Yves Tillé

A Direct Bootstrap Method for Complex Sampling Designs From a Finite Population

In complex designs, classical bootstrap methods result in a biased variance estimator when the sampling design is not taken into account. Resampled units are usually rescaled or weighted in order to achieve unbiasedness in the linear case. In the present article, we propose novel resampling methods that may be directly applied to variance estimation. These methods consist of selecting subsamples under a completely different sampling scheme from that which generated the original sample, which is composed of several sampling designs. In particular, a portion of the subsampled units is selected without replacement, while another is selected with replacement, thereby adjusting for the finite population setting. We show that these bootstrap estimators directly and precisely reproduce unbiased estimators of the variance in the linear case in a time-efficient manner, and eliminate the need for classical adjustment methods such as rescaling, correction factors, or artificial populations. Moreover, we show via simulation studies that our method is at least as efficient as those currently existing, which call for additional adjustment. This methodology can be applied to classical sampling designs, including simple random sampling with and without replacement, Poisson sampling, and unequal probability sampling with and without replacement.

New tests for departures from random behavior in spatial memory experiments

We present statistical tests for departures from random expectation in spatial memory tasks. We consider two common protocols for spatial memory experiments. In the first one, subjects are allowed to search a fixed number of sites. In the second protocol, subjects are allowed to search until they achieve a fixed number of successes. In either of these protocols, the subjects involved may or may not revisit sites that have been previously searched or exploited. This yields four situations to consider: fixed number of sites searched or fixed number of successes, with or without revisits. We derive analytical expressions for the probability mass functions, expectations, and variances associated with each type of null hypothesis. We present three statistical tests of these hypotheses: the Kolmogorov-Smirnov test, the ordinary sign test, and theZ test. We use our results to demonstrate a priori calculation of sample sizes and statistical power and to consider a mixed model of sampling with and without replacement.

Towards optimal regression estimation in sample surveys

The Montanari (1987) regression estimator is optimal when the population regression coefficients are known. When the coefficients are estimated, the Montanari estimator is not optimal and can be extremely volatile. Using design-based arguments, this paper proposes a simpler and better alternative to the Montanari estimator that is also optimal when the population regression coefficients are known. Moreover, it can be easily implemented as it involves standard weighted least squares. The estimator is applicable under single stage stratified sampling with unequal probabilities within each stratum.

Some remarks on Unequal Probability Sampling Designs Without Replacement

A set of demands is presented which should be satisfied by a good unequal-probability sampling method without replacement. First it is shown that some of these demands are contradictory. ln particular, it is shown that a sequential algorithm that ensures strictly positive joint inclusion probabilities does not exist; not does there exist a sequential procedure that yields a result which is not dependent on the order of units in the data file. Next, a way is discussed to build approximations of the joint-inclusion probabilities for sampling design implemented by means of a sequential algorithm and preceded by a random sort of the data file. An original approximation is proposed which resorts to a method of adjustment to marginal totals. Finally, several approximations are compared to systematic sampling, and to Sunters method.

Selection of several unequal probability samples from the same population

The well-known problem of unequal probability sampling of fixed size from a finite population can be generalised in a partitioning problem of a population into subsets, having unequal inlcusion probabilities in each subset. A simple algorithm that allows one to solve this problem is presented. The random partition of a population can be used to easily solve questions related to sample coordination in repeated sample surveys: the management of the overlap and the rotation. An application is developed where the maximal overlap of two samples selected with unequal probabilities is reached. Nevertheless, this method has an inconvenience: the samples must be selected at the same time. In order to overcome this difficulty, another method based on the multi-phase sampling method is also proposed.

Histogram-Based Interpolation of the Lorenz Curve and Gini Index for Grouped Data

In grouped data, the estimation of the Lorenz curve without taking into account the within-class variability leads to an overestimation of the curve and an underestimation of the Gini index. We propose a new strictly convex estimator of the Lorenz curve derived from a linear interpolation-based approximation of the cumulative distribution function. Integrating the Lorenz curve, a correction can be derived for the Gini index that takes the intraclass variability into account.

Sampling with unequal probabilities

Publisher Summary Since the mid-1950s, there has been a well-developed theory of sample survey design inference embracing complex designs with stratification and unequal probabilities. Unequal probability sampling was first suggested by Hansen and Hurwitz in the context of sampling with replacement. Narain, Horvitz, and Thompson developed the corresponding theory for sampling without replacement. A large part of survey-sampling literature is devoted to unequal probability sampling, and more than 50 sampling algorithms have been proposed. Multistage sampling is one of the applications of unequal probability sampling design where the selection of primary units within strata may be done with unequal probability. For example, self-weighted two-stage sampling is often used to select primary sampling units with probabilities that are proportional to the number of secondary sampling units within the primary units; a simple random sample is selected within each primary unit.Since the mid 1950s, there has been a well-developed theory of sample survey design inference embracing complex designs with stratification and unequal probabilities (Smith, 2001). Unequal probability sampling was first suggested by Hansen and Hurwitz (1943) in the context of sampling with replacement. Narain (1951), Horvitz and Thompson (1952) developed the corresponding theory for sampling without replacement. A large part of survey sampling literature is devoted to unequal probabilities sampling, and more than 50 sampling algorithms have been proposed.

Complex national sampling design for long-term monitoring of protected dry grasslands in Switzerland

We describe a probabilistic sampling design of circular permanent plots for the long-term monitoring of protected dry grasslands in Switzerland. The population under study is defined by the perimeter of a national inventory. The monitoring focus is on the species composition of the protected grassland vegetation and derived conservation values. Efficient trend estimations are required for the whole country and for some predefined target groups (six biogeographical regions and eleven vegetation types). The target groups are equally important regardless of their size. Consequently, intensified sampling of the less frequent groups is essential for sample efficiency. The prior information needed to draw a targeted sample is obtained from the sampling frame and external databases. The logistics and generalized delineation of the target population may pose further problems. Thus, investments in fieldwork and travel time should be well balanced by selecting a cluster sample. Second, any access problems in the field and non-target units in the sample should be compensated for by selecting reserve plots as they otherwise may considerably reduce the effective sample size. Finally, the design has to be flexible as the sampling frame may change over time and sampling intensity might have to be adjusted to redefined budgets or requirements. Likewise, the variables and biological items of interest may change. To fulfil all these constraints and to optimally use the available prior information, we propose a multi-stage self-weighted unequal probability sampling design. The design uses modern techniques such as: balanced sampling, spreading, stratified balancing, calibration, unequal probability sampling and power allocation. This sampling design meets the numerous requirements of this study and provides a very efficient estimator.

Systematic sampling is a minimum support design

In order to select a sample in a finite population of N units with given inclusion probabilities, it is possible to define a sampling design on at most N samples that have a positive probability of being selected. Designs defined on minimal sets of samples are called minimum support designs. It is shown that, for any vector of inclusion probabilities, systematic sampling always provides a minimum support design. This property makes it possible to extensively compute the sampling design and the joint inclusion probabilities. Random systematic sampling can be viewed as the random choice of a minimum support design. However, even if the population is randomly sorted, a simple example shows that some joint inclusion probabilities can be equal to zero. Another way of randomly selecting a minimum support design is proposed, in such a way that all the samples have a positive probability of being selected, and all the joint inclusion probabilities are positive.

Small area estimation by splitting the sampling weights

A new method is proposed for small area estimation. The prin- ciple is based upon the splitting of the sampling weights between the areas. A matrix of weights is defined. Each column of this matrix enables us to estimate the total of the variables of interest at the level of an area. This method automatically satisfies the coherence property between the local estimates and the overall estimate. Moreover, the local estimators are cal- ibrated on auxiliary information available at the level of the small areas. This methodology also enables the use of composite estimators that are weighted means between a direct estimator and a synthetic estimator. Once the weights are computed, the estimates can be easily computed for any variable of interest. A set of simulations shows the interest of the proposed