Erros padrão das estimativas de distribuição hiperbólica usando o método delta?

hyperbPi $\pi$ summaryhyperbFit $\mathrm{Var}(X)=\mathrm{SE}(X)^2$ $g_{\alpha}(\zeta, \pi, \delta)$ $g_{\beta}(\zeta, \pi, \delta)$ $\alpha$ $\beta$

\begin{aligned} g_{α} (ζ, π, δ) & = \frac{ζ \sqrt{1 + π^{2}}}{δ} \\ g_{β} (ζ, π, δ) & = \frac{ζ π}{δ} \end{aligned}

$\begin{align} g_{\alpha}(\zeta, \pi, \delta) &=\frac{\zeta\sqrt{1 + \pi^{2}}}{\delta}\\ g_{\beta}(\zeta, \pi, \delta) &= \frac{\zeta\pi}{\delta}\\ \end{align}$

α

$\alpha$

\begin{aligned} \frac{\partial}{\partial ζ} g_{α} (ζ, π, δ) & = \frac{\sqrt{1 + π^{2}}}{δ} \\ \frac{\partial}{\partial π} g_{α} (ζ, π, δ) & = \frac{π ζ}{\sqrt{1 + π^{2}} δ} \\ \frac{\partial}{\partial δ} g_{α} (ζ, π, δ) & = - \frac{\sqrt{1 + π^{2}} ζ}{δ^{2}} \end{aligned}

$\begin{align} \frac{\partial}{\partial \zeta} g_{\alpha}(\zeta, \pi, \delta) &=\frac{\sqrt{1+\pi^{2}}}{\delta}\\ \frac{\partial}{\partial \pi} g_{\alpha}(\zeta, \pi, \delta) &= \frac{\pi\zeta}{\sqrt{1+\pi^{2}}\delta }\\ \frac{\partial}{\partial \delta} g_{\alpha}(\zeta, \pi, \delta) &= -\frac{\sqrt{1+\pi^{2}}\zeta}{\delta^{2}}\\ \end{align}$

β

$\beta$

\begin{aligned} \frac{\partial}{\partial ζ} g_{β} (ζ, π, δ) & = \frac{π}{δ} \\ \frac{\partial}{\partial π} g_{β} (ζ, π, δ) & = \frac{ζ}{δ} \\ \frac{\partial}{\partial δ} g_{β} (ζ, π, δ) & = - \frac{π ζ}{δ^{2}} \end{aligned}

$\begin{align} \frac{\partial}{\partial \zeta} g_{\beta}(\zeta, \pi, \delta) &=\frac{\pi}{\delta}\\ \frac{\partial}{\partial \pi} g_{\beta}(\zeta, \pi, \delta) &= \frac{\zeta}{\delta }\\ \frac{\partial}{\partial \delta} g_{\beta}(\zeta, \pi, \delta) &= -\frac{\pi\zeta}{\delta^{2}}\\ \end{align}$

$\alpha$

V a r (α) \approx \frac{1 + π^{2}}{δ^{2}} \cdot V a r (ζ) + \frac{π^{2} ζ^{2}}{(1 + π^{2}) δ^{2}} \cdot V a r (π) + \frac{(1 + π^{2}) ζ^{2}}{δ^{4}} \cdot V a r (δ) + 2 \times [\frac{π ζ}{δ^{2}} \cdot C o v (π, ζ) - \frac{(1 + π^{2}) ζ}{δ^{3}} \cdot C o v (δ, ζ) - \frac{π ζ^{2}}{δ^{3}} \cdot C o v (δ, π)]

$\mathrm{Var}(\alpha)\approx \frac{1+\pi^{2}}{\delta^{2}}\cdot \mathrm{Var}(\zeta)+\frac{\pi^{2}\zeta^{2}}{(1+\pi^{2})\delta^{2}}\cdot \mathrm{Var}(\pi) + \frac{(1+\pi^{2})\zeta^{2}}{\delta^{4}}\cdot \mathrm{Var}(\delta) + \\ 2\times \left[ \frac{\pi\zeta}{\delta^{2}}\cdot \mathrm{Cov}(\pi,\zeta) - \frac{(1+\pi^{2})\zeta}{\delta^{3}}\cdot \mathrm{Cov}(\delta,\zeta)- \frac{\pi\zeta^{2}}{\delta^{3}}\cdot \mathrm{Cov}(\delta,\pi)\right]$

β

$\beta$

V a r (β) \approx \frac{π^{2}}{δ^{2}} \cdot V a r (ζ) + \frac{ζ^{2}}{δ^{2}} \cdot V a r (π) + \frac{π^{2} ζ^{2}}{δ^{4}} \cdot V a r (δ) + 2 \times [\frac{π ζ}{δ^{2}} \cdot C o v (π, ζ) - \frac{π^{2} ζ}{δ^{3}} \cdot C o v (δ, ζ) - \frac{π ζ^{2}}{δ^{3}} \cdot C o v (π, δ)]

$\mathrm{Var}(\beta)\approx \frac{\pi^{2}}{\delta^{2}}\cdot \mathrm{Var}(\zeta) + \frac{\zeta^{2}}{\delta^{2}}\cdot \mathrm{Var}(\pi) + \frac{\pi^{2}\zeta^{2}}{\delta^{4}}\cdot \mathrm{Var}(\delta) + \\ 2\times \left[ \frac{\pi\zeta}{\delta^{2}}\cdot \mathrm{Cov}(\pi,\zeta) - \frac{\pi^{2}\zeta}{\delta^{3}}\cdot \mathrm{Cov}(\delta, \zeta) - \frac{\pi\zeta^{2}}{\delta^{3}}\cdot \mathrm{Cov}(\pi, \delta) \right]$

Codificação em R

$D$ $\alpha$ $\beta$ $\zeta, \pi, \delta$ $\Sigma$ $3\times 3$ $\zeta, \pi, \delta$ vcov(my.hyperbFit)my.hyperbFit $\alpha$

V a r (α) \approx D_{α} Σ D_{α}^{⊤}

$\mathrm{Var}(\alpha)\approx D_{\alpha}\Sigma D_{\alpha}^\top$

β

$\beta$

Em R, isso pode ser facilmente codificado como este:

#-----------------------------------------------------------------------------
# The row vector D of the partial derivatives for alpha
#-----------------------------------------------------------------------------

D.alpha <- matrix(
  c(
    sqrt(1+pi^2)/delta,                 # differentiate wrt zeta
    ((pi*zeta)/(sqrt(1+pi^2)*delta)),   # differentiate wrt pi
    -(sqrt(1+pi^2)*zeta)/(delta^2)      # differentiate wrt delta
  ),
  ncol=3)

#-----------------------------------------------------------------------------
# The row vector D of the partial derivatives for beta
#-----------------------------------------------------------------------------

D.beta <- matrix(
  c(
    (pi/delta),            # differentiate wrt zeta
    (zeta/delta),          # differentiate wrt pi
    -((pi*zeta)/delta^2)   # differentiate wrt delta
  ),
  ncol=3)

#-----------------------------------------------------------------------------
# Calculate the approximations of the variances for alpha and beta
# "sigma" denotes the 3x3 covariance matrix
#-----------------------------------------------------------------------------

var.alpha <- D.alpha %*% sigma %*% t(D.alpha) 
var.beta <- D.beta %*% sigma %*% t(D.beta)

#-----------------------------------------------------------------------------
# The standard errors are the square roots of the variances
#-----------------------------------------------------------------------------

se.alpha <- sqrt(var.alpha)
se.beta <- sqrt(var.beta)

$\log(\zeta)$ $\log(\delta)$

$\zeta^{*}=\log(\zeta)$ $\delta^{*}=\log(\delta)$ $\zeta$ $\delta$

\begin{aligned} g_{α} (ζ^{*}, π, δ^{*}) & = \frac{\exp (ζ^{*}) \sqrt{1 + π^{2}}}{\exp (ζ^{*})} \\ g_{β} (ζ^{*}, π, δ^{*}) & = \frac{\exp (ζ^{*}) π}{\exp (δ^{*})} \end{aligned}

$\begin{align} g_{\alpha}(\zeta^{*}, \pi, \delta^{*}) &=\frac{\exp(\zeta^{*})\sqrt{1 + \pi^{2}}}{\exp(\zeta^{*})}\\ g_{\beta}(\zeta^{*}, \pi, \delta^{*}) &= \frac{\exp(\zeta^{*})\pi}{\exp(\delta^{*})}\\ \end{align}$

α

$\alpha$

\begin{aligned} \frac{\partial}{\partial ζ^{*}} g_{α} (ζ^{*}, π, δ^{*}) & = \sqrt{1 + π^{2}} \exp (- δ^{*} + ζ^{*}) \\ \frac{\partial}{\partial π} g_{α} (ζ^{*}, π, δ^{*}) & = \frac{π \exp (- δ^{*} + ζ^{*})}{\sqrt{1 + π^{2}}} \\ \frac{\partial}{\partial δ^{*}} g_{α} (ζ^{*}, π, δ^{*}) & = - \sqrt{1 + π^{2}} \exp (- δ^{*} + ζ^{*}) \end{aligned}

$\begin{align} \frac{\partial}{\partial \zeta^{*}} g_{\alpha}(\zeta^{*}, \pi, \delta^{*}) &=\sqrt{1+\pi^{2}}\exp(-\delta^{*}+\zeta^{*})\\ \frac{\partial}{\partial \pi} g_{\alpha}(\zeta^{*}, \pi, \delta^{*}) &=\frac{\pi\exp(-\delta^{*}+\zeta^{*})}{\sqrt{1+\pi^{2}}} \\ \frac{\partial}{\partial \delta^{*}} g_{\alpha}(\zeta^{*}, \pi, \delta^{*}) &=-\sqrt{1+\pi^{2}}\exp(-\delta^{*}+\zeta^{*})\\ \end{align}$

β

$\beta$

\begin{aligned} \frac{\partial}{\partial ζ^{*}} g_{β} (ζ^{*}, π, δ^{*}) & = π \exp (- δ^{*} + ζ^{*}) \\ \frac{\partial}{\partial π} g_{β} (ζ^{*}, π, δ^{*}) & = \exp (- δ^{*} + ζ^{*}) \\ \frac{\partial}{\partial δ^{*}} g_{β} (ζ^{*}, π, δ^{*}) & = - π \exp (- δ^{*} + ζ^{*}) \end{aligned}

$\begin{align} \frac{\partial}{\partial \zeta^{*}} g_{\beta}(\zeta^{*}, \pi, \delta^{*}) &=\pi\exp(-\delta^{*}+\zeta^{*})\\ \frac{\partial}{\partial \pi} g_{\beta}(\zeta^{*}, \pi, \delta^{*}) &=\exp(-\delta^{*}+\zeta^{*})\\ \frac{\partial}{\partial \delta^{*}} g_{\beta}(\zeta^{*}, \pi, \delta^{*}) &=-\pi\exp(-\delta^{*}+\zeta^{*})\\ \end{align}$

α

$\alpha$

V a r (α) \approx (1 + π^{2}) \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot V a r (ζ^{*}) + \frac{π^{2} \exp (- 2 δ^{*} + 2 ζ^{*})}{1 + π^{2}} \cdot V a r (π) + (1 + π^{2}) \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot V a r (δ^{*}) + 2 \times [π \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot C o v (π, ζ^{*}) - (1 + π^{2}) \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot C o v (δ^{*}, ζ^{*}) - π \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot C o v (δ^{*}, π)]

$\mathrm{Var}(\alpha)\approx (1+\pi^{2})\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Var}(\zeta^{*})+\frac{\pi^{2}\exp(-2\delta^{*}+2\zeta^{*})}{1+\pi^{2}}\cdot \mathrm{Var}(\pi) + (1+\pi^{2})\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Var}(\delta^{*}) + \\ 2\times \left[ \pi\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Cov}(\pi,\zeta^{*}) - (1+\pi^{2})\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Cov}(\delta^{*},\zeta^{*}) - \pi\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Cov}(\delta^{*},\pi)\right]$

β

$\beta$

V a r (β) \approx π^{2} \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot V a r (ζ^{*}) + \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot V a r (π) + π^{2} \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot V a r (δ^{*}) + 2 \times [π \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot C o v (π, ζ^{*}) - π^{2} \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot C o v (δ^{*}, ζ^{*}) - π \exp (- 2 δ^{*} + 2 ζ^{*}) \cdot C o v (δ^{*}, π)]

$\mathrm{Var}(\beta)\approx \pi^{2}\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Var}(\zeta^{*})+\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Var}(\pi) + \pi^{2}\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Var}(\delta^{*}) + \\ 2\times \left[\pi\exp(-2\delta^{*}+2\zeta^{*}) \cdot \mathrm{Cov}(\pi,\zeta^{*}) -\pi^{2}\exp(-2\delta^{*}+2\zeta^{*})\cdot \mathrm{Cov}(\delta^{*},\zeta^{*}) -\pi\exp(-2\delta^{*}+2\zeta^{*}) \cdot \mathrm{Cov}(\delta^{*},\pi)\right]$

Codificação em R2

sigma $\zeta^{*}=\log(\zeta)$ $\delta^{*}=\log(\delta)$ $\zeta$ $\delta$

#-----------------------------------------------------------------------------
# The row vector D of the partial derivatives for alpha
#-----------------------------------------------------------------------------

D.alpha <- matrix(
  c(
    sqrt(1+pi^2)*exp(-ldelta + lzeta),            # differentiate wrt lzeta
    ((pi*exp(-ldelta + lzeta))/(sqrt(1+pi^2))),   # differentiate wrt pi
    (-sqrt(1+pi^2)*exp(-ldelta + lzeta))          # differentiate wrt ldelta
  ),
  ncol=3)

#-----------------------------------------------------------------------------
# The row vector D of the partial derivatives for beta
#-----------------------------------------------------------------------------

D.beta <- matrix(
  c(
    (pi*exp(-ldelta + lzeta)),    # differentiate wrt lzeta
    exp(-ldelta + lzeta),         # differentiate wrt pi
    (-pi*exp(-ldelta + lzeta))    # differentiate wrt ldelta
  ),
  ncol=3)

#-----------------------------------------------------------------------------
# Calculate the approximations of the variances for alpha and beta
# "sigma" denotes the 3x3 covariance matrix with log(delta) and log(zeta)
#-----------------------------------------------------------------------------

var.alpha <- D.alpha %*% sigma %*% t(D.alpha) 
var.beta <- D.beta %*% sigma %*% t(D.beta)

#-----------------------------------------------------------------------------
# The standard errors are the square roots of the variances
#-----------------------------------------------------------------------------

se.alpha <- sqrt(var.alpha)
se.beta <- sqrt(var.beta)

COOLSerdash
fonte

C o v (π, ζ)

$\mathrm{Cov}(\pi, \zeta)$

π

$\pi$

ζ

$\zeta$

β

$\beta$

ζ / δ \times π / δ = (ζ \cdot π) / δ^{2}

$\zeta/\delta \times \pi/\delta = (\zeta\cdot\pi)/\delta^{2}$

varcov <- solve(hyperbfitalv$hessian)

π, ζ, δ

$\pi, \zeta, \delta$

Muito obrigado pela sua resposta, mas este é EXATAMENTE o problema, porque a parametrização deste hessian é para log (delta) e log (zeta) e não para delta e zeta! Ver o meu follow-up mensagens: stats.stackexchange.com/questions/67595/... e, especialmente, a resposta de CT Zhu aqui stats.stackexchange.com/questions/67602/...

Jen Bohold

é preciso obter o hessiano de pi, log (zeta), log (delta) e mu para o hessian de pi, zeta, delta e mu. Você sabe como isso pode ser feito?

Jen Bohold

Também tentei fazê-lo com o hessian "errado"; portanto, com log (delta) e log (zeta), depois disso, selecionei a sub-matriz e fiz os cálculos. Os resultados não estavam corretos, porque os valores eram muito grandes, mais ou menos 60.000.

Jen Bohold

Erros padrão das estimativas de distribuição hiperbólica usando o método delta?

Respostas:

log(ζ)log⁡(ζ)\log(\zeta)log(δ)log⁡(δ)\log(\delta)

$\log(\zeta)$ $\log(\delta)$