Quadratic Functional Estimation and Pointwise Inference from Paired Measurements with Unknown Errors Under Polynomial Lower Bounds

We study estimation of the integrated squared latent density from two measurements per subject with independent, unknown measurement errors. The error distributions may differ and need not be symmetric or centered, but their characteristic functions must satisfy prespecified polynomial lower bounds. Gaussian errors and characteristic-function zeros are excluded. A ratio of observed characteristic functions identifies the target without division by the latent characteristic function. For a regularized estimator using the same subjects throughout, we derive an exact linearization and a weighted integrated remainder bound. Under sufficient Sobolev smoothness, these yield a root-n normal limit and mean squared error of order n−1. Conditional projection identities prove positivity of the limiting variance. A consistent variance estimator permits logarithmic and sufficiently slow polynomial cutoffs, giving confidence intervals with pointwise asymptotic coverage under prespecified regularization. Full-pair likelihood bounds identify a low-smoothness obstruction, but the available lower and upper rates do not match. Simulations examine bias, coverage, interval length, and numerical sensitivity. In the simulated regime, logarithmic intervals overcover, whereas some smaller-cutoff settings undercover, emphasizing the distinction between asymptotic validity and finite-sample calibration. The results do not establish adaptation, a sharp smoothness threshold, or semiparametric efficiency.