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... (a) Sei S(Yi– YiE) = 0 and
(b) SX2iei SX2i(Yi– YiE) = 0
where YiE is the estimated value of Yi.
X2i is the same as Xi from before
Proof: S(Yi– YiE) = S(Yi– b^1 - b^2 X2i)
= SYi–S b^1 - Sb^2 X2i
...

# Multi-fractional order estimator

The multi-fractional order estimator (MFOE) is a straightforward, practical, and flexible alternative to the Kalman filter (KF) for tracking targets. The MFOE is focused strictly on simple and pragmatic fundamentals along with the integrity of mathematical modeling. Like the KF, the MFOE is based on the least squares method (LSM) invented by Gauss and the orthogonality principle at the center of Kalman's derivation. Optimized, the MFOE yields better accuracy than the KF and subsequent algorithms such as the extended KF and the interacting multiple model (IMM).The MFOE is an expanded form of the LSM, which effectively includes the KF and ordinary least squares (OLS) as subsets (special cases). OLS is revolutionized in for application in econometrics. The MFOE also intersects with signal processing, estimation theory, economics, finance, statistics, and the method of moments. The MFOE offers two major advances: (1) minimizing the mean squared error (MSE) with fractions of estimated coefficients (useful in target tracking) and (2) describing the effect of deterministic OLS processing of statistical inputs (of value in econometrics)