Fundamental Statistics Learning Note (14)

Wald’s memos

Analyzed data of planes returning from combat missions

  1. rate of survival when hit by enemy gunfire
  2. how to imporve survival rate by reinforcing certain plane parts
    Intuition: section A has lots of hits among survivors, section B has few hits among survivors.
    1. Analysis of rate of survival when hit by enemy fire
      $$\begin{matrix}
      x_{10} && x_{11} && x_{12} && x_{13} \\
      && x_{21} && x_{22} && x_{23}
      \end{matrix}$$
      $N = \sum x_{ij} \equiv$ total planes sent on mission
      $x_{1j} \equiv $ # planes returning with $j$ hits
      $x_{2j} \equiv $ # planes not returned with $j$ hits
      Let $p{ij} \equiv$ probability of being in category $x_{ij}$
      we knoe the total lost is $x_{21}+x_{22}+x_{23}$ but don’t know $x_{21}, x_{22}, x_{23}$ counts individually
      $$L(p_{10},p_{11},p_{12},p_{13}|x_{10},x_{11}, x_{12}, x_{13}, x_{21}+x_{22}+x_{23}) \overset{\text{planes are independent}}= C\cdot p_{10}^{x_{10}}p_{11}^{x_{11}} p_{12}^{x_{12}} p_{13}^{x_{13}}(1-\sum_{j=0}^3 p_{ij}^{x_{21}+x_{22}+x_{23}})$$
      where $C$ is the count orderings which doesn’t depend on $p_{ij}$
      Note that $$x_{21}+x_{22}+x_{23} = N - \sum_{j=0}^3 x_{1j}$$
      $$l(p_{10},p_{11},p_{12},p_{13}) = logC + \sum_{j=0}^3 x_{ij}logp_{ij}+(N - \sum_{j=0}^3 x_{1j})log(1-\sum_{j=0}^3 p_{ij})$$
      $$\Rightarrow \frac{\alpha l}{\alpha p_{10}}=\frac{x_{10}}{p_{10}}+(N-1-\sum_{j=0}^3 x_{ij})\frac{-1}{1-1-\sum_{j=0}^3 p_{ij}} \overset{\text{set}}=0$$
      Similarly, we need the partial derivative for other parameter, $\frac{\alpha l}{\alpha p_{ij}}$, sovling the 4 equations gives MLE: $\hat p_{ij} = \frac{x_{ij}}{n}$
      Defein $r_j = \frac{p_{1j}}{p_{1j}+p_{2j}}$, wald define this to represent probability of surviving hit $j$ given survival $j-1$.

Wald assumed $r_j = q^j, j=1,2,3,$ i.e. hits are iid with survival probability $q$ for each.

Note $1-\sum_{j=1}^3(p_{1j}+p_{2j})=1-p_{10}=\sum_{j=1}^3 \frac{p_{1j}}{r_j}=\sum_{j=1}^3\frac{p_{1j}}{q^j}$

MLE of $q$ by invariance is solution of $1-\hat p_{10} = \sum_{j=1}^3\frac{\hat p_{1j}}{q^j}$

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