getOmega: Generate covariance matrix Omega for quantile residual tests

Description

getOmega generates the covariance matrix Omega used in the quantile residual tests.

Usage

getOmega(data, p, M, params, model = c("GMAR", "StMAR", "G-StMAR"),
  restricted = FALSE, constraints = NULL,
  parametrization = c("intercept", "mean"), g, dim_g)

Arguments

data

a numeric vector class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the order of AR coefficients.

For GMAR and StMAR models:: a positive integer specifying the number of mixture components.
For G-StMAR model:: a size (2x1) vector specifying the number of GMAR-type components M1 in the first element and StMAR-type components M2 in the second. The total number of mixture components is M=M1+M2.

params

a real valued parameter vector specifying the model.

For non-restricted models:

For GMAR model:: Size \((M(p+3)-1x1)\) vector \(\theta\)\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}\)), where \(\upsilon_{m}\)\(=(\phi_{m,0},\)\(\phi_{m}\)\(, \sigma_{m}^2)\) and \(\phi_{m}\)=\((\phi_{m,1},...,\phi_{m,p}), m=1,...,M\).
For StMAR model:: Size \((M(p+4)-1x1)\) vector (\(\theta, \nu\))\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M}\)).
For G-StMAR model:: Size \((M(p+3)+M2-1x1)\) vector (\(\theta, \nu\))\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M}\)).
With linear constraints:: Replace the vectors \(\phi_{m}\) with vectors \(\psi_{m}\) and provide a list of constraint matrices C that satisfy \(\phi_{m}\)\(=\)\(R_{m}\psi_{m}\) for all \(m=1,...,M\), where \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).

For restricted models:

For GMAR model:: Size \((3M+p-1x1)\) vector \(\theta\)\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1})\), where \(\phi\)=\((\phi_{1},...,\phi_{M})\).
For StMAR model:: Size \((4M+p-1x1)\) vector (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M})\).
For G-StMAR model:: Size \((3M+M2+p-1x1)\) vector (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M})\).
With linear constraints:: Replace the vector \(\phi\) with vector \(\psi\) and provide a constraint matrix \(C\) that satisfies \(\phi\)\(=\)\(R\psi\), where \(\psi\)\(=(\psi_{1},...,\psi_{q})\).

Symbol \(\phi\) denotes an AR coefficient, \(\sigma^2\) a variance, \(\alpha\) a mixing weight and \(\nu\) a degrees of freedom parameter. If parametrization=="mean" just replace each intercept term \(\phi_{m,0}\) with regimewise mean \(\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})\). In the G-StMAR model the first M1 components are GMAR-type and the rest M2 components are StMAR-type. Note that in the case M=1 the parameter \(\alpha\) is dropped, and in the case of StMAR or G-StMAR model the degrees of freedom parameters \(\nu_{m}\) have to be larger than \(2\).

model

is "GMAR", "StMAR" or "G-StMAR" model considered? In G-StMAR model the first M1 components are GMAR-type and the rest M2 components are StMAR-type.

restricted

a logical argument stating whether the AR coefficients \(\phi_{m,1},...,\phi_{m,p}\) are restricted to be the same for all regimes.

constraints

specifies linear constraints applied to the autoregressive parameters.

For non-restricted models:: a list of size \((pxq_{m})\) constraint matrices \(C_{m}\) of full column rank satisfying \(\phi_{m}\)\(=\)\(C_{m}\psi_{m}\) for all \(m=1,...,M\), where \(\phi_{m}\)\(=(\phi_{m,1},...,\phi_{m,p})\) and \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).
For restricted models:: a size \((pxq)\) constraint matrix \(C\) of full column rank satisfying \(\phi\)\(=\)\(C\psi\), where \(\phi\)\(=(\phi_{1},...,\phi_{p})\) and \(\psi\)\(=\psi_{1},...,\psi_{q}\).

Symbol \(\phi\) denotes an AR coefficient. Note that regardless of any constraints, the nominal order of AR coefficients is alway p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

parametrization

is the model parametrized with the "intercepts" \(\phi_{m,0}\) or "means" \(\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})\)?

a function specifying the transformation.

dim_g

output dimension of the transformation g.

Value

Returns size (dim_gxdim_g) covariance matrix Omega.

Details

This function is used for quantile residuals tests in quantileResidualTests.

References