unitbeta

The unit used for a list which, for each integer b from 1 to B includes the name of parameter $\beta_b$, the estimated parameter $\hat{\beta}_b$, the estimated asymptotic standard deviation $\hat{\sigma}(\hat{\beta}_b)$ of $\hat{\beta}_b$ derived from the square root of row~$b$ and column~$b$ of the estimated asymptotic covariance matrix \begin{equation} \label{ecovbeta} \widehat{\Cov}(\hat{\bs{\beta}})= \mathbf{T}[-\nabla^2\ell_V(\hat{\bs{\gamma}})]^{-1}[-\nabla^2\ell(\hat{\bs{\gamma}})] [-\nabla^2\ell_V(\hat{\bs{\gamma}})]^{-1}\mathbf{T}' \end{equation} of $\hat{\bs{\beta}}$, the estimated asymptotic standard deviation $\hat{\sigma}_L(\hat{\beta}_b)$ of $\hat{\beta}_b$ derived from the square root of row~$b$ and column~$b$ of the Louis \cite{lo82} estimate \begin{equation} \label{ecovbetalo} \widehat{\Cov}_L(\hat{\bs{\beta}})= \mathbf{T}[\bs{\Phi}_V(\hat{\bs{\gamma}})]^{-1}\bs{\Phi}(\hat{\bs{\gamma}}) [\bs{\Phi}_V(\hat{\bs{\gamma}})]^{-1}\mathbf{T}' \end{equation} of the asymptotic covariance matrix of $\hat{\bs{\beta}}$, and the estimated asymptotic standard deviation $\hat{\sigma}_S(\hat{\beta}_b)$ of $\hat{\beta}_b$ derived from the square root of row~$b$ and column~$b$ of the sandwich estimated asymptotic covariance matrix \begin{equation} \label{ecovbetasa} \widehat{\Cov}_S(\hat{\bs{\beta}})= \mathbf{T}[-\nabla^2\ell_V(\hat{\bs{\gamma}})]^{-1}\bs{\Phi}(\hat{\bs{\gamma}}) [-\nabla^2\ell_V(\hat{\bs{\gamma}})]^{-1}\mathbf{T}' \end{equation} obtained without the assumption that the model holds (Huber, 1967; White, 1980; Haberman, 1989). If the Louis approximation is used for the negative Hessian matrix in implementation of the Newton-Raphson algorithm, then these three estimated asymptotic standard deviations are all the same. If complex sampling is used, then the estimated asymptotic standard deviation $\hat{\sigma}_C(\hat{\beta}_b)$ of $\hat{\beta}_b$ derived from the square root of row~$b$ and column~$b$ of the estimated asymptotic covariance matrix \begin{equation} \label{ecovbetaco} \widehat{\Cov}_S(\hat{\bs{\beta}})= \mathbf{T}[-\nabla^2\ell_V(\hat{\bs{\gamma}})]^{-1}\widehat{\Cov}(\nabla\ell(\bs{\gamma})) [-\nabla^2\ell_V(\hat{\bs{\gamma}})]^{-1}\mathbf{T}'. \end{equation}

In \href{listening9.txt}{listening9.txt}, \nameref{unitbeta} is 11. An example of output appears in \href{listeningbeta.csv}{listeningbeta.csv}. In this example, all expressions for standard errors yield rather similar results. The third column is $\hat{\sigma}(\hat{\beta}_b)$, the fourth column is $\hat{\sigma}_L(\hat{\beta}_c)$, and the fifth column is $\hat{\sigma}_S(\hat{\beta}_b)$. In \href{listeningp2.txt}{listeningp2.txt}, an artificial example of complex sampling is provided in which primary sampling units are defined in terms of a sequence number. In this case, \nameref{unitbeta} is 11. The corresponding file is \href{listeningpbeta.csv}{listeningpbeta.csv}. Because the example is artificial, the added column for $\hat{\sigma}_C(\hat{\beta}_b)$ is quite similar to the other columns.