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Error-in-Variables Estimator

Also EV Estimator · Errors-in-Variables Estimator · Error in Variables · Error-in-Variables · R_EV

The error-in-variables estimator of the resistance divides the average voltage by the average current, R̂_EV = (1/N Σ u_k) / (1/N Σ i_k); it is the only asymptotically unbiased one of the three estimators in the resistance example.

Know these firstEstimator Law of Large Numbers

Properties#

Asymptotically unbiased under the noise model $u_k = u_\text{true} + n_{u,k}$ , $i_k = i_\text{true} + n_{i,k}$ with zero-mean, independent, i.i.d. noise. By the Law of Large Numbers, numerator and denominator converge separately: $\lim_{N \to \infty}\hat R_\text{EV}^{[N]} = \frac{u_\text{true} + \lim_{N\to\infty}\frac1N\sum_k n_{u,k}}{i_\text{true} + \lim_{N\to\infty}\frac1N\sum_k n_{i,k}} = \frac{u_\text{true}}{i_\text{true}} = R_\text{true}.$
Why it works: both averages are linear in the noise, so the zero-mean noise averages out in each. Only one nonlinear operation, the final division, is applied after averaging. The Simple Approach Estimator does the division first and the Least Squares Estimator squares the noisy current, which is why both are biased.
It needs $i_\text{true} \ne 0$ , and the noise on both variables to be zero-mean.

Example#

Sources#

MSI script p. 12 (§1.2.2), MSI_VL01 > 1.2.2 Error-in-Variables (EV)
MSI script p. 20 (§2.5.1, eqs. (2.5)–(2.6)), MSI Slides 02 > 2.5.1 Error in Variables estimator (not published)

Connections

Mentioned in inline links6

Simple Approach Estimator
Root cause: $1/i$ is nonlinear, and $\mathbb{E}\{u/i\} \ne \mathbb{E}\{u\}/\mathbb{E}\{i\}$ (Expected Value). Averaging after the nonlinear division keeps a noise contribution; the Error-in-Variables Estimator divides only after averaging. §
VL01 – Introduction
The concept notes for this lecture now exist. In ## Concepts, replace the plain-text names with links: “Estimator” → Estimator (foundation), “Resistance estimation example” → Resistance Estimation Example, “Error-in-Variables estimator” →… §
Resistance Estimation Example
Estimate the resistance $R$ from $N$ noisy measurements $(u_k, i_k)$ . Three natural estimators, averaging the ratios, dividing the averages and least squares, all converge as $N$ grows, but to three different values (script Fig. 1.4). The… §
Error-in-Variables Estimator — $\sum_k u_k / \sum_k i_k$ : ratio of the averages, asymptotically unbiased §
Asymptotically Unbiased EstimatorPROB
In the resistance example, the Error-in-Variables Estimator is asymptotically unbiased, while the Least Squares Estimator and the Simple Approach Estimator are not. §
EstimatorPROB
From noisy voltage and current measurements $(u_k, i_k)$ , MSI builds three estimators of $R$ : the Simple Approach Estimator $\frac1N\sum_k \frac{u_k}{i_k}$ , the Error-in-Variables Estimator $\frac{\sum_k u_k}{\sum_k i_k}$ and the Least Squ… §
Law of Large NumbersPROB
Main tool for asymptotic analysis: limits of estimators built from averages are computed by replacing each average by its expected value. Ratios of averages converge to ratios of expectations (if the denominator’s limit is nonzero), which… §

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