PSI - Issue 75

Philippe AMUZUGA et al. / Procedia Structural Integrity 75 (2025) 53–64 Author name / Structural Integrity Procedia 00 (2025) 000–000

58

6

The proof of unbiasedness is established as follows:

1 X ⊤ y ]

E [ ˆ β ] = E [( X ⊤ X ) −

1 X ⊤ ( X β

= E [( X ⊤ X ) − = β + ( X ⊤ X ) −

+ ε + η )]

1 X ⊤ E [ ε ]

1 X ⊤ E [ η ]

+ ( X ⊤ X ) −

= β,

since E [ ε ] = E [ η ] = 0 [11]. This result ensures that the estimator ˆ β remains centered on the true value regardless of the amplitude of the Gaussian noise, guaranteeing convergence to β on average across multiple samples. To illustrate this property, a perfectly linear synthetic dataset ( y = 3 x + 2) was perturbed with Gaussian noise representing 30 % of the original standard deviation. As shown in Figure 5, the estimated coe ffi cients remain close to the theoretical values, confirming the predicted robustness.

(a) Perfect data

(b) Gaussian noise at 30 %

Fig. 5: Linear regression on perfect data (a) and with 30 % Gaussian noise (b). Estimated coe ffi cients remain very close to the true values despite the noise.

However, this robustness deteriorates when (i) the noise amplitude becomes dominant, (ii) the error distribution exhibits heavy tails, or (iii) outliers are present. In such cases, alternative methods such as quantile regression or ridge regression may be more suitable [12, 13].

Furthermore, the variance of the estimators increases with the variance of the injected noise:

Var( ˆ β ) = ( σ 2

2 η )( X ⊤ X ) −

1 ,

+ σ

which reduces the precision of ˆ β [2]. Thus, although the estimator remains unbiased, increased variance deteriorates estimation reliability [12, 13]. This robustness relies on key assumptions that must be formalized: • Independence of injected noise : η must be independent of both X and ε , i.e., E [ η | X ] = 0, Cov( ε,η ) = 0. • Zero mean distribution : E [ η ] = 0. Any bias ( E [ η ] 0) would translate into bias in ˆ β .