Constrained estimation using judgment post-stratification

Abstract

Convexity, Maximum likelihood, Ranked-set sampling, Stratified sampling,

Constrained nonparametric estimation of the mean and the CDF using ranked-set sampling with a covariate

“…Consider the constraints on p 2 when the overall CDF value is r = 0.5 and m = 3. Frey and Ozturk (2010) showed that for any fixed value of r , B(r ; 1, 3) ≤ p 2 ≤ B(r ; 3, 1), where B(·; α, β) is the CDF for a Beta(α, β) distribution. With r = 0.5, this leads to constraints 1/8 ≤ p 2 ≤ 7/8.…”

“…Figure 1 shows the spaces K c r and K s r for r = 0.2, 0.4, 0.6, and 0.8 and m = 3. It also, for comparison purposes, shows the space K r of possible values for ( p 1 , p 2 ) under thep = r assumption and the constraints obtained by Frey and Ozturk (2010). Recall that p i ≡ F [i] (y) so that r =p = F(y) is the overall CDF value.…”

“…1, the outer-most set of bounds shows the Frey and Ozturk (2010) constraints, and the inner-most set of bounds is for rankings done according to a covariate that satisfies Assumption 1. We see from the figure that all of the sets are convex and that the new sets of bounds are substantially tighter than the bounds obtained by Frey and Ozturk (2010). The dashed reference line in each plot is the line p 1 = p 2 .…”

“…The value p 3 ≡ F [3] (y) is determined by the constraint that r = 1 3 ( p 1 + p 2 + p 3 ). In each plot, the regions are, from largest to smallest, the Frey and Ozturk (2010) To obtain estimates, we follow the approach used by Frey and Ozturk (2010). Suppose that we want to estimate the vector (F [1] (y), .…”

“…Frey and Ozturk (2010) showed that the in-stratum CDFs for strata that arise from ranking information can be no more extreme, in a certain sense, than the CDFs for true order statistics from the overall distribution. In this paper, we show that if the rankings are done according to a covariate, then constraints tighter than those obtained by Frey and Ozturk (2010) must hold. We also show that under a mild stochastic ordering assumption, still tighter constraints must hold.…”