First derivatives at the optimum analysis (\textit{fdao}): An approach to estimate the uncertainty in nonlinear regression involving stochastically independent variables

Abstract

An important problem of optimization analysis surges when parameters such as $\{{\theta }_j{\}}_{j=1,\dots ,k}$, determining a function $ y=f(x\given\{θ_j\}) $, must be estimated from a set of observables $\{x_i,y_i{\}}_{i=1,\dots ,m}$. Where $\{x_i\}$ are independent variables assumed to be uncertainty-free. It is known that analytical solutions are possible if $ y=f(x\givenθ_j) $ is a linear combination of $\{{\theta }_{j=1,\dots ,k}\}.$ Here it is proposed that determining the uncertainty of parameters that are not \textit{linearly independent} may be achieved from derivatives $ \tfrac{\partial f(x \given \{θ_j\})}{\partial θ_j} $ at an optimum, if the parameters are \textit{stochastically independent}.

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• Precise determination of the mass of the Higgs boson and tests of compatibility of its couplings with the standard model predictions using proton collisions at 7 and 8 TeV
• Randomness of D Sequences via Diehard Testing