomial marginal distribution

---

Fits a Poisson distribution as marginal law

Description

Fits a Poisson distribution as marginal law

Usage

fit_marginalPoisson(x, LM, plotdiag = FALSE)

Arguments

x	vector of equidistant time series data
LM	Lebesgue measure of the estimated trawl
plotdiag	binary variable specifying whether or not diagnostic plots should be provided

Details

The moment estimator for the Poisson rate parameter is given by

$\hat v = \mbox{E}(X)/\widehat{ \mbox{LM}}.$

Value

v: the rate parameter in the Poisson marginal distribution

---

Fits a supIG trawl function to equidistant univariate time series data

Description

Fits a supIG trawl function to equidistant univariate time series data

Usage

fit_supIGtrawl(x, Delta = 1, GMMlag = 5, plotacf = FALSE, lags = 100)

Arguments

x	vector of equidistant time series data
Delta	interval length of the time grid used in the time series, the default is 1
GMMlag	lag length used in the GMM estimation, the default is 5
plotacf	binary variable specifying whether or not the empirical and fitted autocorrelation function should be plotted
lags	number of lags to be used in the plot of the autocorrelation function

Details

The trawl function is parametrised by the two parameters $\delta \geq 0$ and $\gamma \geq 0$ as follows:

$g(x) = (1-2x\gamma^{-2})^{-1/2}\exp(\delta \gamma(1-(1-2x\gamma^{-2})^{1/2})), \mbox{ for } x \le 0.$

It is assumed that $\delta$ and $\gamma$ are not simultaneously equal to zero. The Lebesgue measure of the corresponding trawl set is given by $\gamma/\delta$.

Value

delta: parameter in the supIG trawl

gamma: parameter in the supIG trawl

LM: The Lebesgue measure of the trawl set associated with the supIG trawl

---

Finds the intersection of two trawl sets

Description

Finds the intersection of two trawl sets

Usage

fit_trawl_intersection(
  fct1 = base::c("Exp", "DExp", "supIG", "LM"),
  fct2 = base::c("Exp", "DExp", "supIG", "LM"),
  lambda11 = 0,
  lambda12 = 0,
  w1 = 0,
  delta1 = 0,
  gamma1 = 0,
  alpha1 = 0,
  H1 = 0,
  lambda21 = 0,
  lambda22 = 0,
  w2 = 0,
  delta2 = 0,
  gamma2 = 0,
  alpha2 = 0,
  H2 = 0,
  LM1,
  LM2,
  plotdiag = FALSE
)

Arguments

fct1	specifies the type of the first trawl function
fct2	specifies the type of the second trawl function
lambda11, lambda12, w1	parameters of the (double) exponential trawl functions of the first process
delta1, gamma1	parameters of the supIG trawl functions of the first process
alpha1, H1	parameters of the long memory trawl function of the first process
lambda21, lambda22, w2	parameters of the (double) exponential trawl functions of the second process
delta2, gamma2	parameters of the supIG trawl functions of the second process
alpha2, H2	parameters of the long memory trawl function of the second process
LM1	Lebesgue measure of the first trawl
LM2	Lebesgue measure of the second trawl
plotdiag	binary variable specifying whether or not diagnostic plots should be provided

Details

Computes $R_{12}(0)=\mbox{Leb}(A_1 \cap A_2)$ based on two trawl functions $g_1$ and $g_2$.

Value

The Lebesgue measure of the intersection of the two trawl sets

---

Finds the intersection of two exponential trawl sets

Description

Finds the intersection of two exponential trawl sets

Usage

fit_trawl_intersection_Exp(lambda1, lambda2, LM1, LM2, plotdiag = FALSE)

Arguments

lambda1, lambda2	parameters of the two exponential trawls
LM1	Lebesgue measure of the first trawl
LM2	Lebesgue measure of the second trawl
plotdiag	binary variable specifying whether or not diagnostic plots should be provided

Details

Computes $R_{12}(0)=\mbox{Leb}(A_1 \cap A_2)$ based on two trawl functions $g_1$ and $g_2$.

Value

The Lebesgue measure of the intersection of the two trawl sets

---

Finds the intersection of two long memory (LM) trawl sets

Description

Finds the intersection of two long memory (LM) trawl sets

Usage

fit_trawl_intersection_LM(alpha1, H1, alpha2, H2, LM1, LM2, plotdiag = FALSE)

Arguments

alpha1, H1, alpha2, H2	parameters of the two long memory trawls
LM1	Lebesgue measure of the first trawl
LM2	Lebesgue measure of the second trawl
plotdiag	binary variable specifying whether or not diagnostic plots should be provided

Details

Computes $R_{12}(0)=\mbox{Leb}(A_1 \cap A_2)$ based on two trawl functions $g_1$ and $g_2$.

Value

the Lebesgue measure of the intersection of the two trawl sets

---

Computes the mean of the logarithmic series distribution

Description

Computes the mean of the logarithmic series distribution

Usage

LSD_Mean(p)

Arguments

p	parameter of the logarithmic series distribution

Details

A random variable $X$ has logarithmic series distribution with parameter $0<p<1$ if

$P(X=x)=(-1)/(\log(1-p))p^x/x, \mbox{ for } x=1,2,\dots.$

Value

Mean of the logarithmic series distribution

---

Computes the variance of the logarithmic series distribution

Description

Computes the variance of the logarithmic series distribution

Usage

LSD_Var(p)

Arguments

p	parameter of the logarithmic series distribution

Details

A random variable $X$ has logarithmic series distribution with parameter $0<p<1$ if

$P(X=x)=(-1)/(\log(1-p))p^x/x, \mbox{ for } x=1,2,\dots.$

Value

Variance of the logarithmic series distribution

---

Computes the mean of the modified logarithmic series distribution

Description

Computes the mean of the modified logarithmic series distribution

Usage

ModLSD_Mean(delta, p)

Arguments

delta	parameter $\delta$ of the modified logarithmic series distribution
p	parameter of the modified logarithmic series distribution

Details

A random variable $X$ has modified logarithmic series distribution with parameters $0 \le \delta <1$ and $0<p<1$ if $P(X=0)=(1-\delta)$ and

$P(X=x)=(1-\delta)(-1)/(\log(1-p))p^x/x, \mbox{ for } x=1,2,\dots.$

Value

Mean of the modified logarithmic series distribution

---

Computes the variance of the modified logarithmic series distribution

Description

Computes the variance of the modified logarithmic series distribution

Usage

ModLSD_Var(delta, p)

Arguments

delta	parameter $\delta$ of the modified logarithmic series distribution
p	parameter of the modified logarithmic series distribution

Details

A random variable $X$ has modified logarithmic series distribution with parameters $0\le \delta <1$ and $0<p<1$ if $P(X=0)=(1-\delta)$ and

$P(X=x)=(1-\delta)(-1)/(\log(1-p))p^x/x, \mbox{ for } x=1,2,\dots.$

Value

Mean of the modified logarithmic series distribution

---

Plots the bivariate histogram of two time series together with the univariate histograms

Description

Plots the bivariate histogram of two time series together with the univariate histograms

Usage

plot_2and1hist(x, y)

Arguments

x	vector of equidistant time series data
y	vector of equidistant time series data (of the same length as x)

Details

This function plots the bivariate histogram of two time series together with the univariate histograms

Value

no return value

---

Plots the bivariate histogram of two time series together with the univariate histograms using ggplot2

Description

Plots the bivariate histogram of two time series together with the univariate histograms using ggplot2

Usage

plot_2and1hist_gg(x, y, bivbins = 50, xbins = 30, ybins = 30)

Arguments

x	vector of equidistant time series data
y	vector of equidistant time series data (of the same length as x)
bivbins	number of bins in the bivariate histogram
xbins	number of bins in the histogram of x
ybins	number of bins in the histogram of y

Details

This function plots the bivariate histogram of two time series together with the univariate histograms

Value

no return value

---

Simulates a bivariate trawl process

Description

Simulates a bivariate trawl process

Usage

sim_BivariateTrawl(
  t,
  Delta = 1,
  burnin = 10,
  marginal = base::c("Poi", "NegBin"),
  dependencetype = base::c("fullydep", "dep"),
  trawl1 = base::c("Exp", "DExp", "supIG", "LM"),
  trawl2 = base::c("Exp", "DExp", "supIG", "LM"),
  v1 = 0,
  v2 = 0,
  v12 = 0,
  kappa1 = 0,
  kappa2 = 0,
  kappa12 = 0,
  a1 = 0,
  a2 = 0,
  lambda11 = 0,
  lambda12 = 0,
  w1 = 0,
  delta1 = 0,
  gamma1 = 0,
  alpha1 = 0,
  H1 = 0,
  lambda21 = 0,
  lambda22 = 0,
  w2 = 0,
  delta2 = 0,
  gamma2 = 0,
  alpha2 = 0,
  H2 = 0
)

Arguments

t	parameter which specifying the length of the time interval $[0,t]$ for which a simulation of the trawl process is required.
Delta	parameter $\Delta$ specifying the length of the time step, the default is 1
burnin	parameter specifying the length of the burn-in period at the beginning of the simulation
marginal	parameter specifying the marginal distribution of the trawl
dependencetype	Parameter specifying the type of dependence
trawl1	parameter specifying the type of the first trawl function
trawl2	parameter specifying the type of the second trawl function
v1, v2, v12	parameters of the Poisson distribution
kappa1, kappa2, kappa12, a1, a2	parameters of the (possibly bivariate) negative binomial distribution
lambda11, lambda12, w1	parameters of the exponential (or double-exponential) trawl function of the first process
delta1, gamma1	parameters of the supIG trawl function of the first process
alpha1, H1	parameter of the long memory trawl of the first process
lambda21, lambda22, w2	parameters of the exponential (or double-exponential) trawl function of the second process
delta2, gamma2	parameters of the supIG trawl function of the second process
alpha2, H2	parameter of the long memory trawl of the second process

Details

This function simulates a bivariate trawl process with either Poisson or negative binomial marginal law. For the trawl function there are currently four choices: exponential, double-exponential, supIG or long memory. More details on the precise simulation algorithm is available in the vignette.

---

Simulates a univariate trawl process

Description

Simulates a univariate trawl process

Usage

sim_UnivariateTrawl(
  t,
  Delta = 1,
  burnin = 10,
  marginal = base::c("Poi", "NegBin"),
  trawl = base::c("Exp", "DExp", "supIG", "LM"),
  v = 0,
  m = 0,
  theta = 0,
  lambda1 = 0,
  lambda2 = 0,
  w = 0,
  delta = 0,
  gamma = 0,
  alpha = 0,
  H = 0
)

Arguments

t	parameter which specifying the length of the time interval $[0,t]$ for which a simulation of the trawl process is required.
Delta	parameter $\Delta$ specifying the length of the time step, the default is 1
burnin	parameter specifying the length of the burn-in period at the beginning of the simulation
marginal	parameter specifying the marginal distribution of the trawl
trawl	parameter specifying the type of trawl function
v	parameter of the Poisson distribution
m	parameter of the negative binomial distribution
theta	parameter $\theta$ of the negative binomial distribution
lambda1	parameter $\lambda_1$ of the exponential (or double-exponential) trawl function
lambda2	parameter $\lambda_2$ of the double-exponential trawl function
w	parameter of the double-exponential trawl function
delta	parameter $\delta$ of the supIG trawl function
gamma	parameter $\gamma$ of the supIG trawl function
alpha	parameter $\alpha$ of the long memory trawl function
H	parameter of the long memory trawl function

Details

This function simulates a univariate trawl process with either Poisson or negative binomial marginal law. For the trawl function there are currently four choices: exponential, double-exponential, supIG or long memory. More details on the precise simulation algorithm is available in the vignette.

---

Evaluates the double exponential trawl function

Description

Evaluates the double exponential trawl function

Usage

trawl_DExp(x, w, lambda1, lambda2)

Arguments

x	the argument at which the double exponential trawl function will be evaluated
w	parameter in the double exponential trawl
lambda1	the parameter $\lambda_1$ in the double exponential trawl
lambda2	the parameter $\lambda_2$ in the double exponential trawl

Details

The trawl function is parametrised by parameters $0\leq w\leq 1$ and $\lambda_1, \lambda_2 > 0$ as follows:

$g(x) = w e^{\lambda_1 x}+(1-w) e^{\lambda_2 xz}, \mbox{ for } x \le 0.$

Value

The double exponential trawl function evaluated at x

---

Evaluates the exponential trawl function

Description

Evaluates the exponential trawl function

Usage

trawl_Exp(x, lambda)

Arguments

x	the argument at which the exponential trawl function will be evaluated
lambda	the parameter $\lambda$ in the exponential trawl

Details

The trawl function is parametrised by parameter $\lambda > 0$ as follows:

$g(x) = e^{\lambda x}, \mbox{ for } x \le 0.$

Value

The exponential trawl function evaluated at x

---

Evaluates the long memory trawl function

Description

Evaluates the long memory trawl function

Usage

trawl_LM(x, alpha, H)

Arguments

x	the argument at which the long memory trawl function will be evaluated
alpha	the parameter $\alpha$ in the long memory trawl
H	the parameter $H$ in the long memory trawl

Details

The trawl function is parametrised by the two parameters $H> 1$ and $\alpha > 0$ as follows:

$g(x) = (1-x/\alpha)^{-H}, \mbox{ for } x \le 0.$

Value

the long memory trawl function evaluated at x

---

Evaluates the supIG trawl function

Description

Evaluates the supIG trawl function

Usage

trawl_supIG(x, delta, gamma)

Arguments

x	the argument at which the supIG trawl function will be evaluated
delta	the parameter $\delta$ in the supIG trawl
gamma	the parameter $\gamma$ in the supIG trawl

Details

The trawl function is parametrised by the two parameters $\delta \geq 0$ and $\gamma \geq 0$ as follows:

$gd(x) = (1-2x\gamma^{-2})^{-1/2}\exp(\delta \gamma(1-(1-2x\gamma^{-2})^{1/2})), \mbox{ for } x \le 0.$

It is assumed that $\delta$ and $\gamma$ are not simultaneously equal to zero.

Value

The supIG trawl function evaluated at x

---

Simulates from the trivariate logarithmic series distribution

Description

Simulates from the trivariate logarithmic series distribution

Usage

Trivariate_LSDsim(N, p1, p2, p3)

Arguments

N	number of data points to be simulated
p1	parameter $p1$ of the trivariate logarithmic series distribution
p2	parameter $p2$ of the trivariate logarithmic series distribution
p3	parameter $p3$ of the trivariate logarithmic series distribution

Details

The probability mass function of a random vector $X=(X_1,X_2,X_3)'$ following the trivariate logarithmic series distribution with parameters $0<p_1, p_2, p_3<1$ with $p:=p_1+p_2+p_3<1$ is given by

$P(X_1=x_1,X_2=x_2,X_3=x_3)=\frac{\Gamma(x_1+x_2+x_3)}{x_1!x_2!x_3!} \frac{p_1^{x_1}p_2^{x_2}p_3^{x_3}}{(-\log(1-p))},$

for $x_1,x_2,x_3=0,1,2,\dots$ such that $x_1+x_2+x_3>0$.

The simulation proceeds in two steps: First, $X_1$ is simulated from the modified logarithmic distribution with parameters $\tilde p_1=p_1/(1-p_2-p_3)$ and $\delta_1=\log(1-p_2-p_3)/\log(1-p)$. Then we simulate $(X_2,X_3)'$ conditional on $X_1$. We note that $(X_2,X_3)'|X_1=x_1$ follows the bivariate logarithmic series distribution with parameters $(p_2,p_3)$ when $x_1=0$, and the bivariate negative binomial distribution with parameters $(x_1,p_2,p_3)$ when $x_1>0$.

Value

An $N \times 3$ matrix with $N$ simulated values from the trivariate logarithmic series distribution

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