get_varying_h adjusts differences for overly large degrees of freedom parameters
for finite difference approximation of the derivatives of the log-likelihood function. StMAR and
G-StMAR models are supported.
get_varying_h(p, M, params, model)a positive integer specifying the autoregressive order of the model.
a positive integer specifying the number of mixture components.
a size (2x1) integer vector specifying the number of GMAR type components M1 in the
first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.
a real valued parameter vector specifying the 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 \(\phi_{m}\)=\((\phi_{m,1},...,\phi_{m,p}), m=1,...,M\).
Size \((M(p+4)-1x1)\) vector (\(\theta, \nu\))\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M}\)).
Size \((M(p+3)+M2-1x1)\) vector (\(\theta, \nu\))\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M}\)).
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 \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).
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 \(\phi\)=\((\phi_{1},...,\phi_{M})\).
Size \((4M+p-1x1)\) vector (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M})\).
Size \((3M+M2+p-1x1)\) vector (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M})\).
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 \(2\).
is "GMAR", "StMAR", or "G-StMAR" model considered? In the G-StMAR model, the first M1 components
are GMAR type and the rest M2 components are StMAR type.
Returns a vector with the same length as params. For other parameters than degrees
of freedom parameters larger than 100, the differences will be 6e-6. For the large degrees of
freedom parameters, the difference will be signif(df/1000, digits=2).
This function is used for approximating gradient and Hessian of a StMAR or G-StMAR model. Large degrees of freedom parameters cause significant numerical error if too small differences are used.