1.6. Estimation of Dispersion Parameter

Remark(ML estimation)

If the dispersion parameter is unknown, its ML estimate, \(\hat\phi\) may be obtained as a solution to the equation

\[\begin{eqnarray*} 0&=&\frac{dL}{d\phi}\Big|_{\beta=\hat\beta}\\ &=& -\frac{1}{\phi^2}\sum_{i=1}^n w_i \{\hat\theta y_i -b(\hat\theta) \} + \sum_{i=1}^n\frac{d}{d\phi}c(y_i;\phi/w_i), \end{eqnarray*}\] where \(a(\phi)=\phi/w_i\).

• Example : Normal variance The log-likelihood for a normal theory linear model is \[ l(\hat\beta, \phi; y)= -\frac{1}{\sigma^2}\sum_{i=1}^n (y_i-\hat\mu_i)^2 - \frac{n}{2}\log\phi, \] where \(\hat\mu=x_i'\beta\). Differentiating and solving for \(\hat\phi\) yiedls the MLE \[ \hat\phi=\frac{1}{n}\sum_{i=1}^n(y_i-\hat\mu_i)^2. \]

  

Remark(Pearson and deviance estimates)

Recall \[ \text{Var}(Y_i)= V(\mu_i)a(\phi)=V(\mu_i)\frac{\phi}{w_i}. \] It follows that \[ E\left[ \frac{w_i(y_i-\mu)^2}{V(\mu_i)} \right]=\phi\mbox{ }\mbox{ for all }i. \]

This motivates the Pearson estimate of \(\phi\), given by \[ \tilde\phi= \frac{1}{\text{df}}\sum_{i=1}^n \frac{w_i(y_i-\hat\mu_i)^2}{V(\hat\mu_i)}=\frac{1}{\text{df}}\chi^2(y;\hat\mu), \] where \(\text{df}=n-p-1\) is the degree of freedom for estimating the dispersion parameter.

In the normal theory linear model setting, this estimate is the MSE, the usual unbiased estimate of the variance.

• 정규분포 예제에서, \(b(\theta_i)=\frac{\theta_i^2}{2}\)이고 \(\mu_i=\theta_i\)이다. 때문에 \(b''(\theta_i)=V(\mu_i)=1\)이어서 위의 식이 MSE가 된다.

Similarly, the deviance estimate is defined by \[ \bar{\phi}=\frac{1}{\text{df}}D(y;\hat\mu). \]

  

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