Estimating order scale parameters of two scale mixture of exponential distributions

Estimation of the ordered scale parameter of a two scale mixture of the exponential distribution is considered under Stein loss and symmetric loss. Under certain conditions, we prove that the inadmissibility equivariant estimator exhibits several improved estimators. Consequently, we propose various estimators that dominate the best affine equivariant estimators (BAEE). Also, we propose a class of estimators that dominates BAEE. We have proved that the boundary estimator of this class is a generalized Bayes estimator. The results are applied to the multivariate Lomax distribution and the Exponential Inverse Gaussian (E-IG) distribution. Consequently, we have obtained improved estimators for the ordered scale parameters of two multivariate Lomax distributions and the exponential inverse Gaussian distribution. For each case, we have conducted a simulation study to compare the risk performance of the improved estimators.

💡 Research Summary

The paper addresses the problem of estimating ordered scale parameters (σ₁ ≤ σ₂) for a two‑component scale‑mixture of exponential distributions under two loss functions: a symmetric loss L₁(δ,σ)=δ/σ+σ/δ−2 and Stein’s loss L₂(δ,σ)=δ/σ−log(δ/σ)−1. The model assumes a latent mixing variable τ>0 with an arbitrary distribution H(·). Conditional on τ, the two independent samples X₁,…,X_{p₁}∼Exp(μ₁,σ₁/τ) and Y₁,…,Y_{p₂}∼Exp(μ₂,σ₂/τ) are observed. Sufficient statistics (S₁,X_{(1)}) and (S₂,Y_{(1)}) are derived, where S₁=∑(X_i−X_{(1)}) and similarly for S₂. Conditional distributions are Gamma(p_i−1,σ_i/τ) for S_i and Exponential(μ_i,σ_i/(p_iτ)) for the minima.

First, the authors obtain the best affine equivariant estimators (BAEE) for each σ_i under both loss functions by exploiting the affine group G_{a,b}. Lemma 1 shows that the BAEE have the simple form δ_{1i}=c_i S_i with c_i=E(τ)(p_i−1)(p_i−2)E(1/τ) for L₁, and δ_{2i}=d_i S_i with d_i=1/(p_i−1)E(1/τ) for L₂. These estimators ignore the order restriction and are therefore inadmissible.

To incorporate the order constraint, the ratio W=S₂/S₁ is introduced and a broad class of estimators δ_{φ}=φ(W) S₁ (or φ(W) S₂) is considered. Theorem 1 provides explicit dominating functions φ_{11}(W) and φ_{12}(W) such that the truncated estimators min{φ(W),φ_{11}(W)} S₁ and min{φ(W),φ_{12}(W)} S₁ have risk never larger than the BAEE under L₁ and L₂ respectively. These φ’s involve the moments E(τ) and E(1/τ) together with the sample sizes p₁, p₂.

A more systematic improvement is achieved via the Integral Expression of Risk Difference (IERD) technique. Theorem 2 gives sufficient conditions on φ: it must be non‑decreasing, converge to a specific limit, and dominate a lower bound φ* expressed through incomplete beta functions B(·). When these conditions hold, the estimator δ_{φ} uniformly dominates the BAEE for the corresponding loss. The lower bound φ* is shown to be a generalized Bayes estimator under the non‑informative prior π(σ₁,σ₂,μ₁,μ₂)∝1/(σ₁σ₂) I(σ₁≤σ₂). The authors verify that the Bayes estimator under both L₁ and L₂ coincides with φ*.

Analogous results are derived for σ₂, including a double‑shrinkage estimator of the form α S₂+β S₁, which further reduces risk.

The theoretical developments are applied to two important multivariate models:

1. Multivariate Lomax distribution – obtained by taking τ∼Gamma(shape,scale). Closed‑form expressions for E(τ) and E(1/τ) are available, allowing direct implementation of the proposed estimators.

2. Exponential‑Inverse‑Gaussian (E‑IG) distribution – obtained by letting τ follow an inverse Gaussian law. Again, the required moments are analytically tractable.

For each application, extensive Monte‑Carlo simulations are performed across a range of sample sizes (p₁, p₂) and mixing‑distribution parameters. The relative risk improvement (RRI) of the proposed estimators over the BAEE (and over the unrestricted MLE) is reported. Results consistently show risk reductions of 5–20 %, with the greatest gains when the mixing distribution is highly variable or when sample sizes are small.

In conclusion, the paper demonstrates that the BAEE, though optimal within the affine equivariant class, is inadmissible under order restrictions. By exploiting the ratio of sufficient statistics, applying IERD, and invoking generalized Bayes arguments, the authors construct a suite of estimators that uniformly dominate the BAEE for both symmetric and Stein losses. The methodology is versatile, extending naturally to multivariate Lomax and E‑IG families, and is supported by thorough simulation evidence. This work provides a comprehensive framework for ordered scale‑parameter estimation in mixture‑exponential settings, bridging frequentist risk‑optimality and Bayesian admissibility.