Abstract
This paper mainly addresses maximum likelihood estimation for a response-selective stratified sampling scheme, the basic stratified sampling (BSS), in which the maximum subsample size in each stratum is fixed. We derived the complete-data likelihood for BSS, and extended it as a full-data likelihood by incorporating incomplete data. We also similarly extended the empirical proportion likelihood approach for consistent and efficient estimation. We conducted a simulation study to compare these two new approaches with the existing estimation methods in BSS. Our result indicates that they perform as well as the standard full information likelihood approach. Methods were illustrated using a growth model for fish size at age, including between-individual variability. One of our major conclusions is that the fully observed BSS data, the partially observed data used for stratification, and the sampling strategy are all important in constructing a consistent and efficient estimator.
Highlights
In stratified random sampling (SRS), the population or a random sample of the population is partitioned into relatively homogeneous subgroups, or strata, and random samples are taken independently in each stratum for full observation
Practical implementations of SRS frequently fall into two categories as classified by [1]: 1) basic stratified sampling (BSS) where the maximum second phase subsample size (BSS1) or subsampling fraction (BSS2) in each stratum is prefixed, and 2) variable probability sampling (VPS) in which sequential units are independently generated from a model and classified into strata where they are selected for full observation with pre-specified probabilities. [2] classified BSS2 as VPS, and all the inference methods for VPS are suitable for BSS2
For the linear model with BI variation (22), Tables 1-3 indicate that the full information, full-data and empirical proportion (EP) likelihood approaches have quite close performance, and in general they perform substantially better than all the other approaches in terms of Relative biases (RBias), relative standard errors (RSE) and relative square root mean squared errors (RRMSE) for all estimated parameters
Summary
In stratified random sampling (SRS), the population or a random sample of the population is partitioned into relatively homogeneous subgroups, or strata, and random samples are taken independently in each stratum for full observation. For the examples in the simulation studies and real data analysis, we use a Gamma distribution for age so that our comparison among various inference approaches is less influenced by numerical issues related to integrating over a complicated covariate distribution This is the motivation of this paper. ∑ ∑ ( ) ( ) lw θ = H Nh nh ln f n =h 1= h i 1 yi , xi | θ This weighted log-pseudo-likelihood (7) may provide an unbiased parameter estimating equation, the HT approach is known to be inefficient, and can be seriously so in some situations such as when the sample unit values are not approximately proportional to the inclusion probabilities
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