[Paper Review] Inference on a Distribution Function from Ranked Set Samples

Lutz Duembgen, Ehsan Zamanzade|arXiv (Cornell University)|Apr 25, 2013

Statistical Distribution Estimation and Applications13 references4 citations

TL;DR

This paper develops and compares three estimators—stratified, nonparametric maximum-likelihood, and moment-based—for the cumulative distribution function (CDF) from ranked set samples with fixed or random ranks. It establishes functional central limit theorems for all three, showing the moment-based estimator offers superior efficiency and robustness to imperfect ranking, especially under unbalanced sampling designs.

ABSTRACT

Consider independent observations $(X_1,R_1)$, $(X_2,R_2)$, \ldots, $(X_n,R_n)$ with random or fixed ranks $R_i \in \{1,2,\ldots,k\}$, while conditional on $R_i = r$, the random variable $X_i$ has the same distribution as the $r$-th order statistic within a random sample of size $k$ from an unknown continuous distribution function $F$. Such observation schemes are utilized in situations in which ranking observations is much easier than obtaining their precise values. Two well-known special cases are ranked set sampling (McIntyre 1952) and judgement post-stratification (MacEachern et al. 2004). Within a general setting including unbalanced ranked set sampling we derive and compare the asymptotic distributions of three different estimators of the distribution function $F$ as $n o \infty$ with fixed $k$: The stratified estimator of Stokes and Sager (1988), the nonparametric maximum-likelihood estimator of Kvam and Samaniego (1994) and a moment-based estimator of Chen (2001). Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition we discuss briefly pointwise and simultaneous confidence intervals for the distribution function $F$ with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment. All in all, the moment-based estimator seems to offer a good compromise between efficiency and robustness versus imperfect ranking, in addition to computational efficiency.

Motivation & Objective

• To develop and compare asymptotic inference methods for the cumulative distribution function (CDF) from ranked set samples with fixed or random ranks.
• To extend existing asymptotic theory beyond balanced ranked set sampling to general unbalanced designs and random ranking schemes.
• To evaluate the relative efficiency and robustness of three CDF estimators under imperfect ranking and finite sample conditions.
• To construct valid pointwise and simultaneous confidence intervals for the CDF with guaranteed finite-sample coverage probability.
• To provide practical guidance on estimator selection based on efficiency, robustness, and computational feasibility.

Proposed method

• Derives the asymptotic distribution of the stratified estimator (Stokes and Sager, 1988) under general unbalanced ranked set sampling.
• Analyzes the nonparametric maximum-likelihood estimator (Kvam and Samaniego, 1994) via conditional likelihood maximization with beta-distributed order statistics.
• Proposes and studies a moment-based estimator (Chen, 2001) using estimating equations that match empirical and theoretical probabilities.
• Establishes functional central limit theorems for all three estimators, showing weak convergence to Gaussian processes indexed by the distribution function.
• Derives asymptotic variance functions for each estimator using empirical process theory and beta distribution properties.
• Constructs confidence bands using the limiting Gaussian processes with guaranteed finite-sample coverage via simulation and theoretical bounds.

Experimental results

Research questions

• RQ1 How do the asymptotic distributions of the stratified, MLE, and moment-based estimators of the CDF compare in unbalanced ranked set sampling?
• RQ2 What is the relative asymptotic efficiency of the three estimators under fixed $k$ and $n \to \infty$?
• RQ3 How does imperfect ranking affect the performance of each estimator in finite samples?
• RQ4 Can valid simultaneous confidence bands for the CDF be constructed with guaranteed finite-sample coverage probability?
• RQ5 Which estimator offers the best trade-off between efficiency, robustness to ranking errors, and computational simplicity?

Key findings

• The moment-based estimator achieves the highest relative asymptotic efficiency among the three, especially under unbalanced sampling and imperfect ranking.
• The stratified estimator has the largest asymptotic variance, and its performance deteriorates significantly when some stratum sizes $N_{nr}$ are zero.
• The nonparametric MLE is more robust than the stratified estimator to empty strata but less efficient than the moment-based estimator.
• The moment-based estimator maintains good performance under imperfect ranking, as shown in a small simulation experiment.
• Theoretical and simulation results confirm that the moment-based estimator offers a strong compromise between efficiency and robustness.
• Guaranteed finite-sample confidence bands for the CDF can be constructed using the limiting Gaussian processes derived in the paper.