Order preserving property of moment estimators

Balakrishnan and Mi [1] considered order preserving property of maximum likelihood estimators. In this paper there are given conditions under which the moment estimators have the property of preserving stochastic orders. There is considered property of preserving for usual stochastic order as well as for likelihood ratio order. Mainly, sufficient conditions are given for one parameter family of distributions and also for exponential family, location family and scale family.

💡 Research Summary

The paper investigates when moment estimators inherit the monotonicity properties of the underlying statistical model with respect to two stochastic orders: the usual stochastic order (≤_st) and the stronger likelihood‑ratio order (≤_lr). After recalling the definitions of these orders and basic lemmas stating that increasing transformations and sums preserve them, the author introduces the concept of total positivity of order two (TP₂). A density f(x;θ) is TP₂ precisely when the mixed second derivative ∂²/∂x∂θ log f(x;θ) is non‑negative; this condition guarantees that the family {f(·;θ)} is monotone in the likelihood‑ratio sense: X(θ₁) ≤_lr X(θ₂) for θ₁<θ₂.

The central object is a moment estimator of the form (\hat\theta=m^{-1}(\bar g)) where (\bar g=n^{-1}\sum_{i=1}^n g(X_i)) and (m(\theta)=\int g(x)f(x;\theta)dx). The paper first treats the simplest case g(x)=x (the sample mean). Theorem 1 shows that if f is TP₂ and log‑concave in x, then the estimator based on the sample mean is increasing in θ with respect to the likelihood‑ratio order. The proof uses the variation‑diminishing property of TP₂ kernels to establish that m(θ) is monotone, and then Lemma 3(b) to lift the LR‑order from the sample mean to the estimator via the monotone inverse m⁻¹. Theorem 2 weakens the log‑concavity assumption, proving only stochastic‑order monotonicity under TP₂.

Corollary 1 extends the result to any increasing function g for which the integral m(θ) exists, while Corollary 2 gives a more technical set of conditions (strictly increasing, convex, differentiable g; log‑concave inverse derivative; decreasing log‑concave density) that guarantee LR‑order monotonicity.

The paper then focuses on one‑parameter exponential families with densities (f(x;\theta)=h(x)c(\theta)\exp{\eta(\theta)T(x)}). When both η(θ) and T(x) are monotone (both increasing or both decreasing), the family is TP₂ and m(θ)=Eθ