Review of the literature

Statistical information on land use and rural development can be derived with reference to different observation units, such as households, agricultural holdings, and parcels of land or points. Furthermore, agricultural and rural statistics can be derived through sampling and non-sampling methods. Non-sampling methods mainly include agricultural census and the use of administrative data collected for different purposes. Sampling methods can be based upon a list frame or an area frame, or rely on the combined use of different sampling frames. The use of administrative sources in producing agricultural and rural statistics has been discussed by Carfagna and Carfagna (2010). The main advantages and drawbacks deriving from using list and area frames in agricultural sample surveys have been analyzed by Carfagna and Carfagna (2010) and Cotter et al (2010).

Data collected from sample surveys can be used to derive reliable direct estimates for large areas, making use of auxiliary information from agricultural census, administrative sources, or remotely sensed data. Auxiliary information can be used before the sample selection, in designing the survey, as well as after the sample selection, in the estimation procedure (Bee et al 2010). The ex-ante use of auxiliary information mainly concerns the construction of optimal sample stratification and the definition of a balanced sampling design, which allows for obtaining robust estimators of the totals (Bee et al 2010).

Many different estimators can be applied in different practical circumstances. A very important research question is represented by the choice of the more appropriate estimator for the particular case of study under investigation. Besides, these different estimators should be compared. This issue is not largely analyzed in the specialist literature. In the following, we will only provide some ideas and contributes that can be applied to this context.

Auxiliary information may be incorporated in the estimation procedure by using the Generalized REGression (GREG) estimator, which postulates a regression relationship between the study variable and covariates. The robustness of GREG estimators has been investigated by Beaumont and Alavi (2004). The calibration approach has been contrasted with GREG estimation, as an alternative way to take auxiliary information into account, by Särndal (2007). Calibration estimators make use of calibration weights, which satisfy a set of calibration equations (Deville and Särndal 1992). Calibration equations force the sample sum of the weighted auxiliary variables to match the known population total. Calibration weights generally result in design consistent estimators, i.e. estimators that approach the finite population parameter as the sample size increases. The GREG estimator has been derived as a first approximation to the family of calibration estimators by Deville and Särndal (1992). The use of calibration and regression estimators to combine information from ground and remotely sensed data in agricultural surveys has been discussed by Gallego et al (2010).

The increasing demand for statistical information on land use and rural development on a small scale motivates the use of small area estimation methods (see next Section 6). Dealing with small areas or domains, the area specific sample size could not be large enough to support reliable direct estimates. By contrast, indirect small area estimation enables to produce reliable estimates of characteristics of interest for areas or domains for which only small sample or no samples are available. While direct estimators use only data from the area of interest, small area indirect estimators are based on either implicit or explicit models that relate small areas in such a way that information from other areas contribute to the estimation in a particular small area. A review of small area methods used in the context of agricultural surveys can be found in Rao (2003). Indirect estimation, based on implicit models, includes synthetic and composite estimators. Recent developments in small area estimation include Empirical Best Linear Unbiased Predictors (EBLUP), Empirical Bayes (EB) and Hierarchical Bayes (HB) estimation. These approaches use explicit models, mainly categorized as area level and unit level models, to delineate the relationship among small areas, and show advantages over traditional indirect estimators (see, e.g., Rao 2003, Pfeffermann 2013).

The use of model-based small area estimators raises the question of the robustness of the inference to possible model misspecifications. Furthermore, when a reliable direct estimate is available for an aggregate of small areas, the model-based small area estimates need to be consistent with the direct estimate for the larger area. This condition is crucial when the direct estimate for the larger area has an official nature.

A number of benchmarking procedures intended to ensure the consistency between model-based small area estimates and direct estimates for large areas have been developed (see, e.g., Wang et al 2008, Pfeffermann 2013). The benchmarking procedures make robust the inference forcing the model-based small area predictors to agree with the design-based estimator for an aggregate of the areas (Pfeffermann 2013). Denoted by θ_i a parameter of interest in area i, for i=1,…,m, and assuming that the larger area contains all the m small areas under investigation, a general formulation for the benchmarking equation is given by:

A commonly used benchmarking approach is the ratio or pro-rata adjustment (Pfeffermann 2013) expressed by:

The above described predictors are internally benchmarked. Externally benchmarked predictors can be also derived through an a-posteriori adjustment of model-based predictors. A recent review of these approaches can be found in Wang et al (2008). Additional benchmarking procedures, developed in both a Bayesian and a frequentist framework, are described in Pfeffermann (2013).

Imposing the benchmarking restriction implies the possibility that the small area model is misspecified and the predictors are biased (Wang et al 2008). Benchmarking procedures thus contribute to improve the reliability of small area estimates. The achievement of this objective is essential to ensure higher quality and coverage of agricultural and rural statistics.

The main advantages and drawbacks of the methods described in this topic are summarized in the following Table 5.1.

The term small area generally refers to a small geographical area or a spatial population unit for which reliable statistics of interest cannot be produced due to certain limitations of the available data. For instance, small areas include small geographical regions like county, municipality or administrative division; domains or subpopulations, like a particular economic activity or a subgroup of people obtained by cross-classification of demographic characteristics, are called small areas if the domain-specific sample size is small. SAE is a research topic of great importance because of the growing demand for reliable small area statistics even when only very small samples are available for these areas. The problem of SAE is twofold. The first issue is represented by how to produce reliable estimates of characteristics of interest for small areas or domains, based on very small samples taken from these areas. The second issue is how to assess the estimation error of these estimates.

In the context of agriculture, the term small area usually refers to crop areas and crop yields estimates at small geographical area level. Agriculture statistics are generally obtained through sample surveys where the sample sizes are chosen to provide reliable estimators for large areas. A limitation of the available data in the target small areas seriously affects the precision of estimates obtained from area-specific direct estimators.

When auxiliary information is available, the design-based regression estimator is a classical technique used to improve the precision of a direct estimator. This technique has been widely applied to improve the efficiency of crop area estimates (Flores and Martinez 2000), where the used auxiliary information is given by satellite image data. Unfortunately, direct area-specific estimates may not provide acceptable precision at the SA level, in other terms they are expected to return undesirable large standard errors due to the small size or even zero of the sample in the area. Furthermore, when there are no sample observations in some of the relevant small domains, the direct estimators cannot even be calculated.

In order to increase precision of area-specific direct estimators have been developed various types of estimators that combine both the survey data for the target small areas and auxiliary information from sources outside the survey, such as data from a recent census of agriculture, remote sensing satellite data and administrative records. Such estimators, referred as indirect estimators, are based on models (implicit or explicit) that provide a link to related small areas through auxiliary data, in order to borrow information from the related small areas and thus increase the effective sample size. Torabi and Rao (2008) derived the model mean squared error of a GREG estimator and two-level model-assisted new GREG estimator of a small area mean. They show that due to borrowing strength from related small areas, estimator based on explicit model exhibits significantly better performance relative to the GREG and the new GREG estimators.

Many contributions have been introduced in literature on the topic of SAE. In particular, the paper of Ghosh and Rao (1994), Rao (2002, 2003), and Pfeffermann (2002, 2013) have highlighted the main theories on which the practical use of small area estimator is based on.

Section 6.2 is devoted to a description of the models for small area estimation. Sections 6.3 and 6.4 will contain a review of the foremost SA approaches, namely area level model and unit level model. In Section 6.5 we describe the spatial approach to SAE.

6.2 Models for small area estimation
Indirect small area estimates that make use of explicit models for taking into account specific variation between different areas have received a lot of attention for several reasons:

1. 
   The explicit models used are a special case of the linear mixed model and thus are very flexible for handling complex problems in SAE (Fay and Herriot 1979, Battese et al 1988).
   
2. 
   Models can be validate from sample data
   
3. 
   The MSE of the prediction is defined and estimated with respect to the model.
   

Let consider a partition of the population into D small sub-domains, with N_d the size of U_d. So we have:

Small area estimates that make use of explicit models are generally referred as Small Area Models (SAM) and they can be broadly classified into two types: area level models and unit level models (Rao 2003).

The area level model is constituted by two components: the linking model and the sampling model. In the linking model we assume that θ_d are related to x_d through a linear model as:

In the sampling model we suppose that the direct estimator or its transformation is available and defined as:

Combining equations (6.2) and (6.3), we obtain the Fay-Herriott model (1979):

To predict the random effects under the assumed mixed model (6.4), the Best Linear Unbiased Prediction (BLUP) approach is widely used. The estimator BLUP for under the model (6.4) is (Ghosh and Rao 1994):

where, and is the is the weighted least square estimator of defined as:

Under the EBLUP we use an estimate of as measure of variability of, where the expectation is with respect to the model (6.4). A lot of attention has been given to the estimation of the MSE. Unfortunately closed forms of exist only in particular cases. So, many scholars decided to give importance for identifying accurate approximations for it.

A valid approximation for, if D is large and under the assumption of normality of the errors and e is (Rao 2003):
(6.7)
where, and, with is the asymptotic variance of an estimator of.

Note that the main term in (6.7) shows that may be considerable smaller of if the weight is small or if is small if compared with . This means that the process of SAE depends in large part by the availability of good auxiliary information that contributes to reduce the model variance in respect to.

The assumption of known sampling variances is sometime a problem. You and Chapman (2006) consider the situation where the sampling error variances are individually estimated by direct estimators. A full hierarchical Bayes (HB) model is constructed for the direct survey estimators and the sampling error variances estimators.

Various methods other then EBLUP have been introduced in literature to estimate under the model (6.4). The most common are: Empirical Bayes (EB) and Hierarchical Bayes (HB). Good reviews of these methods have been written by Rao (2003), Ghosh and Rao (1994), and Pfeffermann (2002, 2013).

Area level models such as the Fay–Herriot model are widely used to obtain efficient model based estimators for small areas in agriculture statistics where a rich set of auxiliary variables is mainly available from remote sensing data. Benedetti and Filipponi (2010) addressed the problem of improving the land cover estimate at a small-area level related to the quality of the auxiliary information. Two different aspects associated with the quality of remote sensing satellite data have been considered:

1. 
   the location accuracy between the ground survey and satellite images;
   
2. 
   outliers and missing data in the satellite information.
   

The first problem is addressed by using the area-level model; the small-area direct estimator is related to area-specific auxiliary variables, that is, number of pixels classified in each crop type according to the satellite data in each small area. The missing data problem is addressed by using a multiple imputation.

Furthermore, various extensions have been proposed. Datta et al (1991) proposed the multivariate version of Fay-Herriot model that lead to more efficient estimators. Rao and Yu (1994) suggested an extension of (6.4) for the analysis of time series and cross-sectional data.

6.4 Unit level models
This approach is used when unit-level auxiliary data are available. The model assumes that the values of a study variable are related to unit-specific auxiliary data. More formally, if y is a continuous response variable a basic unit- level model relates the y_dk to the x_dk through a one-fold nested error regression model of the form:
(6.8)
where β is a fixed set of regression parameters, are random sample area effects, and are the residual errors. Furthermore, the are independent from the residual errors e_dks (Battese et al 1988).

If and are respectively, the population mean of a study variable and the population mean of the auxiliary variables for the area d, we assume that . Then the EBLUP estimates of is:

On the other hand, for small sample size and small the EBLUP tends towards the regression synthetic estimator.

Also, unit level models have been often used to obtain efficient model based estimators for small areas in agriculture statistics. Battese et al (1998) first used the unit-level model for the prediction of areas planted with corn and soybeans for 12 counties in north- central Iowa. The area of corn and soybeans in the 37 segments (PSUs) of the 12 counties was determined by interviewing farm operators. Each segment represents approximately 250 hectares. The sample information has been integrated with auxiliary data derived from satellite imagery readings. Crop areas for each segment are estimated from satellite images by counting the number of individual pixels in the satellite photographs. The model used assumes that there is a linear relationship between the survey and satellite data with county-specific random effects.

Very often in agricultural statistics y_dk are not continuous variables. For example, if the statistical units are sampled points, crop area estimates related to the point k in the small area d is given by where is an indicator variable with value if the point k is classified in crop type j and otherwise. In this situation the SA quantities of interest are usually proportions or counts. In such cases, the mixed linear models above described are no longer applicable. MacGibbon and Tomberlin (1989) defined a Generalized Linear Mixed Model (GLMM) for SAE that is widely used for this kind of problems.

Rashid and Nandram (1998) use a rank-based method to estimate the mean of county- level crop production data, when the data are not normally distributed. They use the nested error regression model, to borrow strength from other areas. Then, the estimates of the model parameters are used to construct a predictor of the population mean of a small area, and the mean squared error of the predictor. They applied the methodology using satellite and survey data obtained from 12 counties to estimate crops area.

Datta et al (1998) considered multivariate HB prediction of small area means using a multivariate nested error regression model. Advantages of using a multivariate approach over a univariate approach were demonstrated via simulations. Moreover, they analyse the corn and soybean data in Battese et al (1988) using the multivariate and the univariate models.

The spatial autocorrelation amongst neighboring areas or units can be introduced in the small-area estimation. A possible improvement in the EBLUP method can be achieved by including spatial structure in the random area effects (Cressie 1991).

An area-level model with conditional spatial dependence among random effects can be considered an extension of the Fay-Herriot model (6.4) where area-specific random effect take into account the interaction between neighbouring areas. There are two different approaches to describe the spatial information: conditional autoregressive model (CAR) and simultaneous autoregressive model (SAR).

Denote with N (d) the set of neighbourhood of the small area d, then for the random effect, it is possible to define a CAR spatial model as:

Now, define the Fay-Herriott model in matrix notation. The model (6.4) can be written as:

The Spatial BLUP estimator of is obtained from (6.5) as (Pratesi and Salvati 2008):

The spatial BLUP depend on the unknown variance and . Replacing these parameters with the correspondent estimators, we can define a two-stage estimator denoted as Spatial EBLUP (SEBLUP):

Assuming normality of the random effects, and ρ can be estimated both using ML and REML procedures. For further details about the estimation procedure see Pratesi and Salvati (2008).

It is worth noticing that different estimation methods, such as calibration, regression and SAE estimators, can lead different results both in the magnitudes of coefficients estimates and in the values of the related estimated standard error. The appropriate use of estimators primarily depends on the available data and on the objective of the analysis. For example, according to the availability of auxiliary information (i.e. remote sensed images), it can be used an approach rather than another. Furthermore, the researcher should pay attention to the definition of the used methods in order to compare statistical properties.

In fact, note that calibration and regression estimators are model-assisted methods, and the properties have to be assessed in terms of design. These estimators are design-unbiased. On the other hand, SAE are model-based techniques, and so the statistical properties should be analyzed with reference to the model. So, the analysts should interpret the comparisons with caution.

However, for possible ideas on comparison of SAE techniques the reader can see Section 5.

The main advantages and drawbacks of the methods described in this topic are summarized in the following Table 6.1.