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Say you want to estimate a statistic $\theta$ and have two data sources. A sample from data source A can be treated as a low-variance, somewhat biased estimate of $\theta$. A sample from data source B can be treated as a very high-variance, unbiased estimate of $\theta$. The distributions for A and B are otherwise unknown. The bias of A is unknown, but presumed low enough to use A as an initial estimate of $\theta$ until better information becomes available from B.

At each time step, you are presented with a sample from A and a sample from B, and at each time step you want to give your best estimate of $\theta$ based on the data seen so far. $\theta$ is not time-varying, and the samples from A are i.i.d., as are the samples from B. What is a good algorithm for this?

I'd imagine a good solution should be some time-varying weighted average of the mean of the A samples and the mean of the B samples. The weights should favor the low-variance A samples for the initial estimates, and as more samples accumulate, the weights should gradually switch over to the unbiased B samples. But how can we set the schedule for the weights? We need to balance the error resulting from the bias of A against the error resulting from the high variance of the sample mean from B, the second of which will decrease over time as sample size increases.

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If your two observations at time $t$ are $(y_{1t}, y_{2t})$ you can imagine that they are generated as: $$\pmatrix{y_{1t} \\ y_{2t}} = \pmatrix{k \\ 1}\theta_t + \pmatrix{\epsilon_{1t} \\ \epsilon_{2t}};$$ $y_{2t}$ is the unbiased observation, $k$ is meant to account for the bias of $y_{1t}$ and can be known or unknown. For instance, if you know that the bias around 10% the true value and positive, $k=1.10$, otherwise it can be estimated.

For $\theta_t$ you might write: $$\theta_t = \theta_{t-1} + \eta_t$$ which with the previous equation forms a simple state-space model that you can estimate using a Kalman filter.

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  • $\begingroup$ The bias is assumed not too large, but is unknown. It can only be estimated by comparing the long run sample mean of A with the long run sample mean of B. And if you have enough observations from B to do this accurately, then you already have an accurate estimate of $\theta$ from B alone, and no longer need A. So estimating the bias of A is really most of the problem. $\endgroup$
    – causative
    Commented Jan 13, 2022 at 14:56
  • $\begingroup$ I should also clarify that $\theta$ is not time-varying. $\endgroup$
    – causative
    Commented Jan 13, 2022 at 15:11
  • $\begingroup$ @clarivate, you do not need a beforehand estimation of the bias, you can just estimate $k$ with the other parameters. Time-invariance of $\theta$ can be accommodated setting the variance of $\theta_t$ to zero. $\endgroup$
    – F. Tusell
    Commented Jan 13, 2022 at 15:41
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    $\begingroup$ @causative, yes, you can use a t-test if you do the right correction, thats why I wrote "rodinary t-test". What you say about the variances assumes you pool A and B observations with equal weights, something which you would never want to do. $\endgroup$
    – F. Tusell
    Commented Jan 15, 2022 at 13:31
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    $\begingroup$ If $X$ and $Y$ both have mean $\theta$, that you want to estimate, an unbiased estimator is of the form $\hat\theta = cX+(1-c)Y$, and has variance $c^2\sigma_X^2 + (1-c)^2\sigma_Y^2$. If you minimize this wrt $c$ you arrive to the quite intuitive result that the weights of the observations should be inversely proportional to their variances. This generalize to more than two observations. For a suitable $c$ you whill have a variance which is less than that obtainable from $X$ or $Y$ alone. $\endgroup$
    – F. Tusell
    Commented Jan 16, 2022 at 16:54

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