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Two-stage stratified sampling is a complex design that involves nested sampling units and stratification. This complexity increases when the strata have too few sampled units for variance estimation, necessitating the use of collapsed strata, where multiple strata are combined to ensure an adequate sample size. When collapsing strata, two cases can be distinguished depending on whether a size variable associated with the variable of interest is available at the stratum level. • We present computer-implementable formulas for total, mean, and ratio estimators, along with their corresponding sampling variance estimators, for stratified two-stage simple random sampling without replacement, and we provide ready-to-use algorithms. • We introduce two methods for grouping strata: (1) a deterministic approach that uses stratum codes to define an ordinal variable, which orders the strata, and (2) a stochastic method that aims to minimize within-group inertia, which measures the heterogeneity within the newly formed groups of strata. • We emphasize that, unlike the correlation between a size variable and the variable of interest at the stratum level, the bias of the sampling variance estimator for the collapsed strata technique is not invariant to linear transformations. It follows that a high correlation does not ensure a low-bias estimator of the sampling variance.

Partitioning a set of elements into a given number of classes to find a globally optimal solution can be challenging due to the combinatorial explosion of the problem size. In the univariate case, where elements can be ordered, the number of partitions is significantly lower than in the multivariate case, and the problem is easier to handle. In this article, we focus on the univariate case and propose using complete enumeration to find a globally optimal solution. Although complete enumeration may also be computationally prohibitive as the number of elements and classes increases, it can be feasible in some situations. For such cases, we propose an algorithm that generates all contiguous partitions for a variable number of classes to be used with any objective function or set of constraints. • We compare exact problem sizes and approximate time complexities for multivariate and univariate partitioning. • We fill a technical gap in the literature by providing a valuable tool for researchers or engineers who need to exactly solve unusual univariate partitioning problems. • We use a convenient data structure for representing partitions of elements into classes and an iterative algorithm that simulates nested loops for any depth level, allowing for efficient generation of all possible contiguous partitions.

Assessing the abundance of biological populations is a central technical challenge in ecology, whether fundamental or applied. Although the question faced is simple, it is actually a complicated topic that has produced a vast array of methods. The first step is always sampling the geographical space, ideally by means of a randomized selection process. In that case, with a frequentist interpretation of probability, abundance can be estimated by taking into account randomization, or predicted conditionally on the sample at hand, by specifying a statistical model. This leads to a choice between so-called design-based estimation and model-based prediction. The goal of this methodological article is to contribute to the understanding of fundamental notions regarding these two statistical frameworks by the targeted audience, mainly quantitative ecologists. For this purpose, we illustrate the comparison between design-based estimation in the case of simple random sampling without replacement (SRSWOR) and model-based prediction. As an example, we model count data with a delta-lognormal distribution and rely on uniformly minimum-variance unbiased estimators (UMVUEs) for the prediction of abundance. We investigate the robustness of the predictor by contaminating the delta-lognormal distribution using actual count data. Data from a survey concerning wintering populations in France of two wader species, namely, northern lapwing (Vanellus vanellus) and European golden plover (Pluvialis apricaria) serve as illustrative examples. By means of Monte Carlo simulations, we highlight the lack of robustness of the predictor based on the delta-lognormal distributional model, in terms of both actual bias and precision. We organize the discussion around the illustrative examples in the context of the sampling design, the model and the data.

A variable taking positive values from a lognormal distribution and null values with a given probability is distributed according to the so-called delta-lognormal distribution. Two situations arise depending on whether the data are regarded as a random sample from an infinite population (superpopulation) or from a finite population, itself considered as a random sample from a superpopulation. In the case of an infinite population, estimating the mean can be accomplished using a uniformly minimum-variance unbiased estimator (UMVUE). Likewise, the prediction of the mean in the case of a finite population may be based on the UMVUE. In both cases, one expects a gain in precision when taking into account the shape of the distribution by relying on the UMVUE rather than on the sample mean, which is a nonparametric estimator (or predictor). 1. For the infinite population case, the relative efficiency results presented in this article are more complete and more accurate than those published so far. 2. The article fills a gap regarding the question of relative efficiency in the case of a finite population. 3. Calculations were performed using the exact expression for the variance of the UMVUE of the mean, expressed in terms of the confluent hypergeometric limit function.

Variance estimation in multistage sampling without replacement usually requires considerable computational effort. One option is to implement explicit formulas on a computer, at least for some specific sampling designs. This approach becomes quite cumbersome to handle beyond two stages, both from the formulation and computer implementation points of view. Another option is to provide a general method to compute variance estimates for any number of stages. Such an approach may involve data structures and estimators which are recursively defined. • The solution we present in this article is intended to be both general and computationally efficient by relying on a full-iterative implementation. • The definition of the estimators remains implicit as in the recursive approach, but is expressed in terms of recurrence relations translated into iterative algorithms. • These algorithms rely only on (dense) array data structures. Moreover, most of the necessary computer memory is only used during preliminary steps and is not required when performing the statistical calculations.

Reliable hunting bag statistics are a prerequisite for sustainable harvest management based on quantitative modeling. Estimating the total hunting bag for a given game species is faced with a multiplicity of error sources. Of particular concern is the nonresponse error. We consider that the major cause of nonresponse bias is when the reluctance to respond is related to a null harvest, which leads to a potentially important overestimation. For tackling the nonresponse bias issue, we advocate the repeated subsampling of nonrespondents, with a final phase of personal interview by phone, intended to be without nonresponse. When a 100% response rate is actually reached at the last phase, both total and sampling variance can be estimated without bias, whatever the response rates at the previous phases. The actual case of imperfect response at the last phase is studied using Monte Carlo simulations. For imperfect response at the last phase, we show that the estimators we advocate are biased downwards but that these bias remain very moderate if the response rate at the last phase is high enough, depending on the circumstances. Furthermore, we illustrate that increasing the number of phases improves the nonresponse bias attenuation. In case of a hunting bag collecting scheme prone to a high nonresponse rate, for obtaining a very satisfying nonresponse bias attenuation we advocate relying on the multiphase sampling strategy with two- or three-phases, and a response rate in the last phase of at least 90%.