Estimating a Process Parameter from a Noisy Observable Variance

Summary

The document asks whether an unknown process parameter can be recovered from the sample variance of an observable series that approximately equals a latent process. Given a known variance formula for the latent quantity, the proposed estimator rearranges that formula to solve for the target parameter using the observed sample variance and known constants. The answer says this is reasonable when the observable truly equals the latent variable, and suggests modeling an error term when the relationship is only approximate.

A second response highlights that variance calculations for sums involve covariance terms, especially when estimating variance from changes in two jointly observed components. The key limitation is therefore the observation equation: approximation error, measurement noise, and dependence between components can affect the variance estimate. The document does not establish an estimator’s sampling properties or provide a procedure for estimating the latent-variable error.

Key ideas

• If the observable sum equals the latent quantity, its sample variance can estimate the latent variance.
• Known constants in the variance formula can then be used algebraically to estimate the target parameter.
• An approximate observation relationship may require an explicit residual or measurement-error model.
• Variance of a sum depends on the covariance between its components.
• The document does not assess finite-sample accuracy or estimator bias.

Tags

# How to estimate the variance of this stochastic process?


# How to estimate the variance of this stochastic process?












I have an unobservable stochastic quantity $\lambda(t)$, which I analytically know the variance of, that is

$$\text{Var}(\lambda(t))= \frac{\theta \sigma^2}{2\kappa}$$

My goal is to get an estimate of $\sigma^2$.

I can observe S and K historically at times $t=1,2,..$, and know that the following approximately holds

$$ \lambda(t) \approx S(t)+K(t)$$

Does it make sense to simply take the sample variance of the time series $(S(t)+K(t))$, let's call it $\hat{\sigma}^2_S$, and say that a reasonable estimate of $\sigma^2$ is $2 \kappa\frac{\hat{\sigma}^2_S}{\theta}$?

It should be mentioned that the distribution of $\lambda$ is not pretty, but I have the mean and variance.

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/32844

If $$ \Lambda = S + K $$ then you can look at samples of $S+K$ and estimate the variance of $\Lambda$ by the variance of $S+K$. If $\theta$ and $\kappa$ are known constants then you can do the algebra to derive $\sigma^2$.

The question is how the above equality holds. If it holds almost surely, then you are done. If it holds in probability then you are done too.

Maybe it helps to look at

$$ \Lambda_t = S_t + K_t + \epsilon_t $$ and model that resiudal $\epsilon$.

## Answer by Mats Lind (score 0)

https://quant.stackexchange.com/a/29522

$$ \text{Var}(dL) = \text{Var}(dK) + \text{Var}(dS) + \text{cov}(dS, dK) $$ I hope u have some simultaneous observations on $dK$ and $dS$ to estimate the covariance or a good assumption about it.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.