On uniformly consistent estimation of parametric lorenz curves
Let be n non-negative random variables with a common Lorenz curve L(p, θ) and let be an estimator of θ. Here we establish that if L(p, θ) is continuous in θ and Tn is strongly consistent for θ, then L(p,Tn) is uniformly strongly consistent for L(p, θ). This result supplements the uniform strong consistency of empirical Lorenz curves, proved by Goldie (1977) and generalizes the uniform strong consistency of maximum likelihood estimator of Lorenz curve of each of Pareto, exponential and lognormal distributions, established by Moothathu (1985c). To illustrate the main result we derive a Uniformly strongly consistent estimator of a Lorenz curve; based on random variable which are neither independent nor identically distributed.