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 $1-\hat p_{10} = \sum_{j=1}^3\frac{\hat p_{1j}}{q^j}$

If you like my article, please feel free to donate!