Chantal Enguehard

Mathieu Mangeot

LMF for a selection of African Languages

Introduction

Electronic resources are scarce regarding less-resourced languages, so it is wise to take published dictionaries and convert them into a standard format usable by automated tools for natural language processing. We introduce the notion of lessresourced languages and then discuss the methodology of conversion that we have defned and implemented. The fourth part presents examples of conversion from the initial published format to the LMF format. The last part describes some diffculties encountered when representing certain information into LMF format.

Less-resourced languages

Definition

Although a precise inventory of all the existing natural languages is diffcult to achieve, there are currently approximately 6,000 languages spoken by humans, but only 200-300 are written. The transition from oral to written is complex and can not be limited to a simple transcription of sounds. It is necessary to conduct studies to achieve a linguistic description of the language in order to determine the transcription system to be used, to choose the most appropriate signs, then to write the spelling and syntactic rules, etc. Finally, languages are more or less wellresourced in terms of their support by tools: adapted keyboard, spell-checker, speech synthesis, machine translation, etc. A classifcation based on the estimation of the electronic resources and tools defnes three classes: well-resourced languages or τlanguages (e.g.: English, French), languages with moderately-resourced languages μ-languages (e.g.: Portuguese or Swedish), and less-resourced languages or πlanguages (e.g.: Bambara or Kanuri) [Berment].

The term less-resourced languages covers contrasting situations. We mention here three of them:

-it is the offcial language of a country as is Irish (or Gaelic Irish) in Ireland.

-it is a language without offcial status, that became a regional language: for example Basque and Breton in France; Ladin in Italy, Cornish in the United Kingdom.

-it is a national language of a country whose offcial language (used at school, or to write the laws) is different and often comes from a former colonizer state [Calvet]. This is the case of African languages on which we have worked and that are spoken in Niger, Mali and Burkina Faso. In these three countries, the offcial language is French.

Socio-economic context

We focus on fve African languages: Bambara, Kanuri, Hausa, Zarma and Tamajaq. They are less-resourced languages which socio-economic context is characterized by limited resources:

-there are few linguists having a less-resourced language as their mother tongue and exercising their professional activity on that language.

-the budget for the development of linguistic resources is low.

The governmental investment dedicated to language planning and, in particular, the development of electronic language resources are therefore very limited. The few studies that are conducted are characterized by a discontinuity in the time and spatial spread, which affects their sustainability and reuse [Streiter].

linguistic resources

Because of the scarcity of linguistic research, descriptions of these languages are incomplete and many questions remains. There are few dictionaries and they are generally not made by professional lexicographers. In addition, it is unusual to revise and make corrections on a published dictionary. This contrasts sharply with the published dictionaries of well-resourced languages like French or English. For example, Larousse or Harrap's are frms employing dozens of professionals who regularly review their dictionaries for several decades. Therefore, the dictionaries on which we worked contain numerous errors or incompleteness and are likely to evolve.

Building electronic lexical resources

Developing lexical resources ex nihilo requires large budgets, qualifed and available professionals, and the ability to lead a project for several years. These conditions can not be met in many countries. However, there are some published dictionaries (often bilingual) that can be exploited to build a frst version of an electronic resource in a few weeks and at low cost.

Dictionaries written by a single author

Many of the dictionaries written by a single author are bilingual because their author, originally from another language, aims to promote a language. Some were written by clerics in charge of people evangelism in colonized countries ("peres blancs" in Africa, Portuguese Jesuits in Asia). For example, we worked on the Bambara-French dictionary of Father Charles Bailleul [Bailleul]. There are also dictionaries developed by literate people, often linguists, wishing to serve their mother tongue. This is the case of elementary Hausa-French dictionary written by Abdou Minjinguini [Minjinguini] and the monolingual zarma dictionary written by Issouf Alzouma Oumarou [Oumarou].

Dictionaries built by projects

Dictionaries built by projects have several authors. The group of authors usually defnes some principles about the structure and the defnition of closed lists of values such as grammatical classes.

For example, we worked on dictionaries written in fve national languages West-Africa languages for the DiLAF project [Enguehard b].

From published dictionaries to LMF

Objectives

Our goal is to convert published dictionaries to make them available to the natural language processing (NLP) scientifc community. We choose LMF as fnal format because it is an ISO standard that favors the re-utilization of the data (this is a key-point when working on less-resourced languages as stated also by the RELISH project [Windhouwer]). The actual conversion of several dictionaries with thousands of entries constitutes an experimentation in order to test the operationality of this format and, optionally, to suggest improvements.

Methodology

Lexicographers and NLP experts must collaborate to convert a published dictionary into a structured electronic format. Thus we defne the tasks performed by each collaborator of such a project.

The conversion methodology we defned proceeds in several steps and requires successive transformations of the published dictionary to three different XML fles called copy, pivot and target formats. We also take into account the fact that lexicographers will revise and develop the produced resources.

The copy format is a structural copy of the published dictionary in a valid XML format. The transformation of the published dictionary to the copy format is performed by lexicographers 2 with the support of NLP experts. This step requires solving many problems, including the conversion of special characters to Unicode, 2 As each dictionary includes thousands of entries that would be tedious to manually tag, the conversion methodology includes the training of lexicographers in handling regular expressions so that they are able to automate themselves a part of this task.

the identifcation of each information part, the defnition of a set of markup tags and fnally the explicit tagging of information by placing tags [Enguehard b]. When a frst valid version of the copy format is available, various checks are performed using simple programs (counting the number of occurrences of each tag, checking the embeddedness of the markups, counting the number of closed lists values like parts of speech, etc.) and errors are reported to lexicographers who can make the corrections. The use of a CSS stylesheet associated with the display of the copy format also allows a browser to introduce facilities for valuable consultation: the relationship of synonymy and antonymy are represented by href links, which allows to easily control their consistency. Finally, the markup tag names are often expressed in the language of the dictionary which facilitates the appropriateness of the new format. The copy format does not alter the structure of the original format but improves readability by explicitly labelling every part of information.

The pivot format respects the normative core of LMF. It is obtained by structural changes of the copy format by applying an XSLT3 program. It may be necessary, for example, to change the place of a morphological information that was described in a semantic block. Most important changes may be necessary like the merge of two lexical entries, or the separation of a lexical entry with two semantic blocks into two lexical entries with a single semantic block. These treatments are performed by perl programs. Markup tag names are preserved from the copy format.

T h e target format follows the syntax of the informative part of the LMF standard. It is obtained by processing the pivot format with an XSLT program. As the pivot format meets the standard LMF format, the transformations from the pivot format towards the target format are limited to changing the name of an element, to add an additional level element with a child, and to convert a text node in an attribute value (see examples below). NLP experts develop conversion programs to process the transformation from copy format to pivot format, and from pivot format t o target format. When they conceive these programs they get the opportunity to detect new errors and inconsistencies that are reported for subsequent corrections.

Finally, the copy format dictionary is aimed to disappear in favour of the pivot format dictionary. It can be easily understood by lexicographers as they themselves chose the markup tag names. On the contrary, the target format dictionary is more diffcult to understand and to modify. The pivot format dictionary can be then uploaded on an online lexical resources management platform such as Jibiki [Mangeot c] in order to be readable and editable online by lexicographers who will be able to correct and enhance it directly (by adding new lexical entries, various information, translations, examples, etc..). It would then be easy to generate a new target format dictionary by processing again the adequate program on the pivot format dictionary.

Illustrations

Here are some examples of the implementation of the above methodology.

Definition of the copy format

The copy format defned by the lexicographers is close to the initial structure of the dictionary. This is to make explicit the nature of the information: defnition, lexical label, phonetic, synonyms, French equivalent, etc... Four of the fve dictionaries on which we have worked are intended for an audience of students in elementary courses. They are written primarily in the dictionary language, only the presence of one or several French equivalents for each input gives them a bilingual character. In these dictionaries, lexical categories are expressed in the language of the dictionary (see Table 1) and may vary according to the characteristics of the language. For example, in Kanuri, the class of each verb is specifed; in Zarma most verbs are defned as transitive or intransitive. The information contained in lexical entries are different according to the dictionary because of the language represented or the choices made by the dictionary authors. For instance, in the Tamajaq dictionary, an annexation state is indicated for some lexical entries, while this information does not exist in other languages, and phonetics is not specifed; in the Hausa dictionary colloquial expressions and many variants spelling are reported. Some examples of markup names are presented in Table 2. As the names of lexical categories were written in the language of each dictionary, it seemed natural to defne also markup names in the same language.

Language

Language

Tag name English equivalent

From original format to copy format

Two examples, one Kanuri, the other Tamajaq can illustrate this frst step.

The fgures 7.2 and 7.3 show two entries. In their original version, the special characters initially entered with an artisanal font 4 are not readable. In the Unicode version these special characters have been transformed to meet Unicode. The transformation of the Tamajaq special characters was specially complicated because the artisanal font that was used changes the glyphs of the letter 'p' (missing in the Tamajaq alphabet [République du Niger]) into the glyphs of the letter 'ǝ'. As the character 'p' is susceptible to occur in French equivalents, using a regular expression was essential for rapid replacement of 'p' in 'ǝ' only in parts written in Tamajaq.

From copy format to pivot format

The lexical entries in Figures 7.4 and 7.5 are automatically transformed into the pivot format in which appear explicitly a lemma block (with its spelling and pronunciation) and a semantic block. Adjustments can be carried out directly by the lexicographers in this format. In the Kanuri example, the synonym is designated by its phonetic and should be replaced by an article identifier.

. Tamajaq lexical entry aṣaruf in pivot format

In the Tamajaq example, and additional program automatically replaced the notation of the annexion state by a single vowel 'a' (meaning that the first vowel of the lemma must be replaced by 'a' to determine the annexion state) by the new form of the lemma aṣaruf.

From pivot format to target format

The conversion from the pivot format to the target format is automatically processed by XSLT programs. There is one program per dictionary.

Examples of Figure 7.6 and 7.7 are automatically transformed into target format that meets the syntax of the informative part of the LMF standard.

<LexicalEntry id="bannadu2"> <Lemma> <feat att="writtenForm" val="bannadu"/> <feat att="phoneticForm" val="bànnàd̠ ú"/> </Lemma> <feat att="partOfSpeech" val="kkye3."/> <Sense id="1"> <Equivalent> <feat att="language" val="fra"/> <feat att="writtenForm" val="éduquer(mal)"/> </Equivalent> <Definition> <feat att="writtenForm" val="Diwiro yal alamdu."/> </Definition> <Context> <TextRepresentation> <feat att="language" val="kau"/> <feat att="writtenForm" val="Gǝnanjun bannaje, ku tadanju rakce kǝlanju rojiwawo."/></TextRepresentation> <TextRepresentation> <feat att="language" val="fra"/> <feat att="writtenForm" val="Durant son jeune âge il l'a mal éduqué, aujourd'hui son fils n'arrive pas à se prendre en charge."/> </TextRepresentation> </Context> <SenseRelation targets="là̠ nd̠ ú"> <feat att="type" val="synonym"/> </SenseRelation> </Sense> </LexicalEntry> <LexicalEntry id="aṣaruf"> <Lemma> <feat att="writtenForm" val="aṣaruf"/> </Lemma> <feat att="partOfSpeech" val="sn. yy."/> < W o r d F o r m w r i t t e n F o r m = " a ṣ a u f " contextualVariation="annexion"/> < W o r d F o r m w r i t t e n F o r m = " a ṣ u r u f " grammaticalNumber="plural"/> <Equivalent> <feat att="language" val="ttq"/> <feat att="writtenForm" val="təsureft"/> </Equivalent> <Sense id="1"> <Equivalent> <feat att="language" val="fra"/> <feat att="writtenForm" val="pardon"/> </Equivalent> <Definition> <feat att="writtenForm" val="Agamay n əkənni dəffər erk arat."/> </Definition> <Context> <TextRepresentation> <feat att=language="tmh"/> <feat att="writtenForm" val="Musa as yəwat eməjinet dəffər ənki yəgmay daɣ-as aṣaruf."/> </TextRepresentation> </Context> <SenseRelation targets="tətubt"> <feat att="type" val="synonym"/> </SenseRelation> </Sense> </LexicalEntry>

Difficulties and proposals

The actual conversion of dictionaries into LMF format was the opportunity to meet the diffculties that we detail below. We also include solutions or elements of refection.

Data category

Language names and associated ISO 639-3 codes

It may be that difficult to identify a language and the associated ISO 639-3 code. Currently, ISO refers to the ethnologue 5 website that relies on a small number of studies mainly carried out by the staff of the Summer Institute of Linguistics (SIL). For example, the page dedicated to the language Tamahaq (for Tamajaq) includes only one bibliographic reference (an article on the music of the Tuareg) and cites no extract of text. However, a significant number of academic research have been conducted on this language and should be included in the bibliography. Thus, we suggest to enrich this languages catalog with some academic research articles.

Parts of speech list

We encountered parts of speech that are not included in the Parts of speech list of the ISO Data Category Registry (DCR) 6 . For example, the part of speech "ideophone" appears in the list of parts of speech in Hausa and Kanuri dictionaries. It is also used in Somali [Assowe]. For this latter language, there are also other parts of speech ("verbal affix", "focus marker", "sentence marker", etc.) that are missing in the ISO list. Zarma language does not distinguish between masculine and feminine, but distinguish between the definite and indefinite, etc. Thus, it appears necessary to enrich this parts of speech list or to allow a modular definition of this list with a sublist for each language.

LMF structure

Absence of macrostructure

The LMF standard represents a lexical resource in a unique fle (see Figure 7.10). Thus, it is not possible to represent complex dictionaries macrostructures and their links between volumes, such as Papillon pivot structure [Mangeot b] or PIVAX structure [Mangeot d]. In [Mangeot a], we defne a volume as an alphabetically ordered set of entries of the same language and a dictionary as a set of volumes. An entry of one volume can be linked to an entry of another volume. <LexicalResource> <GlobalInformation entrySource="Prolex"/> <Lexicon languageSymbol="fra"> <LexicalEntry partOfSpeech="noun">...</<LexicalEntry> <LexicalEntry partOfSpeech="noun">...</<LexicalEntry> ….. <LexicalEntry partOfSpeech="noun">...</LexicalEntry> </Lexicon> </LexicalResource> Figure 7.10. The beginning of the ProLex lexicon in LMF format [Maurel] 7.5.2.2 Objects of different nature at the same level

We think that, in order to be clearly understandable, an XML format should avoid to put objects of different nature at the same indentation level. The siblings of an element must be of the same nature. The LMF format does not respect this principle. In figure 7.10, the object <GlobalInformation> which is a metainformation about the lexicon is a sibling of the object <Lexicon> which is the resource itself.

Informative part of LMF -Free text

In XML, it is customary to include items from closed lists as attribute values, and frame the free texts by markup tags. This general principle is not respected in the informative part of LMF since all the information is stored in textual attributes. This choice has the effect of prohibiting the minimal information display via a browser for example.

-Examples of use representation

The dictionaries we worked on being bilingual, we faced the problem of representing information in different languages. In the general structure of an article, the lexical entry is clearly distinguished from its equivalents in other languages. In contrast, the representation of the equivalent of an example in the same form as the example itself, only by specifying another language, does not distinguish the example itself from its translation (see Figure 7.11). LMF offers the possibility to represent a translation with the "Multilingual notations extension" mechanism which makes the assumption that each equivalent exists in the dictionary of its language. But this is not always the case. For instance, the French equivalent of the Kanuri lexical entry bannadu i s éduquer (mal) which is not a French lexical entry. This phenomena is common when two languages concern cultures with differences in food, religion, cooking, dressing, etc. and because there are distinct linguistic structures 7 . Thus we decide to add a convention to read the occurrences of 7 Here are some examples issued from the Kanuri-French dictionary (with a translation in English): adinnamdu -aller vers l'est (to go towards east); albayi -pochette touareg (Tuareg bag); asar -troisième prière (third pray); bare -il ne faut pas (it is not allowed to); TestRepresentation: "When the language of the TextRepresentation is different from the language of the dictionary, the TextRepresentation is a translation of the TextRepresentation expressed in the language of the dictionary". <Context> <TextRepresentation> <feat att="language" val="kau"/> <feat att="writtenForm" val="Gǝnanjun bannaje, ku tadanju rakce kǝlanju rojiwawo."/> </TextRepresentation> <TextRepresentation> <feat att="language" val="fra"/> <feat att="writtenForm" val="Durant son jeune âge il l'a mal éduqué, aujourd'hui son fils n'arrive pas à se prendre en charge."/> </TextRepresentation> </Context>

Figure 7.11. A usage example and its equivalent in another language

Finally, an example may need to be explained, a simple translation being not enough to make it understandable. This is the case of many Bambara proverbs in the dictionary [Bailleul]. The author has often included a loan translation and an explanation giving the meaning of the proverb. We choose to simply represent such an explanation by using the "explanation" category (see an example in Figure 7.12). proverb: jalaki tɛ baji la literal translation: on ne condamne pas l'eau du fleuve 8 explanation: c'est de ta propre faute !9 <Context> <TextRepresentation> <feat att="language" val="bam"/> <feat att="writtenForm" val="jalaki tɛ baji la."/> </TextRepresentation> <TextRepresentation> <feat att="language" val="fra"/> <feat att="writtenForm" val="on ne condamne pas l'eau du fleuve."/> <feat att="explanation" val="c'est de ta propre faute !"/> </TextRepresentation> </Context> In English: River water can not be condemned.

Less-resourced languages are sometimes written for a short time and orthographic forms may vary. Also, some words have different spellings. They are neither synonyms nor geographical variants.

Adding annotations

Dictionaries on which we have worked are incomplete, often being the first version, they contain errors. In addition, their use by NLP researchers should raise new linguistic questions. Thus, it appears necessary to provide the ability to add annotations that could be collected later and addressed to the concerned linguists community. Annotations about inaccuracies of the dictionary can easily lead a linguist to make new corrections (e.g.: an entry marked as synonymous has three meanings, or a synonymous is missing in the dictionary). Annotations about more fundamental problems could feed the thoughts of the linguists community (e.g.: some words seem to hesitate between two lexical categories and are labeled with both).

Conclusion

The actual conversion of multiple published dictionaries into the LMF format has put into practice the DiLAF methodology of conversion we defned. This methodology is suitable for less-resourced languages and integrates the limitations in working time and fnancial resources. The fnal conversion into LMF allows to distinguish limitations regarding the completeness of the list of parts of speech and the consequences of structuring information in the form of attribute values. We have identifed some desirable developments for the future as an opportunity to enrich the list of parts of speech or the defnition of new markup tags to annotate dictionaries evolution. After practicing the LMF standard [LMF] for encoding our dictionaries, we think that LMF would gain in usability with a simple examplifed tutorial of how to encode an existing resource into LMF.

Bibliography

[Assowe] Assowe, Houssein Ahmed. 2011 Etude linguistique et approches de l'étiquetage morphosyntaxique du somali. Mémoire de Master 2. Université Michel de Montaigne Bordeaux 3.

[Bailleul] Bailleul, C. 1996. Dictionnaire bambara-français, édition 1996.

[Berment] Berment, V. 2004. Méthodes pour informatiser des langues et des groupes de langues peu dotées. Ph.D. thesis, Université Joseph Fourier.

Figure 7

 7 Figure 7.1. Conversion process

Figure 7 . 2 .Figure 7 . 3 .

 7273 Figure 7.2. Kanuri lexical entry bannadu (2) in initial published format then in Unicode format

Figure 7 . 5 .

 75 Figure 7.5. Tamajaq lexical entry aṣaruf in copy format

Figure 7 . 6 .

 76 Figure 7.6. Kanuri lexical entry bannadu (2) in pivot format

Figure

  Figure 7.7. Tamajaq lexical entry aṣaruf in pivot format

  Figure 7.8. Kanuri lexical entry bannadu (2) in target format

Figure 7 . 9 .

 79 Figure 7.9. Tamajaq lexical entry aṣaruf in target format

Figure 7 . 12 .

 712 Figure 7.12. Example of a proverb in Bambara and its representation -orthographic variants

  8

Table 1 .

 1 Examples of parts-of-speech

		Lexical Category	Abbreviation	English equivalent
	Hausa	kamantau	k.	adjective
	Kanuri	alama njoma	alnj	adjective
	Zarma	taka sifa	tsif.	adjective

Table 2 .

 2 Examples of tag names (copy and pivot formats)

	Hausa	ma_ana	definition
	Kanuri	maana	definition
	Tamajaq	almaɣna	definition
	Zarma	feeriji	definition
	Kanuri	maana_tiloa	synonym
	Tamajaq	anammelu	synonym
	Zarma	himacare	synonym
	Kanuri	bowodu	phonetic
	Zarma	ciiyaŋ	phonetic

  The same lexical entries, transformed into the copy format are shown in Figures 7.4 and 7.5.

	<article>
	<kalma lamba="2">bannadu</kalma>	lexical entry number 2
	<bowodu>[bànnàd̠ ú]</bowodu>	phonetic
	<naptu_curo_nahauyen>kkye3.</naptu_curo_nahauyen>	part of speech
	<maana>Diwiro yal alamdu.</maana>	definition
	<misal>	example
	<version tǝlam="kau">Gǝnanjun bannaje, ku tadanju	example in Kanuri
	rakce kǝlanju rojiwawo.</version>
	<version tǝlam="fa">Durant son jeune âge il l'a mal	e q u i v a l e n t o f t h e
	éduqué, aujourd'hui son fils n'arrive pas à se prendre en	example in French
	charge.</version>
	</misal>	synonym
	<maana_tiloa>là̠ nd̠ ú</maana_tiloa>	in French
	<kalakta tǝlam="fa">éduquer (mal)</kalakta>
	</article>
	Figure 7.4. Kanuri lexical entry bannadu (2) in copy format
	The lexicographer that transforms bannadu corrected the part of speech in kkey3
	(third class verb) and added a French equivalent of the example.
	<albab>	article
	<təzugəst>aṣaruf</təzugəst>	lexical entry
	<təmuṣt>sn. yy.</təmuṣt>	part of speech
	<təfaransist>pardon</təfaransist>	equivalent in French
	<almaɣna>Agamay n əkənni dəffər erk arat.</almaɣna>	definition
	<əlmisal>Musa as yəwat eməji-net dəffər ənki yəgmay daɣ-as	example in Tamajaq
	aṣaruf.</əlmisal>
	<anammelu>tətubt.</anammelu>	synonym
	<əsəfsəs>a.</əsəfsəs>	annexion state
	<igət>aṣuruf.</igət>	plural
	<tastəqW>təsureft.</tastəqW>	Tawəlləmət variant
	</albab>

eXtensible Stylesheet Language Transformations.

http://www.ethnologue.com 6 http://www.isocat.org/rest/dcs/119.html

In English: It is your fault!

Acknowledgements

The DiLAF project to convert published dictionaries to LMF format is funded by the Fonds Francophone des Inforoutes of the Organisation Internationale de la Francophonie.


Toky Randrianasolo
email: toky.randrianasolo@ifsttar.fr

Yves Tillé
email: yves.tille@unine.ch

Small area estimation by splitting the sampling weights

Keywords: AMS 2000 subject classifications: primary 62D05 Indirect estimator, matrix calibration, weights

A new method is proposed for small area estimation. The principle 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 calibrated 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 method.

Introduction

Three main families of estimators are usually used by statisticians to increase the quality of estimates at the level of small areas: direct estimators and indirect estimators based on implicit or explicit models (see, for instance, [START_REF] Rao | Small Area Estimation[END_REF].

The direct estimators are based on the survey data provided only by the considered area. When available, the auxiliary information only depends on the units of the area. The family of direct estimators gathers the estimator proposed by [START_REF] Horvitz | A generalization of sampling without replacement from a finite universe[END_REF] also called π-estimator, the generalized regression estimators [START_REF] Särndal | Model Assisted Survey Sampling[END_REF] and the calibration estimators [START_REF] Deville | Calibration estimators in survey sampling[END_REF]. The main problem with this family of estimators is the increasing variance when the area size decreases. A wise choice of auxiliary information can reduce the variance.

The indirect estimators can depend on all the sampled units for the estimation of a particular area. These estimators are based on the notion of deriving strength from space because a unit of a given area can help for the estimation of any area. Two sub-families of estimators are distinguished: synthetic and composite estimators.

According to [START_REF] Gonzalez | Use and Evaluation of Synthetic Estimates[END_REF], "an unbiased estimate is obtained from a sample survey for a large area; when this estimate is used to derive estimates for sub-areas on the assumption that the small areas have the same characteristics as the larger area, we identify these estimates as synthetic estimates". Practically, these estimators are based on the hypothesis of the equality of a parameter within the areas and in the overall population. The synthetic estimator can thus be a prediction by means of a linear model of a total based upon the assumption that the regression coefficient does not vary from an area to another. These estimators generally have a low variance since they depend on all the observations, i.e. the size of the sample is thus large. Unfortunately, these estimators can miss a specificity of a given area. The composite estimators are weighted means of a direct estimator and a synthetic one. For each area, the weight of the mean can be chosen by minimizing the mean squared error.

The indirect estimators built from explicit modeling are based on linear mixed models, generalized linear mixed models and Bayesian procedures which derive Best Linear Unbiased predictors (blup), Empirical Best Linear Unbiased Predictors (Eblup) and empirical Bayes estimators (see, among others, [START_REF] Fuller | Transformation for estimation of linear models with nested error structure[END_REF]Prasad andRao, 1990, 1999;[START_REF] Rao | Small Area Estimation[END_REF]. The most famous model using linear mixed models is the one developed by [START_REF] Fay | Estimates of Income for Small Places : An Application of James-Stein Procedures to Census Data[END_REF]. The authors begin by modeling a function of the mean in a given area, explained on the one hand, by the auxiliary information, and on the other hand, by a random part explaining the variability across areas that are not considered in the auxiliary information. Then, they show that the blup is a composite estimator.

In small area estimation, an important factor to bear in mind is that local estimates are not always consistent with the overall population estimate. Indeed, in general, the sum of estimates at the level of small areas does not coincide with the estimate at the level of the overall population. In order to satisfy a benchmarking property, [START_REF] Prasad | On Robust Small Area Estimation Using a Simple Random Effects Model[END_REF] propose a two-step procedure to obtain a pseudo-Eblup of a small area mean. They combine area models using survey weights with unit level models. Using a nested error regression model, [START_REF] You | A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights[END_REF] developed a method that provides coherent estimates thanks to a skillful variable change in the regression and the use of sampling weights so as to build a pseudo-Eblup. Also, under a nested error regression model, [START_REF] You | Pseudo hierarchical Bayes small area estimation combining unit level models and survey weights[END_REF] use a pseudo-hierarchical Bayes approach to obtain posterior estimators of small area means. [START_REF] Ugarte | Benchmarked estimates in small areas using linear mixed models with restrictions[END_REF] propose an Eblup based upon a linear mixed model with restrictions. They force the sum of small area estimates to equal the calculated estimate of the overall population using a synthetic estimator.

In this paper, a new approach is proposed. The method consists of splitting the sampling weights of the overall population estimator to construct local estimators. Each weighting system corresponds to a small area. The idea is to define weights for the areas that depend on the sampled units as well as the other small areas. Each statistical unit can contribute to all the small areas.

In order to satisfy a benchmarking principle, the sum of the weights of a particular unit relative to each area must be equal to its global weight, which automatically implies that the global estimator is the sum of the local estimates. Furthermore, the weights are calibrated in such a way that, in each area, the estimates of the totals are equal to the population total for the auxiliary variables that are known at the level of the small areas.

The main tool of this method is a matrix Q for which the number of rows is equal to the number of units and the number of columns is equal to the number of small areas. This matrix embodies the way the global weights are split into the areas. The benchmarking principle is obtained by the simple fact that the sum of the elements of each row is equal to 1. For each unit, two kinds of contributions are distinguished: its contribution to its own area ("autocontribution") and its contribution to the other areas ("extra-contribution"). A consequence of the benchmarking principle is that the more a unit contributes to its own area, the less it contributes to the other ones. A composite estimator is built with an "area" part by a direct estimation and with an "extra-contribution" part, given the Q probability matrix. At the level of each small area, these two parts are balanced with a parameter (tuning constant) obtained by minimizing the statistical dispersion of the variable of interest.

This paper is structured as follows. In Section 2, the notation is defined. The direct estimators are presented in Section 3. In Section 4, the weight splitting estimation is developed. Next, in Section 5, the composite estimator is presented. The choice of the tuning constant is discussed in Section 6. A simulation study is presented in Section 7, and the paper ends with some brief concluding remarks in Section 8.

Notation

Consider a finite population U of N statistical units belonging to D disjoint areas {A 1 , . . . , A d , . . . , A D } of sizes {N 1 , . . . , N d , . . . , N D }. The units can be identified by a label k ∈ {1, . . . , k, . . . , N }. Consider also J auxiliary variables x 1 , . . . , x j , . . . , x J . We are interested in estimating the overall total

t y = k∈U y k .
of the interest variable y. Moreover, we want to estimate the total of y in each area, i.e. for area

A d t d y = k∈U∩A d y k .
The estimation is based upon the availability of J auxiliary variables. The values of the jth variable for all the units are denoted x 1j , . . . , x kj , . . . , x N j . The values of the J variables of unit k are denoted by the column vector of R J

x k = x k1 ⋯ x kj ⋯ x kJ ⊺ .
The J variables of the population are represented by N × J matrix

X U = x 1 ⋯ x k ⋯ x N ⊺ .
The totals of the J variables for the population are represented by the row vector of size J

t x = ∑ k∈U x k1 ⋯ ∑ k∈U x kj ⋯ ∑ k∈U x kJ .
The totals of the J variables for the population belonging to area A d are represented by the row vector of size J

t d x = ∑ k∈U∩A d x k1 ⋯ ∑ k∈U∩A d x kj ⋯ ∑ k∈U∩A d x kJ ,
and the totals of the J variables for the population belonging to each of the areas are represented by the matrix of size D × J

t A x = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ t 1 x ⋮ t d x ⋮ t D x ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (2.1)
Below, matrix t A x is supposed to be known. This matrix can thus be used to improve estimation of the totals in domains t d y . A sample s is a subset of U , and a sampling design p(s) is a probability distribution on all possible samples that can be drawn from U , such that p(s) ≥ 0, and s⊂U p(s) = 1.

For a given sampling design p(s), a sample s is the realization of a random sample S, i.e. Pr(S = s) = p(s) for all s ⊂ U. We note {n 1 , . . . , n d , . . . , n D } the sizes of the areas {#(S ∩ A 1 ), . . . , #(S ∩ A d ), . . . , #(S ∩ A D )}. The first order inclusion probability of unit k is denoted by π k = Pr(k ∈ S).

Direct estimation

At the level of an area, direct estimation consists of building an estimator of t d y without using any information outside of the given area. Then, in a direct estimation, a unit can only contribute to its own area. For instance, the [START_REF] Horvitz | A generalization of sampling without replacement from a finite universe[END_REF] estimator (or π-estimator) which directly uses the sample weights 1 π k is a direct estimator. A calibrated estimator is also a direct estimator.

Let t y = ∑ k∈U y k be the total of the quantitative variable y. The " [START_REF] Horvitz | A generalization of sampling without replacement from a finite universe[END_REF] estimator" or "π-estimator" of t y is defined by

ty,π = k∈S y k π k .
At the level of area A d , the [START_REF] Horvitz | A generalization of sampling without replacement from a finite universe[END_REF] estimator of the total of the variable y denoted by t d y = ∑ k∈U∩A d y k is given by the quantity:

td y,π = k∈S∩A d y k π k .
This method of calibration was formalized by [START_REF] Deville | Calibration estimators in survey sampling[END_REF] who gave a common framework for the calibration estimation and the properties of these estimators. Suppose that a vector of totals t x of J auxiliary variables is known at the level of the population. The calibration estimator of the total of the quantitative variable y depends on a weighting system w k . The estimator is defined by ty,w

= k∈S w k y k ,
where the w k weighting system depends on the sample S and satisfies the calibration equation:

k∈S w k x ⊺ k = t x . (3.1)
The calibration equation (3.1) means that the weighting system of the calibrated estimator must reproduce exactly the values of the totals of the auxiliary variables that are known at the population level.

The weights w k are computed in such a way to be as close as possible to the Horvitz-Thompson weights d k = 1 π k . Moreover, the weights must satisfy the calibration equation (3.1). In order to find such weights, [START_REF] Deville | Calibration estimators in survey sampling[END_REF] propose several pseudo-distances denoted by G k (w k , d k ) that is assumed to be positive, derivable, strictly convex with regard to w k and such that G k (d k , d k ) = 0 for all k ∈ U. The w k weights are obtained by minimizing the quantity

k∈S G k (w k , d k ) q k
subject to the constraints of the calibration equation given in (3.1). The q k -1

are coefficients which determine the importance of each unit in the calculation of the distance.

Many different distances can be used and are discussed in [START_REF] Deville | Calibration estimators in survey sampling[END_REF]. In general, if g k (w k , d k ) denotes the derivative of G k (w k , d k ) with respect to w k , then the weights are defined by

w k = d k F k (q k λ ⊺ x k ), (3.2)
where λ is the vector of Lagrangian multiplier and

d k F k (.) is the reciprocal function of g k (., d k ).
The value of the Lagrangian multipliers λ can be identified by inserting (3.2) in (3.1) and by solving the calibration equation by the Newton-Raphson method. Below, we mainly use the raking-ratio method which is defined by means of the Kullback-Leibler measure. In this case,

q k = 1, k ∈ S G k (w k , d k ) = w k log w k d k + d k -w k , g k (w k , d k ) = log w k d k ,
and

w k = d k F k (q k λ ⊺ x k ) = d k exp(λ ⊺ x k ).
At the level of small areas, a direct estimator cannot be used when the sample size within the area is small because its variance becomes very large. For a given area A d of size n d , the variance of a direct estimator is O(1 n d ). The smaller the size n d , the larger the variance. Hence, the quality of small area direct estimates is debatable. When the size of a given area is not large enough to have a satisfactory direct estimation, we attempt to improve the quality of the estimates by borrowing information at the level of the other areas. And so we use the method of splitting the sampling weights.

Weight splitting or extra-contribution estimation

Constraints on the split weights

The proposed method consists of splitting the weights into the areas. In a direct estimator, only the weights of the units that belong to an area A d can contribute to a local estimation at the level of A d , which is called "auto-contribution part".

In the weight splitting method, any unit can contribute to the estimation of any area through a certain weight, which is called "extra-contribution part". Suppose that a weighting system has already been computed for the overall estimation. These weights can be the inverse of the inclusion probabilities or can be obtained by means of a calibration procedure on the total t x . The main idea of the proposed weight-splitting approach is to build a weight w kd which depends both on unit k and area A d . This weight is defined as the product of the basic w k weight with a splitting coefficient q kd that distributes the weights in the areas, i.e. w kd = w k q kd , for all k ∈ S and for d = 1, . . . , D.

The D × J matrix t A

x of the D totals of the auxiliary variables, given in (2.1) is supposed to be known. The knowledge of this auxiliary information at the level of the areas is used to calibrate the weighting system.

More precisely we would like to have weights for the areas that are calibrated on the totals of the areas, i.e.

k∈S w kd x ⊺ k = k∈S w k q kd x ⊺ k = t d x , for all d, = 1, . . . , D. (4.1)
Moreover, we want to impose a coherence between the sum of small areas estimates and the overall estimates. Since w kd = w k q kd , the coherence

D d=1 k∈S w kd x ⊺ k = k∈S w k x ⊺ k can be obtained if D d=1 w kd = D d=1 w k q kd = w k , for all k ∈ S.
The splitting coefficients must thus satisfy

D d=1 q kd = 1. (4.2)
To sum up, the n × D weights q kd must satisfy D × J + n constraints:

• the D ×J constraints of calibration on the totals of the areas given in (4.1),

• the n constraints of consistency given in (4.2).

Note also that the constraints of consistency given in (4.2) are based on an important practical interpretation. If, in small area estimation, the estimate is written in the form of a weighting system, the constraints of consistency imply that units that strongly contribute to other areas contribute less to their own area. Extra-contribution works like an exchange of information, if there are few units in an area, this area needs to borrow strength from other areas, but in exchange, the units of this small area must contribute more to the other areas.

The weights q kd can be gathered in a Q matrix with n rows and D columns. The sum of the elements in each row is thus equal to 1:

Q = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ q 11 . . . q 1d . . . q 1D ⋮ ⋮ ⋮ q k1 . . . q kd . . . q kD ⋮ ⋮ ⋮ q n1 . . . q nd . . . q nD ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
, which can be written

Q1 D = 1 n ,
where 1 D (resp. 1 n ) is a column vector of D ones (resp. of n ones). The constraints given in (4.1) can be rewritten with a matrix notation:

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ q 11 . . . q k1 . . . q n1 ⋮ ⋮ ⋮ q 1d . . . q kd . . . q nd ⋮ ⋮ ⋮ q 1D . . . q kD . . . q nD ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ w 1 . . . 0 . . . 0 ⋮ ⋱ ⋮ ⋮ 0 . . . w k . . . 0 ⋮ ⋮ ⋱ ⋮ 0 . . . 0 . . . w n ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ x ⊺ 1 ⋮ x ⊺ k ⋮ x ⊺ n ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ t 1 x ⋮ t d x ⋮ t D x ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .
In short, the two following properties must be satisfied:

1. the sum of the rows of

Q must equal 1, 2. Q ⊺ diag(w 1 , . . . , w k , . . . , w n )X S = t A
x , where X S = (x kj ) k∈S,j=1,...,J . Each coefficient q kd embodies the contribution of unit k to the estimator for area A d .

Computation of matrix Q

In order to compute a matrix Q which satisfies both the coherence property and the totals at the level of each area, we propose a simple algorithm which repeats two successive calibrations. The columns on the totals of the areas are calibrated on the values of the x-variables. Next, the rows of the matrix are calibrated again to ensure that the sum is equal to 1. These steps are repeated until convergence. This method is a generalization of the raking-ratio method that enables us to calibrate a contingency table on marginal totals (see for instance [START_REF] Ireland | Contingency tables with given marginals[END_REF][START_REF] Arora | An investigation of the properties of raking ratio estimator: I. With simple random sampling[END_REF].

More specifically, the algorithm begins with the initialization of the matrix by

Q {0} = ⎛ ⎜ ⎝ N1 N . . . N d N . . . N D N ⋮ ⋮ ⋮ N1 N . . . N d N . . . N D N ⎞ ⎟ ⎠ .
This first matrix of splitting coefficients simply shares the weights proportionally to the size of the areas in the population. Next, at step 2t, for t = 1, 2, 3, . . . the following two operations are repeated:

• Each column of Q {2t-2} is calibrated on the vector of known totals of each area by solving in λ for each area A d , d = 1 . . . D the equations system:

t d x = k∈S d k q {2t-2} kd x ⊺ k exp (x k λ d )
The coefficients of the new matrix Q {2t-1} can be obtained by:

q {2t-1} kd = q {2t-2} kd exp(x k λ d ).
• Then the sum of the rows is calibrated so as to equal 1:

q {2t} kd = q {2t-1} kd ∑ D d=1 q {2t-1} kd .
The iteration stops when the sum of the rows is almost equal to 1 after a column calibration, or more specifically when

n k=1 D d=1 q {2t-1} kd -1 < ε,
where ε is a sufficiently small positive real. The use of an exponential calibration function guarantees that the weights q kd remain nonnegative at each step of the method. Once the Q matrix is computed, the totals of areas for any variable of interest can be estimated. At the level of a given area A d , the extra-contribution estimator of the total of a quantitative variable y is defined by td y,q = k∈S w k q kd y k .

(4.

3)

The sum of the area estimators is always equal to the estimator in the population.

Composite estimator

The weight splitting method is a synthetic estimator because all the statistical units can contribute to each area. In order to avoid to miss some area specificities, a composite estimator can be obtained by mixing the direct estimator with the extra-contribution estimator so as to propose a method where each area is estimated by a part of auto-contribution built from a direct estimation and by a part of extra-contribution built thanks to the Q matrix.

A matrix C of new composite weights is constructed by means of weights α d for d = 1, . . . , D. The procedure starts with the construction of a n × D matrix G = (g kd ), where

g kd = α d q kd + (1 -α d )1 {k∈A d } , k ∈ S, d = 1, . . . , D,
where 1 {C} equals 1, if condition C is true and 0 otherwise. The coefficients α d depend on the areas. The smaller the area, the larger the α d . It is desirable that, in large areas, the estimator depends more on the units of these areas. Whereas in small areas, the estimator depends more on the units outside of these areas.

Next, matrix G is calibrated again on the two sets of constraints in such a way that the totals of the areas are reproduced for the auxiliary variables and that the sum of the rows equals 1. The algorithm described in Section 4.2 is thus applied again. We obtain matrix C = (c kd ) the elements of which can be written as c kd = g kd h kd where the h kd are the matrix calibration adjustments.

Considering a quantitative variable of interest y, the composite estimator of the total of y at the level of A d is a weighted average given by:

td y,c = k∈S c kd w k y k = α d k∈S h kd q kd w k y k + (1 -α d ) k∈S∩A d h kd w k y k . (5.1)
The estimator is a weighted average of two terms. The first one is a synthetic estimator that depends on all the statistical units of the sample. The second one is a direct estimator that only depends on the selected units in the small area.

6. Determination of a tuning constant α d

Approximation of the variance of the composite estimator

In order to obtain a reasonable value for the tuning constants, one can use heuristic reasoning. Since the first term of the composite estimator given in (5.1) depends on all the units, we assume that its variance can be written σ 2 d,1 n. Since the second term only depends on the units that belong to A d , we assume that its variance is equal to σ 2 d,2 n d .

Moreover, if we assume that the correlation coefficient between the first and the second term is equal to ρ, the variance of the composite estimator is equal to the following quantity:

Var( td y,c ) = α 2 d σ 2 d,1 n + (1 -α d ) 2 σ 2 d,2 n d + 2α d (1 -α d )ρ σ d,1 σ d,2 √ n d n .
If we assume that the covariance term is negligible,

Var( td y,c ) ≈ α 2 d σ 2 d,1 n + (1 -α d ) 2 σ 2 d,2 n d . (6.1)
This kind of approximation is done for composite estimators for instance in Rao (2003, p. 57).

By setting the derivative of (6.1) with respect to α d to zero, we obtain:

α d σ 2 d,1 n -(1 -α d ) σ 2 d,2 n d = 0.
The value for α d that minimizes (6.1) is then given by:

α d (n d ) ≈ 1 1 + σ 2 d,1 σ 2 d,2 n d n
.

If

θ d = σ 2 d,1 σ 2 d,2
, we can see that when n d tends to 0 (resp. +∞), then α d (n d ) tends to 1 (resp. 0). If σ 2 d,1 and σ 2 d,2 do not depend on d, then we obtain a simplification:

α d (n d ) ≈ 1 1 + θ n d n .
6.2. Eblup and pseudo-Eblup under a mixed model

Nested error linear regression model

Let us now assume that we are interested in small area means. [START_REF] Henderson | Best Linear Unbiased Estimation and Prediction under a Selection Model[END_REF]; [START_REF] Battese | An Error-Components Model for Prediction of County Crop Areas Using Survey and Satellite Data[END_REF]; [START_REF] Prasad | The Estimation of the Mean Squared Error of Small Area Estimators[END_REF]; [START_REF] You | A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights[END_REF]; [START_REF] Rao | Small Area Estimation[END_REF] proposed a nested error linear regression model to estimate small area means. Using the [START_REF] Prasad | The Estimation of the Mean Squared Error of Small Area Estimators[END_REF]; [START_REF] You | A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights[END_REF]; [START_REF] Rao | Small Area Estimation[END_REF] notation, we can consider the mixed model approach

y dk = x ⊺ dk β + v d + ε dk (6.2)
where k = 1, . . . , n d , d = 1, . . . , D, v d are independent centered normal variables with variances σ 2 v , ε dk are independent centered normal variables with variances σ 2 ε . Moreover, the v d are assumed to be independent from the ε dk .

The mean for area A d , denoted Y d , can be approximated by the parameter

µ d = X ⊺ d β + v d
where

X d = 1 N d N d k=1
x dk .

You and [START_REF] You | A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights[END_REF] proposed a combination of the basic unit level model (6.2) with sample weights and obtained the following weighted area level model

y dw = n d k=1 y dk π dk ∑ n d l=1 1 π dl = x ⊺ dw β + v d + ε dw (6.3) where E(ε dw ) = 0 and Var(ε dw ) = σ 2 ε n d k=1 (π dk n d l=1 1 π dl ) -2 = σ 2 ε δ dw .

Eblup under a mixed model

When σ 2 v and σ 2 ε are known, it follows from [START_REF] Rao | Linear Statistical Inference and Its Applications[END_REF]; [START_REF] Henderson | Best Linear Unbiased Estimation and Prediction under a Selection Model[END_REF]; [START_REF] Battese | An Error-Components Model for Prediction of County Crop Areas Using Survey and Satellite Data[END_REF]; [START_REF] Prasad | The Estimation of the Mean Squared Error of Small Area Estimators[END_REF]; [START_REF] You | A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights[END_REF]; [START_REF] Rao | Small Area Estimation[END_REF] that the best linear unbiased predictor (blup) of µ is given by

μd = γ d y d + (X d -γ d x d ) ⊺ β, (6.4)
where β is the generalized least square estimator of β,

y d = 1 n d n d k=1 y dk , x d = 1 n d n d k=1
x dk and

γ d = 1 + σ 2 ε σ 2 v n d -1
.

The variances σ 2 v and σ 2 ε can be estimated using two ordinary least squares regressions and the method of moments (see, for instance, [START_REF] Fuller | Transformation for estimation of linear models with nested error structure[END_REF][START_REF] You | A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights[END_REF][START_REF] Rao | Small Area Estimation[END_REF].

The expression of μd in (6.4) can be re-written as

μd = γ d y d + (X d -x d ) ⊺ β + (1 -γ d )X ⊺ d β.
The empirical best linear unbiased predictor estimator (Eblup) of μd , denoted μd , is then obtained by replacing σ 2 v and σ 2 ε by σ2 v and σ2 ε in the expression of

γ d μd = γd y d + (X d -x d ) ⊺ β + (1 -γd )X ⊺ d β,
where β = β(σ 2 u , σ2 ε ). It follows that the expression of the total estimator is obtained by

td y,Eblup = N d μd , = γd N d y d + (t d x -N d x d ) ⊺ β + (1 -γd )t d x ⊺ β, = γd td y,Eblup,direct + (1 -γd ) td
y,Eblup,synth (6.5)

Pseudo-Eblup under a mixed model

The pseudo-blup estimator is obtained from a combination of the model (6.2) with the sample weights. From assuming that σ 2 v and σ 2 ε are known, the pseudoblup estimator of µ d from the aggregated model (6.3) is given by

μdw = γ dw y dw + (X dw -x dw ) ⊺ βw + (1 -γ dw )X dw βw , (6.6)
where

γ dw = 1 + σ 2 ε δ dw σ 2 v -1 and βw = D d=1 n d k=1 x dk π dk (x dk -γ dw x dw ) ⊺ -1 D d=1 n d k=1 (x dk -γ dw x dw ) y dk π dk .
The pseudo-empirical best linear unbiased predictor estimator (pseudo-Eblup) of μdw , denoted μdw , is then obtained by replacing σ 2 v and σ 2 ε by σ2 v and σ2

ε μdw = γdw y dw + (X dw -x dw ) ⊺ βw + (1 -γdw )X dw βw ,
where βw = βw (σ 2 u , σ2 ε ). It also follows that the expression of the total estimator is obtained by

td y,p-Eblup = N d μdw , = γdw N d y dw + (t d x -N d x dw ) ⊺ βw + (1 -γdw )t d x ⊺ βw , = γdw td y,p-Eblup,direct + (1 -γdw ) td y,p-Eblup,synth (6.7)
For a given area A d , when the weights are calibrated with the known size N d i.e.

n d k=1 1 π dk = N d ,
and when the unit level model (6.2) includes the intercept term, the estimator td y,p-Eblup satisfies the benchmarking property without any adjustment (see for instance [START_REF] You | A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights[END_REF][START_REF] Rao | Small Area Estimation[END_REF].

In the case that the weights are not calibrated with the known size N d , a preliminary calibration on the weights can be done in order to obtain a coherent pseudo-Eblup estimator. It follows that under a simple random sampling without replacement, a preliminary calibration on the weights leads to the equality γ d = γ dw . In fact, under a simple random sampling without replacement, the sampling weights are all equal: 1 π dk = N n for all k. Then, new weights calibrated on the known size N d are obtained:

1 π dk,w = N d n d . It follows that δ dw = ∑ n d k=1 ( N d n d ∑ n d l=1 N d n d
) 2 = 1 n d and then γ d = γ dw .

Composite form of the Eblup and the pseudo-Eblup

The Eblup and the pseudo-Eblup can be seen as composite estimators. They are weighted averages of a regression synthetic estimator and a pseudo-direct estimator. Similarly to the proposed method, when n d tends to 0 (resp. +∞), γ d and γ dw tend to 0 (resp. +∞): when the size of an area is large enough, more weight is attached to the direct estimation part and vice versa. Then, an estimation of the tuning constant α d can be obtained using an analogy with the parameters γ d and γ dw of the blup:

α{1} d = 1 -γd , (6.8)
or α{2} d = αdw = 1γdw . (6.9)

7. Simulation study

Simulated data with a mixed model

In order to test the performance of the proposed methodology, we ran a set of simulations.

Simulated population

A population of N = 2, 000 units, with D = 20 disjoint areas of sizes (N 1 , . . . , N D ), is created from a linear mixed model given in (6.2): 1 gives an overview of a generated population.

x dk = 1 x dk ⊺ with x dk , iid ∼ N (20, σ 2 x = 9), β = 12 0.4 ⊺ , v d iid ∼ N (0, σ 2 v = 4), ε dk iid ∼ N (0, σ 2 ε = 1). The Figure
15 20 25 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 10 15 20 25 30
x y factor(area)

• • • • • • • • • • • • • • • • • • • • 1

Precision comparison between the weights splitting estimator and some classical small area estimators

From the generated population, B = 10, 000 samples of size n = 200 are drawn by a simple random sampling without replacement. Within areas, the Eblup estimator, the pseudo-Eblup estimator and the proposed estimator (with their respective direct and synthetic components) are computed to estimate the totals t d y , for d = 1, . . . , 20. The relative root mean square error (%RRMSE) is used to quantify the performance of the estimators. For a given estimator td y , the %RRMSE d is obtained as follows

%RRMSE d = 100 × √ MSE d t d y , (7.1)
where MSE d is the sum of the square of the bias and the variance

MSE d = 1 B B b=1 td y,b -t d y 2 + N -n N -1 1 B -1 B b=1 td y,b - 1 B B b=1 td y,b 2 .
Table 1 shows a comparison between the proposed method and the Eblup estimator. Table 2 shows a comparison between the proposed method and the is not fixed because the sampling design is a simple random sampling without replacement from the overall population. That is why, the second columns of Table 1 and Table 2 represent the means n d (for d = 1, . . . , D) of each area size through the B = 10, 000 drawings. The tuning constants αd and αdw also are reestimated in each simulation run. That also is why, the third columns of Table 1 and Table 2 represent the means αd (resp. αwd ) (for d = 1, . . . , D) of each area tuning constant through the B = 10, 000 drawings. As discussed in Section 6.2.3, the equalities αd = αwd and αd = αwd are obtained because the sampling design is a simple random sampling without replacement from the overall population.

In Table 1, the direct estimation component of the Eblup estimator appears to be generally better than the global calibration which is the direct estimation part of the weights splitting method. This can be explained by the fact that the direct estimation component of the Eblup is a regression estimator built from the parameter β of the synthetic estimation component of the Eblup. The synthetic part of the Eblup and the extra-contribution part seem to have equivalent performance. In spite of the weakness of the direct estimation part of the proposed composite estimator relative to the direct estimation part of the Eblup, the two composite estimators perform equivalently thanks to the matrix calibration computed during the weights splitting method procedure. The result shown in Table 1 is quite difficult to interpret, because the weightsplitting estimator incorporates sampling weights and is benchmarked, while the Eblup estimator does not. Table 2 seems to be fairer in terms of comparison because both the pseudo-Eblup estimator and the proposed estimator respect the benchmarking property and use sampling weights. As previously, whereas the synthetic part of the pseudo-Eblup and of the proposed method seem to perform equivalently, the direct estimation component of the pseudo-Eblup performs better than the direct estimation part of the weights splitting method. This also can be explained by the construction of the direct estimation of the pseudo-Eblup with the parameter βw which is derived from the synthetic part of the pseudo-Eblup. Despite this disadvantage, when the calibration and the extra-contribution parts of the weights splitting method are mixed and are re-calibrated on the rows (benchmarking constraints) and on the columns (calibration constraints on the areas), the pseudo-Eblup estimator and the weights splitting estimator appear to perform equivalently. These obtained similar efficiencies can also partly be explained by the same weight attached to the synthetic component of the pseudo-Eblup and attached to the extra-contribution component of the proposed composite estimator.

Resampling procedure to estimate the variance of the weights splitting estimator

From the generated artificial population (see Figure 1), a resampling procedure can be performed to estimate the variance of the weights splitting estimator. Given a sample drawn from the generated population, B samples are drawn from the simulated expectation of the weights splitting bootstrap variance estimator. Table 3 gives a comparison between these two quantities. It shows that the two quantities are very closed.

Once the bootstrap variances computed, construction of confidence intervals can be processed. Table 4 reports the numbers of times (%) the true total values t d y (for d = 1 . . . D), from the generated population, lie within the 95% confidence intervals built with bootstrap variances. Table 4 shows that given an area, the number of times the true total value lies within the 95 % confidence interval increases with the size of the area.

Data with county crop areas

We compare the pseudo-Eblup and the weights splitting procedures applied to a real data given by [START_REF] Battese | An Error-Components Model for Prediction of County Crop Areas Using Survey and Satellite Data[END_REF] and taken up by [START_REF] You | A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights[END_REF]; [START_REF] Rao | Small Area Estimation[END_REF]. These authors wanted to estimate the mean of hectares of corn per segment for D = 12 counties in north-central Iowa. Each county is divided into area segments and the areas under corn are in the area segments. The authors used a sample s of n = 36 segments and assumed simple random sampling within areas. The population is assumed to follow the linear mixed model given in (6.2) where the function of interest is the number of hectares of corn per segment per county, and the auxiliary information is the number of pixels seen as corn and as soybeans.

Table 5 reports the pseudo-Eblup and weights splitting estimates of hectares of corn with their respective coefficients of variation. The variances are obtained by a resampling procedure of 1, 000 iterations (see Algorithm 2). The two es- Battese, Harter and Fuller (1988, p. 29) Algorithm 2 Resampling procedure for the variance estimation by bootstrap td y,c = ty,w = 817087, 2. Table 5 gives quite similar estimates with similar efficiencies.

Concluding remarks

The simulations show that, even when the data are really generated by a mixed model, the proposed estimator does not seem worse than the Eblup and pseudo-Eblup estimators. The proposed estimators offer several advantages. The regional estimates are coherent with the overall estimates. The estimator takes the sampling weights into account. The estimator can be written as a weighting system and can thus be applied on any variable of interest.

As for all the composite estimators, the proposed estimator clearly states that the parameters α d (for d = 1 . . . D) depend on the variable of interest. In the case of a set of variables of interest belonging to a specified theme, the constant α d can be chosen as the mean of the tuning constants obtained from each variable of interest.

The next step will consist of computing a variance estimator of the composite estimator. Since the proposed estimator is obtained by successive matrix calibrations, the computation of this variance is complex, which is thus a challenging objective. If a closed form of variance estimator is intractable, resampling procedures for simple random sampling as seen in Section 7.1.3 can be considered; resampling procedures for sampling design with unequal inclusion probabilities can be based on [START_REF] Antal | A Direct Bootstrap Method for Complex Sampling Designs From a Finite Population[END_REF] methodology.

The proposed method is not particularly robust for the resistance to outliers. It could be an interesting further topic of research.

Fig 1 .

 1 Fig 1. Generated population from the linear mixed model given in 7.1.1: the different colors indicate the areas).

  Consider B = 1, 000 the number of iterations Consider n = 36 the sample size for b = 1 to B do Draw a sample s b of size n by a simple random sampling with replacement within areas from the sample s (the area sample sizes are fixed and are the same as those in s). Compute the pseudo-Eblup and weights splitting estimators from the sample s b . end for Compute the variances of the B obtained pseudo-Eblup and weights splitting estimators with considering the finite population correction factor. timators automatically respect the benchmarking property. Indeed, we have ∑ D d=1 N d μdw = ty,π + (t xtx,π ) ⊺ βw = 815016, 3 and ∑ D d=1

Table 1

 1 Computed %RRMSE of the Eblup estimator, of the proposed estimator, and their respective component estimators, from 10,000 drawn samples of size 200 by a simple random sampling without replacement from the generated population (see Figure1) Eblup estimator. For each simulation run, each size n d (for d = 1, . . . , D)

				Eblup components	Weights Splitting method
	Area	n d	αd	Direct	Synthetic	Eblup	Calibration	Extra	Composite
	1	3.45	0.14	5.26	24.68	5.95	49.14	24.10	9.70
	2	10.03	0.04	1.94	10.62	1.92	27.72	10.09	2.96
	3	5.66	0.07	2.68	8.65	2.55	37.40	8.14	3.24
	4	17.40	0.02	1.10	1.34	1.07	20.50	0.67	0.98
	5	4.69	0.09	2.58	1.48	2.32	41.11	1.52	1.91
	6	12.53	0.03	1.29	3.60	1.26	25.07	3.03	1.31
	7	18.67	0.02	1.09	2.03	1.07	19.91	2.20	1.06
	8	2.81	0.17	3.94	11.88	4.02	53.72	12.22	5.53
	9	7.28	0.05	2.02	7.82	1.96	32.72	7.31	2.62
	10	15.01	0.02	1.41	12.35	1.41	22.41	11.83	2.61
	11	4.41	0.10	2.20	7.38	2.14	43.12	7.73	3.05
	12	18.15	0.02	1.14	7.11	1.12	20.45	6.59	1.59
	13	3.51	0.14	2.97	1.10	2.57	48.92	0.71	2.08
	14	7.76	0.05	1.76	1.11	1.67	31.96	0.65	1.46
	15	4.88	0.09	2.46	14.11	2.63	40.50	14.44	5.08
	16	12.40	0.03	1.28	5.36	1.25	24.86	5.69	1.60
	17	16.82	0.02	1.22	1.62	1.19	21.13	0.95	1.12
	18	11.60	0.03	1.25	3.65	1.22	25.65	3.95	1.39
	19	6.48	0.06	1.84	5.57	1.76	35.13	5.89	2.19
	20	16.46	0.02	1.08	10.76	1.10	21.53	11.11	2.23
	pseudo-

Table 2

 2 Computed %RRMSE of the pseudo-Eblup estimator, of the proposed estimator, and their respective component estimators, from 10,000 drawn samples of size 200 by a simple random sampling without replacement from the generated population (see Figure1)

			pseudo-Eblup components	Weights Splitting method
	Area	n d αdw Direct Synthetic pseudo-Eblup Calibration Extra Composite
	1	3.45 0.14	5.25	24.58	5.97	49.14	24.10	9.70
	2 10.03 0.04	1.94	10.49	1.92	27.72	10.09	2.96
	3	5.66 0.07	2.67	8.51	2.54	37.40	8.14	3.24
	4 17.40 0.02	1.10	0.86	1.07	20.50	0.67	0.98
	5	4.69 0.09	2.58	1.16	2.30	41.11	1.52	1.91
	6 12.53 0.03	1.29	3.38	1.26	25.07	3.03	1.31
	7 18.67 0.02	1.09	1.83	1.07	19.91	2.20	1.06
	8	2.81 0.17	3.94	11.90	4.03	53.72	12.22	5.53
	9	7.28 0.05	2.02	7.68	1.96	32.72	7.31	2.62
	10 15.01 0.02	1.41	12.23	1.41	22.41	11.83	2.61
	11	4.41 0.10	2.20	7.38	2.14	43.12	7.73	3.05
	12 18.15 0.02	1.14	6.97	1.12	20.45	6.59	1.59
	13	3.51 0.14	2.97	0.49	2.55	48.92	0.71	2.08
	14	7.76 0.05	1.76	0.49	1.66	31.96	0.65	1.46
	15	4.88 0.09	2.47	14.14	2.64	40.50	14.44	5.08
	16 12.40 0.03	1.28	5.33	1.25	24.86	5.69	1.60
	17 16.82 0.02	1.22	1.22	1.19	21.13	0.95	1.12
	18 11.60 0.03	1.25	3.58	1.22	25.65	3.95	1.39
	19	6.48 0.06	1.84	5.55	1.76	35.13	5.89	2.19
	20 16.46 0.02	1.08	10.78	1.10	21.53	11.11	2.23

Table 3

 3 Variance of the weights splitting estimators vs Mean of the bootstrap variances of the weights splitting estimators

	Area	Var{( td y,c ) b * }	Ẽsim {σ b * d }
	1	1264.0	989.3
	2	1566.2	1550.0
	3	612.8	570.0
	4	1021.0	1163.5
	5	324.6	274.9
	6	778.4	810.6
	7	1288.8	1573.6
	8	574.2	432.9
	9	713.8	699.0
	10	2555.2	2768.5
	11	505.2	409.1
	12	2088.3	1981.9
	13	218.0	164.5
	14	485.1	472.0
	15	1442.2	1264.0
	16	1134.5	1290.5
	17	1275.3	1383.5
	18	813.8	837.5
	19	528.9	543.1
	20	3438.0	3849.4

Table 4

 4 Numbers of times (%) the true total values t d y (for d = 1 . . . D), from the generated population (see Figure1), lie within the 95% confidence intervals built with bootstrap variances

	Area	N d	Numbers of times (%)
	1	34	75
	2	100	84
	3	57	78
	4	174	93
	5	47	84
	6	125	91
	7	187	93
	8	28	65
	9	73	85
	10	150	87
	11	44	75
	12	182	87
	13	35	88
	14	78	91
	15	49	73
	16	124	89
	17	168	94
	18	116	87
	19	65	86
	20	164	87

Table 5

 5 Estimated hectares of corn with coefficients of variation LandSat data from Table1 in

			pseudo-Eblup	Weights Splitting Method
	County	n d	Estimate	c.v. (%)	Estimate	c.v. (%)
				Corn
	Cerro Gordo	1	120.5	2.6	121.8	3.0
	Hamilton	1	125.3	2.7	122.7	3.2
	Worth	1	106.3	7.8	108.3	6.4
	Humboldt	2	107.3	8.4	111.1	5.7
	Franklin	3	143.8	4.5	142.8	5.3
	Pocahontas	3	111.5	5.0	111.8	6.0
	Winnebago	3	112.1	5.5	113.8	4.6
	Wright	3	121.3	3.8	120.2	3.3
	Webster	4	115.1	3.3	114.7	4.5
	Hancock	5	124.5	3.4	124.2	3.1
	Kossuth	5	106.6	3.2	109.3	3.8
	Hardin	5	143.5	3.3	141.0	3.0
	Source:

Acknowledgments

The authors would like to thank an anonymous referee for useful comments and suggestions.

this initial sample (for instance B = 500). The B weights splitting estimators are computed from these B new samples. The variance by bootstrap is obtained from computing the empirical variance of the B weights splitting estimators. The considered sampling design is always a simple random sampling without replacement.

In order to test the efficiency of the variance estimation by bootstrap, the Algorithm 1 is considered.