# Extrapolating the amount of data missing from the amount of data partially missing

Suppose I have a fleet of two-car trains riding around, and that each car is equipped with a data recording device. Unfortunately, some of the recording devices aren't working. I don't know either the exact size of the fleet or the percentage of failed recording devices. I'd like to make a reasonable guess about how many cars I'm missing data from.

In particular:

• Total fleet size $S$ is unknown.
• Failure rate (failed units / total cars) $F$ is unknown.
• I have data from both cars from $S_2$ trains and from exactly one car from $S_1$ trains.
• Therefore, I know that there are at least $S_1$ failed units. What I don't know is $S_0$, the number of trains with failed units on both cars.
• Let's assume that the distribution of cars with failed units is random, and whether one car has a failed unit is independent of whether it's mate does.

Does the following make sense for a first-order approximation?

• Guess that the failure rate $F'$ is equal to the proportion of missing cars that I know about to total cars that I know about: $F' = S_1 / (2*(S_2 + S_1))$
• Assume that the likelihood of a train having two failed units is $F'^2$.
• Therefore $S_0 = F'^2 * (S_0 + S_1 + S_2)$
• Therefore $S_0 = (S_2 + S_1) * F'^2 / (1 - F'^2)$
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You can use LaTeX for formulas, enclose them in dollar signs as in normal tex document. What is the reasoning of your second-last equation. As far as I understood $S_2$ is the number of trains with non-failed units, so your equation does not make sense then. –  mpiktas Dec 22 '10 at 20:14
Thanks for TeXing that up for me. This is my first time on a TeX-enabled SE, and the WYSIWYG for TeX wasn't working for me, so I left it out. Also, I fixed the last two equations, in which I accidentally put in $S_2$ where I meant $S_0$. –  user2491 Dec 22 '10 at 20:39

This is a quick partial response to outline some options and correct some errors.

You are implicitly seeking a method of moments estimator. Under your assumptions, letting $f$ be the failure rate and $n$ be the fleet size, the expectations of the $S_i$ (which are governed by a multinomial distribution) are

\eqalign{ \mathbb{E}_{f;n}[S_0] = &f^2 n \cr \mathbb{E}_{f;n}[S_1] = &2 f (1-f) n \cr \mathbb{E}_{f;n}[S_2] = &(1-f)^2 n. }

From the algebraic relation

$$\mathbb{E}_{f;n}[S_0] = \frac{\mathbb{E}_{f;n}[S_1]^2}{4 \mathbb{E}_{f;n}[S_2]}$$

(which implies the failure rate should be about twice your estimate) and assuming $S_2 \ne 0$ you can derive the method of moments estimator

$$\hat{S}_0 = \frac{S_1^2}{4 S_2}.$$

This is likely to be biased, especially if $S_2$ is small, which impels us to consider other estimators.

More generally, this problem can be thought of as looking for a "good" estimator for $S_0$ of the form $\hat{S}_0 = t(S_1, S_2)$ in an experiment in which the outcomes are the sum of $n$ Binomial($1-f$, $2$) variables and $S_1$ and $S_2$ are the counts of the single and double "successes," respectively. To this end you need to supply a loss function $\Lambda(s,t)$ and analyze the statistical risk $r$, which is the expected loss

$$r_t(f,n) = \mathbb{E}_{f;n}[\Lambda(S_0, t(S_1, S_2))].$$

The loss function quantifies the cost of estimating that $S_0$ equals $t(S_1, S_2)$; usually the estimate is not perfect and there is a cost associated with that. The risk for a particular estimator $t$ is the expected loss; it depends on the unknown failure rate $f$ and the unknown fleet size $n$. Thus the exercise comes down to finding procedures with acceptable risk functions. If you have some quantitative information about the likely values of $f$ and $n$ you can exploit that either directly, by limiting the domain of the risk to the likely values, or with a Bayesian analysis in which you compute the expected value of $r_t$ under some assumed prior distribution of $(f,n)$. At this point the risk becomes solely a function of the procedure $t$ and it's "merely" a matter of finding a risk-minimizing procedure.

In any event, to make further progress you need to supply a loss function (or some reasonable approximation thereof). I would hesitate to recommend or use the method of moments estimator (derived above) without knowing something about your loss.

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Thanks! In my case, the true value of $n$ is on the order of 30, and the worst case I've seen is $S_2$ around 20 and $S_1$ around 10, making $\hat{S_0}/(S_1 + S_2)$ on the order of a few percent. Does that qualify as $S_2$ being small enough to make $\hat{S_0}$ very biased? I'm not sure how to best express my loss function, but it's negligible for $(S_0 - \hat{S_0})/n$ of, say 5% or less, but rises nonlinearly as that error gets away from 5%. The bottom line is that if the method of moments estimator is "close enough," I'd be biased toward using it rather than something more sophisitcated. –  user2491 Dec 22 '10 at 21:32
@Isaac Simulations suggest the bias in your case isn't bad at all: around 0.5 on average. It also seems the variance of this MM estimator is approximately the square of the estimate itself. For your data this means (if the model is correct!) that you can be confident $S_0$ is between 0 and 4 or so. –  whuber Dec 22 '10 at 21:59