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Theoretical analysis demonstrates parameter estimation without prior distributional assumptions, indicating that maximum likelihood estimation is a restricted special case.
AbstractSince Fisher introduced Maximum Likelihood Estimation in 1922, statisticalinference has imposed a distributional assumption f (x|θ) before examining a singleobservation. For 104 years this imposition has been treated as necessary. It is not.This paper introduces Zulfia Estimation Theory (ZET): estimation withoutimposed distributions. The estimator is not the parameter that maximises an imposedlikelihood. It is the parameter that closes the Zulfia Gap:ˆθZET : ∆(θ, x) = 0where ∆(θ, x) = x − MZ [θ] is the Zulfia Deviation — generated by the data, neverimposed. We prove that MLE is a special case of ZET when the imposed distributionaccidentally matches the generated deviation structure. ZET is always structurallycorrect. MLE is correct only when its assumption is correct. The confusion of 104years ends here
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DR. ZULFIQAR ALI KHAN (2026) studied this question.
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