Innovative covariance-based framework: symmetry assessment and exponentiality testing under multiplicative distortion measurement Errors

This paper proposes a new covariance-based measure to evaluate the symmetry of continuous random variables, focusing on covariance between the square root of density and distribution functions, suitable for uniform distributions. It applies the measure to test exponential distribution adherence and, in undistorted scenarios, introduces two nonparametric exponential parameter estimators, one achieving asymptotic efficiency akin to maximum likelihood estimators. The paper derives asymptotic properties of the measure, uses empirical likelihood for inference, and extends analysis to a multiplicative distortion model with an unknown smoothing function. Through conditional mean calibration, calibrated variables are obtained, and estimators for exponential parameters and the measure are constructed, maintaining asymptotic efficiency for the estimated measure. Empirical likelihood confidence intervals and tests for exponentiality under distortion are developed. Monte Carlo simulations and a real dataset validate the proposed estimators and test statistics.