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If I have a series of measurements without any calibration data for an instrument, are there statistical techniques to recover how much error (variance) is attributable to the instrument and how much is error intrinsic to the process under observation?

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  • $\begingroup$ If there's any information you've got regarding the instrument or any expected model for the data, some separation can be made. $\endgroup$
    – Spätzle
    Commented Nov 7, 2017 at 16:45
  • $\begingroup$ @Spätzle - otherwise, it's hopeless? $\endgroup$
    – Frank
    Commented Nov 7, 2017 at 17:03
  • $\begingroup$ IMHO If you've got no clue, we can't do anything. But if you know anything - even a basic relation these samples should comply, maybe $y=ax$ or any other form of $y=f(x)$ - there's a lot to be done. $\endgroup$
    – Spätzle
    Commented Nov 7, 2017 at 17:14
  • $\begingroup$ Thanks! I'm thinking that in some cases, the "instrument" might be a component of a measurement chain, where you can't really calibrate. $\endgroup$
    – Frank
    Commented Nov 7, 2017 at 20:23

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The short answer is no, and the long answer is "it depends". If I observe the sum of two random variables, but I have no information on the individual random variables themselves, then the error model is unidentifiable. However, if I know something about the distributions of the two errors, then I can gather some information about them based on the observed sum.

Here's an example: I tell you that two random numbers, $a$ and $b$, sum to 7, and ask you what your guess is for $a$ and $b$. You have no idea: the support for the answers is the entire real line! The probability you guess the correct answer is $0$. However, if I tell you that $a$ and $b$ come from two fair dice, then you have some prior information to work with. Your guesses are likely far more accurate.

Motivated by our example, I recommend you take a look at Bayesian methods for measurement error and mixed effect models. Mixed effect models allow you to "soak up" excess variability into a random effect. Here are a few good places to start:

Unfortunately, I'm not an expert in this part of statistics, so I can only point you in the direction I would go and the kind of questions I would then ask. I do hope that this information helps you ask the right questions to get a better-informed answer.

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  • $\begingroup$ Thanks a lot. Very much appreciated, and references/pointers are very useful! $\endgroup$
    – Frank
    Commented Nov 7, 2017 at 22:48

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