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Simple Approach Estimator

Also SA Estimator · Simple Approach · Average of Ratios · R_SA

The simple approach estimator of the resistance averages the single-measurement ratios, R̂_SA = (1/N) Σ u_k / i_k; it may not converge, and where it does it overestimates R by about the factor 1 + σ_i²/i_true².

Know these firstEstimator Expected Value Law of Large Numbers

Properties#

May not converge: if the current noise lets $i_k$ take the value zero (or come arbitrarily close to it), the average contains infinitely large terms.
Biased even when it converges. Assume no $i_k$ is zero and the noise is small enough for the Taylor series of $1/i_k$ around $i_\text{true}$ to converge: $\frac{1}{i_k} = \frac{1}{i_\text{true}\left(1 + \frac{n_{i,k}}{i_\text{true}}\right)} \approx \frac{1}{i_\text{true}}\left(1 - \frac{n_{i,k}}{i_\text{true}} + \left(\frac{n_{i,k}}{i_\text{true}}\right)^2 - \left(\frac{n_{i,k}}{i_\text{true}}\right)^3 + \ldots\right).$ Multiplying by $u_k = u_\text{true} + n_{u,k}$ and averaging, the terms linear in the noise and the product $n_{i,k} n_{u,k}$ vanish, but $n_{i,k}^2$ averages to $\sigma_i^2$ (Law of Large Numbers): $\lim_{N \to \infty}\hat R_\text{SA}^{[N]} \approx \frac{u_\text{true}}{i_\text{true}}\left(1 + \frac{\sigma_i^2}{i_\text{true}^2}\right) > R_\text{true}.$ So under these optimistic assumptions the estimator overestimates $R$ , in contrast to the Least Squares Estimator, which underestimates it.
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.

Example#

Sources#

MSI script p. 11 (§1.2.1), MSI_VL01 > 1.2.1 Simple Approach (SA)
MSI script p. 21 (§2.5.3, eqs. (2.11)–(2.13)), MSI Slides 02 > 2.5.3 Simple Approach (not published)

Connections

Mentioned in inline links7

Error-in-Variables Estimator
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 L… §
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… §
Simple Approach Estimator — $\frac1N\sum_k u_k/i_k$ : average of the ratios, biased upwards (and may not converge) §
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… §
Expected ValuePROB
$\mathbb{E}\{u/i\} \ne \mathbb{E}\{u\}/\mathbb{E}\{i\}$ , because $1/i$ is nonlinear in the noisy current. This is why averaging the ratios $u_k/i_k$ does not converge to $R_\text{true}$ (Simple Approach Estimator). §
Random Variables and DistributionsPROB
Uniform vs. normal. Both have a mean and a variance, but the uniform distribution is bounded (all mass on $[a, b]$ , variance $(b-a)^2/12$ ), while the normal one has unbounded support. For estimators this matters: with normal current noise… §

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