S Carpentier
email: scarpentier@univ-paris12.fr

R Moilleron

C Beltran

D Herve

D Thevenot

Quality of dredged material in the River Seine basin (France).I . Physico-chemical properties

Keywords: Dredged material, Sediment quality, Pore-water, River Seine basin

In rivers, sediments are frequently accumulating persistent chemicals, especially for those that are more contaminated as a consequence of pressure related to environmental pollution and human activity. The Seine river basin (France) is heavily polluted from nearby industrial activities, and the urban expansion of Paris and its suburbs within the Ile de France region and the sediments present in the Seine river basin are contaminated. To ensure safe, navigable waters, rivers and waterways must be dredged. In this paper, the quality of the sediment dredged in 1996, 1999 and 2000 is discussed. Physico-chemical characteristics of the sediment itself and of the pore-water are presented. Seine basin sediments show very diverse compositions depending on the sampling site. Nevertheless, a geographic distribution study illustrated that the Paris impact is far from being the only explanation to this diversity, the quality of this sediment is also of great concern. The sediment once dredged is transported via barges to a wet disposal site, where the dredged material is mixed with Seine water in order to be pumped into the receiving site. This sort of dumping might be responsible for the potential release of contaminants to the overlying water from the significantly contaminated sediments.

Introduction

Dredging rivers is necessary to maintain sufficient sailing depth [START_REF] Hauge | Remediation of contaminated sediments in Oslo harbour, Norway[END_REF], to reduce the risk of flooding [START_REF] Hakstege | Strategies for management of contaminated sediments within the Meuse River-System[END_REF] or to increase outflow during dry periods [START_REF] Ellery | Environmental change over two decades since dredging and excavation on the lower Boro River, Okavango Delta, Botswana[END_REF]. Dredging is also a crucial operation for some engineering works [START_REF] Vale | Mobility of contaminants in relation to dredging operations in a mesodital estuary (Tagus estuary, Portugal)[END_REF]. Dredging can also be used as a remedial treatment to improve the environmental quality of the river ecosystem [START_REF] Besser | Assessment of sediment quality in dredged and undredged areas of the Trenton Channel of the Detroit River, Michigan USA, using the sediment quality triad[END_REF][START_REF] Winkels | Optimal cost-effective sampling for monitoring and dredging of contaminated sediments[END_REF][START_REF] Fioole | Surfis: a tool for designing and optimizing dredging schemes[END_REF]. Increasing pressure from environmental pollution and human disturbance leads to water quality deterioration. With the influx of polluted water, contaminants are absorbed onto suspended particles and subse-quently accumulated in the underlying sediment [START_REF] Hauge | Remediation of contaminated sediments in Oslo harbour, Norway[END_REF][START_REF] Lau | Contaminant release from sediments in a coastal wetland[END_REF].

Dredging methods vary depending on the volume, texture, water content and grain size of the sediments to be removed. Dredging technologies can be roughly divided into three main categories: mechanical, hydraulic and pneumatic dredges [START_REF] Pollice | Evaluation of available technologies for dredging and disposal of contaminated harbour sediments[END_REF][START_REF] Norman | Manutention et traitement des materiaux de dragage contam-ínes provenant des ports et voies navigables interieures ´'MDC[END_REF]. Mechanical dredges are used to remove bottom sediment through direct application of mechanical force. Once dredged, excavated sediment is generally placed into a barge for transportation to the disposal site. Such dredges can be considered as the most cost effective, especially when the dredging site is far from the disposal site, even if leakages of fine-grained material occur when the bucket is lifted from the river. Hydraulic dredges are all equipped with a 'head' that cuts or sweeps the material and conveys it to a hydraulic suction pipeline. The pipe either ends on a barge or is directly connected to a disposal facility on the shore. Typically, the amount of re-suspended sediment generated using hydraulic dredges is less than the amount generated by mechanical dredges [START_REF] Pollice | Evaluation of available technologies for dredging and disposal of contaminated harbour sediments[END_REF]. In fact, hydraulic dredges, such as cutterhead dredges, cause 2-5 mg l TSS y1 background concentrations, whereas mechanical dredges, such as clamshell dredges, cause TSS background concentrations ranging from 10 to 12 mg l [START_REF] Havis | Sediment resuspension by selected dredges[END_REF]. Pneumatic dredges use y1 compressed air instead of centrifugal pumps to remove slurry through a pipeline. They can pump material with a relatively high solids content with little generation of turbidity. The method that causes the least environmental impact on the water quality during dredging operations is the hydraulic method. However, mechanical processes allow the dredging of sediments containing high solids concentration, which is an important criteria for the performance of a dredging operation [START_REF] Hauge | Remediation of contaminated sediments in Oslo harbour, Norway[END_REF].

Once dredged, DM can be treated, valorised or disposed. Before any management decision, it is necessary to characterise the DM chemically, physically and mechanically [START_REF] Boutouil | Devenir des vases et boues de dragage: solidification et stabilisation[END_REF]. However, economic and political forces, and management, (i.e. lack of disposal sites) can interfere with the decision following the characterisation of DM. When contamination is too high to allow reuse, DM can either be disposed in controlled storage sites or treated. Disposal sites are of two types [START_REF] Perrin | La gestion des produits de dragage en provenance des voies d'eau interieures[END_REF]:

• terrestrial disposal sites where DM are dumped mechanically or hydraulically towards basins or natural cavities and • under water disposal sites such as disused quarries, deep zones or some river meanders.

Recently, the number of available disposal sites has rapidly declined due to urbanisation (Austin, 1995;[START_REF] Michelsen | Integration of sediment cleanup waterfront development, and habitat improvements through comprehensive port planning[END_REF] and because of the large amounts of sediment that have to be dredged [START_REF] Cuypers | Characterisation of PAH-contaminated sediments in a remediation perspective[END_REF]. In many cases, dredging for navigation and development projects has been hindered due to the presence of contaminated sediments and a lack of disposal sites [START_REF] Michelsen | Integration of sediment cleanup waterfront development, and habitat improvements through comprehensive port planning[END_REF].

Material and methods

Dredging in the Seine basin

The Seine river basin (France) study area includes the rivers Marne, Oise, Yonne and Seine, until Rouen, and covers an area of 68840 km . In 2 the Seine basin, mechanical dredging from a boat was chosen to remove, on average, 150 000 m of 3 sediment per year (Fig. 1). Dredging in the Seine basin is mainly related to navigation. Consequently, dredging operations are nearly continuous along the rivers. As a result, hydraulic dredges cannot be used in this case since they need a deposit site within 2.5 km. Obviously, such a configuration does not exist all along the Seine river system. Therefore, mechanical dredging is, in the case of the Seine basin, the most adapted option.

Within the 6 million m DM extracted in France 3 (internal waters only), 70% are dumped into deposit sites or along the riverside, 10% are valorised as embankments and 7% are spread on agricultural land. The rest is either used as an embankment material, dumped into the sea or treated. Since the dredging operations in the Seine basin occurs in urban areas, it is difficult to dump the DM on the riverside, as is currently performed in the north of France for example. Thus, DM is presently dumped at wet disposal sites downstream from Paris. From 1992 to 1998, the site at Triel and from 1999 to 2001, the site at Rouillard (Verneuil sur Seine) have been filled with the DM extracted from the River Oise, downstream from Compiegne, and from the River Seine downstream from Vitry and until Vernon (Fig. 1). It is of great importance to check the quality of the DM before it is dumped, in order to provide its efficient management. This is presented in this paper. As far as micro-pollutants are concerned, heavy metals and organic pollutants such as PAHs, PCB, pesticides etc. will be presented in a further publication.

Sediment collection

Samples were collected from the bottom of the Seine and Oise rivers (Fig. 1) by core sampler (20-50 cm depth, depending on the sediment structure) in order to assess the sediment quality before dredging. Sixty four samples were collected in May and June 1996 (31 samples), April 99 (18 samples) and May 2000 (15 samples) in strategic areas determined by the Navigation Services of the Seine basin, according to the dredging needs the following year, resulting from the bathymetric data of the river. Once cored, sediments were transferred into fully filled closed glass containers (1l ) and were kept for 48 h at 4 8C before homogenisation and analysis.

Methods

The parameters analysed on the sediment are listed in Table 1, together with the parameters measured in the pore-water of the same sediment samples. Pore-water was obtained after centrifugation of a known amount of sediment (3400 rpm at 4 8C, during 20 min) in order to obtain a sufficient volume to carry out analyses. The supernatant was then filtrated through a 0.45-mm pore filter to remove any remaining suspended solids. Most of the parameters were measured following the French AFNOR standard procedures (Table 1). All samples were analysed by laboratories certified by the French Ministry of the Environment, i.e. certified by the COFRAC (comite francais d'ac-´çreditation). For international quality control, the ĆOFRAC calibration certificates co-operate with other European calibration services (the European co-operation for Accreditation-EA). The harmonisation of the accreditation criteria and procedures are orientated towards the European standards for calibration and testing laboratories, and their assessment and accreditation (EN ISOyIEC 17025 and EN 45 000 series).

Results and discussion

For economic reasons, the Navigation Services of the Seine basin, in charge with the sampling and analysis, divides the Seine basin sites to be checked into three groups, and all of them are analysed within 3 years of sampling (here, 1996, 1999 and 2000). However, for a given year, the sampling sites are randomly defined, hence temporal evolution will not be taken into account in this paper.

Sediment quality

French regulations are not clear about the management of dredged materials extracted from rivers. There is presently no available limit that can be used for deciding the fate of DM. This type of regulation is only available for marine and estuarine sediments (French bylaw of the 14y06y2000). Consequently, waste regulations andyor sewage sludge regulations are often used as a reference, as well as the Dutch sediment decree (circular on target values and intervention values for soil remediation, Ministry of Housing, Spatial Planning and Environment, updated 4th February, 2000). The general characteristics of the sediment samples (Table 2) vary significantly from one sample to the other and demonstrate the great variability of the sediment characteristics all along the Seine basin. This makes the management of the DM quite difficult. The main variation is observed on the grain size, which shows that the percentage of fine particles (-50 mm) presents a coefficient of variation of 106% (Table 3). In fact, some areas show very sandy material (V01, V02, V15, V17, V23, S6, S8, S10, S11), whereas others contain a great proportion of silt (V22, V28, O10, 014, S12, S16). This grain size is in fact strongly linked to the organic content of the sediment. Fine sediments (grain size below 50 mm) are usually considered to adsorb organic and metallic pollutants more than coarse fractions [START_REF] Evans | The PAH and organic content of sediment particle size fractions[END_REF][START_REF] Warren | Suspended particulate grain size dynamics and their implications for trace metal sorption in the Don River[END_REF][START_REF] Perrin | La gestion des produits de dragage en provenance des voies d'eau interieures[END_REF]. Such large variability is also observed for organic matter, whether determined with loss on ignition (OM) or the total organic carbon (TOC).

On average, the density of the sediment is a reasonably stable value (average, 1.47; CV, 14%). We can therefore consider this density as a characteristic of the sediment present in the entire Seine basin studied.

Over the whole Seine basin, the sediment contains between 14 and 75% water. In terms of DM management, this type of material cannot always be treated as a typical urban or industrial waste. As far as waste is concerned, the French regulation of the 18 December, 1992, modified by the regulation of the 18 February, 1994, after the French law no. 75-633 about industrial waste, considers a waste as stabilised if its siccity (i.e. dry matter content) is above 35%. This is the case for most samples collected in the Seine basin except 12 from the 64 samples (Table 2).

If we consider previous studies performed in the Seine basin on suspended solids collected from December 1994 to March 1995 [START_REF] Meybeck | La Seine en son bassin, Fonctionnement ecologique d'un systeme fluvial anthropise[END_REF], 40-45% of the suspended matter collected by sediment traps were fine particles (-50 mm). However, the sampling method was completely different to the one performed in the present study (core sampling) where the average result found for grain size below 50 mm is 20%. For that reason, we shall presume that sediment trap samples represent the very superficial sediment of the Seine basin. This could explain why the fine particles percentages are much smaller when collected with a core sampler (20%). In a previous study carried out in 1992, Seine sediments downstream from Paris had an average fine particles content reaching 36% [START_REF] Garban | Exchanges at the sediment-water interface in the River Seine, downstream from Paris[END_REF]. However, this value was obtained with sediments arising from only three sites on the Seine downstream from Paris, whereas sediments analysed in the present study were sampled in 64 locations, all along the Seine basin including Marne, Oise, Yonne, and Seine upstream and downstream from Paris, hence over a much larger area, presenting a large variability potential (CVs106%).

When organic matter is assessed, former studies carried out on the Seine river downstream from Paris, show a loss on ignition in sediments collected in 1992 reaching 11% [START_REF] Garban | Exchanges at the sediment-water interface in the River Seine, downstream from Paris[END_REF]. The organic layer of the Paris combined sewer system, which is resuspended and discharged into the Seine river during storms (combined sewer overflows CSO) shows loss on ignition values reaching 60% [START_REF] Ahyerre | Bilans et mecanismes de migration de la pollution órganique en reseau d'assainissement unitaire. PhD thesis Úniv Paris 6[END_REF]. Total suspended solids of the Seine river downstream from CSO outlets and Paris, present loss on ignition of approximately 20% [START_REF] Estebe | Impact de l'agglomeration parisienne et de ses rejets `de temps de pluis sur les concentrations en metaux des ḿatieres en suspension et des sediments en Seine en periode `é stivale[END_REF]. In the present study, the sediments collected in 1996, 1999 and 2000 show an average value of 8%. OM values (loss on ignition) in the Seine basin seem to display average to low contents compared to the different sources of urban organic matter.

In the North of France, Ruban et al. (1998) found high concentrations of total organic carbon (TOC average, 83 g kg ) in the sediments of y1 rivers from a very industrialised region of Lille [START_REF] Ruban | Migration of micropollutants in a dredging amended soil in Northern France[END_REF], which is twice as much as the average value found in the Seine basin. [START_REF] Stegmann | Biogas production in dredged material[END_REF], found concentrations ranging from 160 to 200 g kg in dredged y1 sediments and sediment sampled directly from the bottom of the Elbe river in Germany. These comparisons tend to indicate that the sediment of the River Seine basin present concentrations a lot lower than those found in heavily polluted areas, such as the north of France or the River Elbe in Germany. The organic matter content being frequently linked to the micropollutant contamination, the Seine river basin DM should not present a great threat as long as their management is concerned. Nevertheless, it is of considerable importance to determine how polluted the same sediments are as far as micropollutants are con- cerned. Moreover, checking the properties of the pore-waters of such sediments is of great importance for the management of DM, since this material is bound to undergo important physical and chemical changes during its dredging, transport and dumping, and therefore the pore-water is likely to be released in the overlying or underground water.

Pore-water quality

Thirty-one samples of pore-water were also analysed in the sediments collected during the 1996, 1999 and 2000 campaigns (Table 4 andTable 5). As far as the organic content is concerned, the variability observed in the sediment is also found in the pore-water. The average TOC Table 5 Number of samples (n), median (d50), first and last decile (d10, d90), minimum and maximum values (min, max), mean, standard deviation (S.D.) and coefficient of variation in % (CV) of the sediment pore-water characteristics when sampled in the Seine basin in 1996, 1999 and 2000 content of the pore water is 83 mg l (CVs y1 100%) when the River Seine water, downstream from Paris, showed an average TOC of 7 mg l y1 [START_REF] Seidl | Organic matter transport and degradation in the River Seine (France) after a combined sewer overflow[END_REF]. The most striking parameter, which characterises the pore-water, is the ammonium ion. Our results concerning the Seine river basin show relatively high concentrations (61 mg l , on average). The ammonium values obtained pore-waters due to nitrification of organic nitrogen. The consequence of high concentrations of NH q 4 in pore-waters is a release of NH when sediments q 4 are re-suspended [START_REF] Stumm | Aquatic chemistry, an introduction emphasizing chemical equilibria in natural waters[END_REF]. Moreover, it has been found that mortality of C. Fluminea juveniles in river sediments was more closely associated with the presence of unionised ammonia than with the levels of metals, pesticides and PCBs. Similar results were reported for the bivalve Anodonta Imbecillis [START_REF] Cataldo | Environmental toxicity assessment en the Parana river delta (Argentina): simultaneous evaluation of selected pollutants and mortality rates of Corbicula fluminea (Bivalvia) early juveniles[END_REF].

TOC COD BOD 5 NO y 3 NO y 2 NH q 4 TKN P (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg
As the management of DM in the Seine basin involves re-suspension of the sediment with large quantities of Seine water (9y1v yv), this method dilutes the NH contained in the pore-water in the 4 wet disposal site and should cause ammonium contamination of the waters. This ammonium input might cause adverse effects to the overlying waters in the disposal site itself, to the underground waters and to receiving waters if there is a discharge system in the disposal site. This phenomenon has been observed in the case of a disposal site of the River Seine basin used from 1999 to 2001 [START_REF] Carpentier | Fate of dredged ćontaminated sediments in an underwater disposal site[END_REF].

Geographic distribution and correlation analysis

Geographic distribution with the different results (TOC, density, humidity, fine particles) obtained for the sediment quality before dredging were determined using Mapinfo Geographic Informa-᭨ tion System. The Seine basin maps did not reveal any significant effect of the geographic position. Indeed, the fact that some sediments were sampled upstream Paris did not prevent them from being rich in organic matter compared to other samples collected downstream from Paris, as we should expect considering the anthropic effect suggested by this heavily industrialised and urbanised area. Nevertheless, we found (Fig. 2) that sediment organic matter as expressed by loss on ignition, as well as other parameters related to the organic characteristics of the sediment, (i.e. TOC), was more abundant directly downstream from Paris, until the crossing of Seine and Oise rivers, where polluting activities are concentrated. In fact, the impact of Paris conurbation, i.e. the opposition between upstream of and downstream from Paris, leading to the statement that sediments downstream from Paris should be more contaminated Fine particles (% -50 mm). a than those upstream of Paris, was not observed and did not explain the variability of the organic characteristics of the sediment.

In order to assess the possible role of one or several parameters to control other parameters, linear correlation analysis was performed. Correlation analysis with sediment general parameters (Table 6) did not reveal any major trend, apart from very usual ones such as a negative correlation between density and humidity (rsy0.86) or density and TOC (rsy0.70) or a positive correlation between humidity and OM (rs0.69). Surprisingly, we observed weak correlation between TOC and OM (rs0.36), or OM assessed by loss on ignition and fine particles content (rs0.51).

Concerning the sediment pore-waters (Table 6), usual correlations were found between BOD and 5 COD (rs0.95), NH and TKN (rs0.99),o r q 4 between TOC and COD (rs0.99). This type of correlation should probably be more informative when physico-chemical parameters and micropollutants data are simultaneously taken into account (in a further publication).

The combined linear correlation of sediment parameters and pore-water parameters (Table 6) gave high coefficients for TOC in the sediment and NO observed in the pore waters (rs0.71). y 3 This correlation could be related to the biological activity that links these parameters (nitrifying bacteria may be more abundant in more organic media).

Conclusions

During three campaigns performed in 1996, 1999 and 2000, sediment samples were collected and analysed before dredging in order to assess their major composition, in the Seine river basin, upstream and downstream from Paris. This quality is important in the case of the Seine basin, as the sediments are dumped into disused quarries. Hence, the composition of these sediments was estimated in order to assess their possible impact after disposal. As far as the particulate matter is concerned, compared to other industrialised river basins, the sediments are not heavily loaded in organic matter. Pore-waters contain high concentrations of ammonium ions, compared to other river basins, that might lead to a management concern, because the dumping of such sediment might result in an increase of ammonium concentration in the receiving waters, i.e. disused quarries, underground water or Seine water. At the disposal sites, where the water is stagnant, high inputs of ammonium ion, as well as phosphorus inputs, could lead to eutrophication concern. Such an impact was in fact observed on the DM disposal site of the River Seine basin, i.e. the Rouillard pond [START_REF] Carpentier | Fate of dredged ćontaminated sediments in an underwater disposal site[END_REF]. Therefore, the analysis of ammonium in the sediment pore-waters should be considered as an important parameter to assess the quality of sediments before dredging. However, the main characteristic of the sediment samples is its wide variability from one site to the other. Consequently, the main outcome of this work is that the sediment collection and analysis in each dredged area should be performed before dredging, as often as possible. Indeed, along the Seine basin, the river sediment shows a wide range of properties due to point and non-point sources, but also due to hydrology or navigation, which one is unable to control. Depending upon such parameters, the amount of organic matter and nitrogen of sediment samples, different management procedures for DM could be decided, i.e. dumping in wet disposals, separating fine particles from coarse ones, so the fine ones can be treated, and the coarse ones valorised. Such a procedure might be necessary when national or European regulations appear in relation to the European Framework Directive on water management.

Fig. 1 .

 1 Fig. 1. Part of the Seine basin concerned in this study and site codes corresponding to the sediment samples collected during 1996, 1999 and 2000 campaigns.

y1

  in a previous study carried out on sediment samples from the Seine basin cored in 1992, gave average values reaching 55 mg l (Garban et al., y1 1995), which fits within the same concentration range as the results obtained in the present study. Indeed, NH accumulates in anoxic sediment q 4

Fig. 2 .

 2 Fig. 2. Organic matter contents (loss on ignition, %) in the sediment collected along the Seine basin during 1996, 1999 and 2000 campaigns.

Table 2

 2 General characteristics of the Seine basin sediment collected in 1996, 1999 and 2000, before the dredging operation

	Site	Year of	TOC	Density	Humidity	OM	Fine
	code	collection	(gk g ) y1		(%)( %)	particles
							(% -50 mm)
	V01	1996		1.68	27	2	1
	V02	1996		1.66	28	2	3
	V03	1996		1.19	70	5	21
	V04	1996		1.66	28	3	3
	V05	1996		1.5	40	5	8
	V06	1996		1.51	39	13	4
	V07	1996		1.5	40	8	16
	V08	1996		1.32	56	6	9
	V09	1996		1.59	33	3	5
	V10	1996		1.47	60	8
	V11	1996		1.38	50	8	10
	V12	1996		1.63	30	2	5
	V13	1996		1.77	21	1	4
	V14	1996		1.91	14	1	1
	V15	1996		1.88	15	1	1
	V16	1996		1.48	42	4	15
	V17	1996		1.89	15	1	2
	V18	1996		1.7	26	2	9
	V19	1996		1.55	36	4	12
	V20	1996		1.57	34	3	14
	V21	1996		1.35	53	9	5
	V22	1996		1.22	68	11	60
	V23	1996		1.76	22	2	3
	V24	1996		1.41	48	5	21
	V25	1996		1.44	45	7	13
	V26	1996		1.55	36	4	53
	V27	1996		1.58	33	3	47
	V28	1996		1.38	50	6	51
	V29	1996		1.46	43	4	28
	V30	1996		1.39	49	5	29
	V31	1996		1.3	58	8	64
	S1	1999	60		64	10	8
	S2	1999	47		63	10	9
	S3	1999	21		61	10	17
	S4	1999	25		61	12	4
	S5	1999	35		67	10	13
	S6	1999	33		44	6	2
	S7	1999	57		68	16	11
	S8	1999	81		53	6	3
	S9	1999	29		62	12	8
	O1	1999	12		51	6	16
	O2	1999	14		37	4	7
	O3	1999	34		65	10	24
	O4	1999	32		64	10	18
	O5	1999	37		69	11	20
	O6	1999	28		49	6	11
	O7	1999	28		64	10	12
	O8	1999	64		69	12	13
	O9	1999	37		71	12	23
	S10	2000	73	1.26	61	23	1
	S11	2000	38	1.04	19	22	1
	S12	2000	50	1.16	75	25	74

Table 4

 4 General characteristics of the pore-water from the sediment collected in the Seine basin in 1996, 1999, 2000 before the dredging operation

	Site	TOC	COD	BOD	NO y 3	NO y 2	NH q 4	TKN	P
	code	(mg l )							y1
	S1	65
	S2	296
	S3	174
	S4	74
	S5	52
	S6	47
	S7	390
	S8	57
	S9	139
	V1		246	83	-1	0.06	128	104	1.4
	V2		199	39	-1	0.06	146	119	1.5
	V3		-30	nd	8	-0.02	1.2	-2	0.3
	V4		31	-3	1.4	0.07	23	19	0.6
	V5		-30	-3	4	2.57	-0.03	-2	1.2
	V6		-30	-3	11	0.28	9	8	1.2
	V7		-30	3	14	1.71	2.8	2.6	0.8
	V8		156	27	-1	0.04	144	121	2.1
	O1	71.5
	O2	54
	O3	87
	O4	42
	O5	75
	O6	72
	O7	74
	O8	66
	O9	53
	S10	32.5	80		1.8	0.19	65	50	1.2
	S12	245	661		-1	-0.02	93	73	0.9
	S14	49	105		1.2	0.04	136	110	0.9
	S15	39	83		-1	0.17	76	62	0.5
	S16	58	144		1.2	0.03	185	148	1.4
	O10	25	104		-1	0.12	6.6	7.7	0.9
	O11	34.5	73		-1	0.05	28	22	0.9
	O12	36	82		-1	0.13	32	27	1.2
	O13	30	74		-1	0.11	44	48	1.7
	O14	37	91		1.3	0.14	55	43	1.6
	O15	31	76		-1	0.04	18	18	1
	O16	43	111		1.4	0.21	53	46	1.3
	O17	37	74		-1	0.14	42.5	32	1.5

y1 (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg l ) y1 (mg l )

  l )

									y1
	n	31	21	8	21	21	21	21	21
	d10	33	-30	-3	-1	0.04	2.8	2.6	0.6
	d50	54	82	3	1.4	0.12	44.3	43	1.2
	d90	174	199	56	10.4	0.42	144	119	1.6
	Min	25	-30	-3	-1	-0.02	-0.03	-2	0.3
	Max	390	661	83	14	2.6	185	148	2.1
	Mean	83	112	22	3.5	0.3	61	51	1.1
	S.D.	84	141	31	4.6	0.6	56	45	0.4
	CV (%)	101	124	144	131	209	91	90	37

Table 6

 6 Linear correlation coefficient (r) between parameters analysed in the Seine basin sediments (S) and in their pore-waters (P) in1996,

	1999 and 2000
		TOC	Density Humidity OM	F.p. a	TOC	COD	BOD 5 NO y 3	NO y 2	NH q 4	NTK P
		(S)( S)( S)( S)( S)( P)( P)( P)( P)( P)( P)( P)( P)
	TOC	1
	(S)
	Density	y0.70	1
	(S)
	Humidity	0.44 y0.86	1
	(S)
	OM	0.36 y0.77	0.69	1
	(S)
	F.p. a	y0.03 y0.56	0.53	0.51	1
	(S)
	TOC	0.20 y0.45	0.31	0.01	0.10	1
	(P)
	COD	0.25 y0.44	0.55	0.39	0.50	0.99	1
	(P)
	BOD 5	nd	nd	0.75	0.64	0.49	nd	0.95	1
	(P)
	NO y 3	0.71 y0.33	y0.65	y0.64 y0.54 y0.16 y0.31 y0.56	1
	(P)
	NO y 2	0.33	0.05	y0.52	y0.48 y0.43 y0.47 y0.26 y0.38	0.50	1
	(P)
	NH q 4	0.45 y0.69	0.74	0.49	0.59	0.30	0.44	0.79 y0.45 y0.39 1
	(P)
	NTK	0.43 y0.68	0.74	0.49	0.59	0.29	0.43	0.78 y0.46 y0.39 0.99	1
	(P)
	P	y0.03	0.18	0.34	0.15	0.33 y0.21	0.09	0.50 y0.31 y0.07 0.45	0.47 1
	(P)

Acknowledgments

The authors would like to thank the teams of the Service de Navigation de la Seine (SNS) and of Entreprise Morillon Corvol Courbot (EMCC) for their human, technical and financial help as well as the Piren-Seine program (GDR, CNRS) for financial support.


Wael Zouaoui
email: wzouaoui@laas.fr

Yann Labit
email: ylabit@laas.fr

Carolina Albea
email: calbea@laas.fr

Buffer dynamic management for energy-aware network

Keywords: Green Network, Adaptive system, Quality of Service, Energy, ADREAM, Adaptive Link Rate

Energy is a major issue in networks especially in local network implementation. Some technological solutions have been developed to treat this problem e.g. Re-engineering, Dynamic Adaptation, Smart Sleeping. We present an original and alternative adaptive solution from a proposed extended Adaptive Link Rate (ALR) module, which relates energy and router queuelength. Applying a controller already proposed by authors, the router queue-length is adapted in order to follow a reference given by the extended ALR module, that guarantees an energy reduction. The approach is validated in the network simulator "NS-2". Within ADREAM building at LAAS-CNRS, this "control green" will allow to improve energy consumption regarding traffic load and available energy in communication networks.

INTRODUCTION

Since few years, the number of communication system users has significantly increased, thereby energy consumption have also increased. According to [START_REF] Koomey | Worldwide electricity used in data centers[END_REF], the number of Internet users reached 2.4 billion people in 2012. Consequently, the energy consumption in communication networks represents 5.5% of annual world electricity production and a rate of 2% of carbon dioxide emissions (CO 2 ). The perspective for 2015, with probably 2.8 billion of Internet users, is to achieve a traffic 4 times larger and to double energy consumption (approximately 25% larger per year) [START_REF] Zhang | Effects of cloud horizontal inhomogeneity and drizzle on remote sensing of cloud droplet effective radius: Case studies based on large-eddy simulations[END_REF]. This issue requires to research new ways to reduce energy consumption.

In wired networks, energy reduction often diminishes Quality of Service (QoS). This performance, energy reduction trade-off requires to define an efficient strategy to decrease the network energy consumption, what is a real challenge. Some interesting hardware and protocol solutions have been proposed to reduce energy consumption in communication networks. In [START_REF] Bolla | Energy efficiency in the future internet: A survey of existing approaches and trends in energy-aware fixed network infrastructures[END_REF], the author's survey mention some techniques which aim is to improve energy efficiency of computing and network resources. They classify these methods as follows: a) Re-Engineering: This approach aims at modifying and re-engineering network architectures to use more energyefficient elements (green routers, switches ...). This solution is complex and expensive due to fundamental changes in network [START_REF] Baldi | Time for a "greener" internet[END_REF], [START_REF] Noguera | Power reduction in network equipment through adaptive partial reconfiguration[END_REF], [START_REF] Roberts | A radical new router[END_REF], [START_REF] Gupta | Using low-power modes for energy conservation in ethernet lans[END_REF].

b) Dynamic Adaptation: This solution modulates network resources capacities based on varying traffic loads and service requirements. "Performance Scaling" and "Idle Logic" use this approach with the goal of reducing energy consumption in network [START_REF] Yamada | Power efficient approach and performance control for routers[END_REF], [START_REF] Zhang | Realtime performance analysis of adaptive link rate[END_REF], [START_REF] Nedevschi | Reducing network energy consumption via sleeping and rateadaptation[END_REF], [START_REF] Gunaratne | Reducing the energy consumption of ethernet with adaptive link rate (alr)[END_REF]. c) Smart Sleeping: Energy consumption is reduced by switching off some network equipment when they are inactive, and to smartly wake them up only if necessary. There are two kinds of smart sleeping: "on/off and sleeping technology", to switch off the inactive or idle network components [START_REF] Hu | On the design of green reconfigurable router toward energy efficient internet[END_REF], [START_REF] Christensen | Ieee 802.3az: the road to energy efficient ethernet[END_REF]; and "proxying technology" that uses a proxy to answer to nonurgent messages instead of employing a node, which is wakedup when required [START_REF] Agarwal | Sleepserver: A software-only approach for reducing the energy consumption of pcs within enterprise environments[END_REF], [START_REF] Sabhanatarajan | Smart-nics: Power proxying for reduced power consumption in network edge devices[END_REF].

In the industry domain, some important companies in the sector have already started a "green revolution" (development of "green" network equipments). Among these companies, we cite the following ones: Cisco (EnergyWise) that focuses on reducing power utilization of all devices connected to a network [START_REF][END_REF]; Alcatel-Lucent (GreenTouch) based on all-optical networking systems with the goal to master the technology for future green networks [START_REF]The green touch initiative[END_REF]; IBM (Big-Green), which goal is to reduce data centers' energy consumption [START_REF]The big-green project[END_REF]; ICT4EE (Information and Communication Technologies for Energy Efficiency) forum (GeSI, DigitalEurope, JBCE and Europe TechAmerica), whose objective is the enhancement of energy efficiency in ICT systems [START_REF]ICT4EE. The ict for energy efficiency forum[END_REF]; NetGear that offers a wide range of innovative routers and energy-aware switches [START_REF]Netgear green[END_REF] and Hewlett-Packard, Intel, Microsoft, Phoenix Technologies, Toshiba (Advanced Configuration and Power Interface) that provides an open standard for device configuration and power management by the Operating System [START_REF] Acpi | Advanced configuration and power interface[END_REF].

In this paper, an extended ALR module is proposed to show the existing relationship between energy and router queuelength. Then, we apply a control algorithm developed by us [START_REF] Zouaoui | Dynamic adaptive management of buffer for QoS guarantees and energy reduction[END_REF] to a router. This control adapts the traffic that crosses a node by using length-queue where packets are temporarily stored. The adapted length-queue converges to the suboptimal reference given by ALR module, which guarantees an energy reduction. The choice of the selected nodes is a problem in itself, and it is not the focus of this work. The proposed solution is validated in NS-2 [START_REF] Fall | The ns manual," notes and documentation on the software ns2-simulator[END_REF] and exploited in ADREAM (Architectures for Dynamic Resilient Embedded Autonomous and Mobile systems) project (building devoted to cyber-physical systems at LAAS-CNRS, figure 1). We consider two functioning modes (figure 2): -Normal mode: During a daytime, the system administrator ensures the proper functioning of the communication system, the introduction of new materials, computer security and daily network administration. With no software or hardware anomaly, the administrator faces a normal operating mode.

-The energy aware mode: to make an intelligent management computer system (from an energy point of view), the administrator programs the energy aware mode for a sensible time interval (night) and/or in cooperation with users (e.g. time slot 12H-14H). This mode of operation is a degraded mode (e.g. trade off between energy saving and quality of service parameters such as transmission delay information). Likewise, energy consumption measures are given by ECOFEN [START_REF] Orgerie | Energy-efficient bandwidth reservation for bulk data transfers in dedicated wired networks[END_REF] (implemented module in NS-2).

I. ALR MODEL AND ECOFEN MODULE

Firstly, lets model the energy consumption of the network equipment. For this purpose, we consider a router model designed in NS-2. NS-2 is currently one of the most widely used simulators in the network community [START_REF] Fall | The ns manual," notes and documentation on the software ns2-simulator[END_REF]. This simulator allows us to simulate large networks with realistic traffic and to obtain QoS measures.

The ECOFEN module (Energy Consumption Model for End-to-end Networks) was developed in NS-2. This module provides the instantaneous energy consumption of each equipment taking into account the traffic and the type of employed equipment (router, switch, network card, etc.) [START_REF] Orgerie | Energy-efficient bandwidth reservation for bulk data transfers in dedicated wired networks[END_REF]. This module is based on the ALR model (Adaptive Link Rate) which adapts transmission rates depending on link use-rate.

Lets take previous model for studying energy consumption in a router. For this issue, we adapt input traffic, w n , of the controlled node. We artificially and temporarily delay a certain number of packets in a buffer to a pre-defined value q ref . The length-queue reference q ref is given according to the input traffic in the router. This selection may provide some problems in network QoS, in terms of "delay". In other words, there is an energy consumption reduction and transmission delay tradeoff. From the ECOFEN model given in [START_REF] Orgerie | Energy-efficient bandwidth reservation for bulk data transfers in dedicated wired networks[END_REF], the new extended model is

P =                      P0 if wn = 0 a1wn + P1 -bq ref if wn ∈]0; W1] . . . aiwn + (Pi -aiWi-1) -bq ref if wn ∈]Wi-1; Wi] . . . a N wn + (Pn -a N W N -1 ) -bq ref if wn ∈]W N -1 ; W N ] (1)
with :

• a i ∈ R + : the slope of the different linear section.

• w n ∈ R + : the input traffic of each network element.

• W i ∈ R + : the use-rate thresholds.

• P i ∈ R + : the start power of each different stage.

• q ref ∈ R + : the queue length reference.

• b ∈ R + : a coefficient of the queue length reference q ref .

Figure 3 shows the modified power model for a router. In this model, the input traffic depends on the consumed energy by the network element. Therefore, a reliable adaptation of the queue length reference is mandatory in order to reduce energy consumption with QoS guarantees. This selection of q ref depends on two key parameters:

• the update time-window for the queue length reference.

T q ref = T w β , (2)
where T w is the time-window related to the node input traffic. β is the number of changes of the value of q ref during the period T w . This allows to update more or less frequently q ref depending on the traffic variability.

• The queue length reference

q ref = ŵn γ , ( 3
)
where ŵn is the average input traffic on a fixed time-window T w . γ is a step (weighting coefficient) for adapting the amplitude of the reference function w n .

Remark. The judicious choice of the average input traffic ŵn , on a fixed time window T w , take into account the high variability of this traffic.

II. ANALYTIC & SIMULATION RESULTS

The proposed control law in discrete-time is:

u k = sat C 0 {u k-1 + Kx k }, (4)
where

K = [K 1 K 2 ]
are control gains computed following the mechanism proposed in [START_REF] Zouaoui | Dynamic adaptive management of buffer for QoS guarantees and energy reduction[END_REF].

x k = [e k e k-1 ] T
, where e k is the error between queue-length reference and queue-length in time-period k. The high-bound saturation C depends on network capacity. We choose: C = 25000pkts/s, T s = 0.05s.

From [START_REF] Zouaoui | Dynamic adaptive management of buffer for QoS guarantees and energy reduction[END_REF], it comes that: K 1 = -21 and K 2 = 20. To validate this approach, we take a simple topology composed of two nodes and one source, as shown in Figure 4. In !"#$"%" After choosing the couple (β, γ), some analog simulations of the node 4 are performed here. In Figure 6, the evolution of the input traffic similar to the traffic rate in Figure 4 and the output traffic calculated from the control law are presented. Likewise, it is shown in Figure 7, the queue length evolution. Note that the network traffic is unsaturated, so losses are avoided. Figure 8 (up) shows the power consumption with and without control law. Note, there is a significant improvement in power consumption from 9 : 30h to 11 : 30h and from 14 : 30h to 18 : 30h which represents 60% of the total simulation time (20h). During this time period, packets are stored in the queue (with size q ref ), the output traffic is smaller than the input traffic. Moreover, power is smaller than a power threshold, hence the reduction of the energy consumption. Given the ALR model, if the output traffic is smaller than a threshold W i (see Figure 3), the consumed energy will be smaller than without control. On the other hand, for the others periods of time, queue length releases packets already stored. Hence, the output traffic is larger than the input traffic, what leads to a slight energy overconsumption, the difference is less than the reduction. Note that the minimum power consumption of the ALR model is P 0 = 155W (Figure 8). This value is obtained when there is not traffic crossing the controlled node (w n = 0 M b/s). This is a minimum power consumption of the operational node without traffic.

β = 1 β = 2 β = 3 β = 4 β = 5 β = 6 β = 7 β = 8
In Figure 8 (bottom), the two phases of energy consumption (reduced and increased energy consumption) are easily seen. In this simulation, it is shown that the global energy consumption for a time period of 20h is reduced 16.49% (energy gain) in the ALR model. The ratio between reduced and increased energy consumption is up to 60%.

III. CONCLUSION

In this article, an alternative adaptive solution for reducing energy consumption in networks has been presented. This solution is developed from a proposed ALR that relates router energy and queue-length. By using a control protocol developed by us, we achieve to adapt the router queue-length in such a way that it converges to the reference given by the extended ALR model, and thus, energy is reduced. Simulations in and NS-2 validate our algorithm. The shown energy gain let to call our algorithm "control green".

Fig. 1 .

 1 Fig. 1. ADREAM building at LAAS-CNRS.

Fig. 2 .

 2 Fig. 2. The two modes (normal and degraded).

Fig. 3 .

 3 Fig. 3. Modified ALR model of power consumption in a network element (router model).

Figure 4 ,

 4 we show the traffic rate crossing a switch located in the ADREAM building of LAAS-CNRS in the daytime. Several simulations with different values of β and γ are done to select the best couple that gives the maximum of power gain (G W ) and maximum ratio of time that power gain is minimized (T G W ). Taking 21 values for γ from 500 to 1500 with a pitch of 50. For each value of γ, we assign 8 values for the update window β ∈ [1, 8]. In Figure 5, we present the power gain and the ratio of time when we have power gain for different values of β function of γ. It is clearly seen the influence of theses parameters on energy consumption. The best value of G W and T G W are achieved with the couple (β, γ) = (8, 850).

Fig. 4 .

 4 Fig. 4. ADREAM switch traffic

Fig. 5 .

 5 Fig. 5. Power gains (up) and Time power gain (bottom) with differents values of β and γ

Fig. 6 .

 6 Fig. 6. Input (red) and output (blue) traffic

Fig. 7 .

 7 Fig. 7. Queue length evolution (blue) and queue length reference (red).

Fig. 8 .

 8 Fig. 8. Power consumption (up) with (blue) and without (red) flow control, on ALR model of a network node -Power gain (Bottom)).


Jean-François Bercher
email: jf.bercher@esiee.fr

On generalized Cramér-Rao inequalities, and an extension of the Shannon-Fisher-Gauss setting

Keywords: Cramér-Rao inequality, Fisher information, generalized q-entropy, generalized Gaussians, de Bruijn identity

We begin with two possible extensions of Stam's inequality and of de Bruijn's identity. In both cases, a generalized q-Gaussian plays the same role as the standard Gaussian in the classical case. These generalized q-Gaussians are important in several areas of physics and mathematics. A generalized Fisher information also pops up, playing the same role as the classical Fisher information, but for the extended identity and inequality.

In the estimation theory context, we give several extensions of the Cramér-Rao inequality in the multivariate case, with matrix versions as well as versions for general norms. We define new forms of Fisher information, that reduce to the classical one in special cases.

In the case of a translation parameter, the general Cramér-Rao inequalities lead to an inequality for distributions, which involves the same generalized Fisher information as in the generalized de Bruijn's identity and Stam's inequality. This Cramér-Rao inequality is saturated by generalized q-Gaussian distributions. This shows that the generalized q-Gaussians also minimize the generalized Fisher information among distributions with a fixed moment. Similarly, the generalized q-Gaussians also minimize the generalized Fisher information among distributions with a given q-entropy.

Introduction

Classical information theoretic inequalities, as described for instance in [START_REF] Cover | Elements of Information Theory[END_REF][START_REF] Dembo | Information theoretic inequalities[END_REF], interrelate information measures. These inequalities have proved to be useful for communication theoretic problems and engineering applications. They are also connected to uncertainty relations in physics, and to functional inequalities in mathematics. Thus new, or refinements of existing information theoretic inequalities, could be fruitful in several fields. Even before recalling the exact expressions of all the involved information measures, we find useful to first briefly describe their main interrelations.

The Shannon-Fisher-Gauss setting -The Gaussian distribution is well known to play a central role with respect to classical information measures and inequalities. More precisely, in a particular setting, that we will call the Shannon-Fisher-Gauss setting, these three quantities are interrelated in many ways. For instance, the Gaussian distribution maximizes the entropy over all distributions with the same variance; see [START_REF] Dembo | Information theoretic inequalities[END_REF]Lemma 5]. This can be stated as a moment-entropy inequality [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF]:

m 2 [f ] 1 2 N [f ] ≥ m 2 [G] 1 2 N [G] , (1)
where m 2 [f ] is the second order moment, N [f ] is the entropy power, defined by

N [f ] := 1 2πe exp 2 n H 1 [f ]
, where H 1 [f ] is the Shannon entropy of f , and G denotes the standard Gaussian. Similarly, the Cramér-Rao inequality, see e.g. [START_REF] Dembo | Information theoretic inequalities[END_REF]Theorem 20], shows that the minimum of the Fisher information among all distributions with a given variance is attained for the Gaussian distribution:

I 2,1 [f ] m 2 [f ] ≥ I 2,1 [G] m 2 [G] = n. (2)
Stam's inequality [START_REF] Stam | Some inequalities satisfied by the quantities of information of Fisher and Shannon[END_REF] shows that the minimum of the Fisher information over all distributions with a given entropy also occurs for the Gaussian distribution:

I 2,1 [f ] N [f ] ≥ I 2,1 [G] N [G] (3)
while de Bruijn's identity states that Fisher information is related to the derivative of the entropy, with respect to the variance of an additive Gaussian perturbation: if X t = X + √ 2tN , and N ∼ G then

d dt H[f t ] = I 2,1 [f t ]. (4)
In these expressions, the involved Fisher information is the Fisher information associated to a location parameter model. We will give the general definition and further results in what follows.

Clearly, it is both important and attracting to ask whether some similar inequalities, involving generalized information measures exist. A natural option is to consider the Rényi entropy, or the Tsallis entropy, which depend on a real parameter q, instead of the Shannon entropy. Then the answer to the question is positive: a series of information theoretic inequalities does exist which leads to a q-entropy-q-Fisher-q-Gaussian setting. This setting involves not only generalized q-entropies, but also a generalized Gaussian distribution and a suitable extension of Fisher information. It is one of the objectives of this paper to give extensions of inequalities (1)-(4), thus defining the q-entropy-q-Fisher-q-Gaussian setting as an extension of the Shannon-Fisher-Gauss triplet. Let us first recall some definitions, notations, and introduce the generalized q-Gaussian distribution.

Regarding q-entropies and generalized q-Gaussian distribution -Let f (x) be a probability distribution defined on X⊆ R n . For q ≥ 0,

M q [f ] = ˆX f (x) q dx (5)
is the information generating function. With this notation,

S q [f ] = 1 1 -q (M q [f ] -1) (6)
is the Tsallis entropy, and

H q [f ] = 1 1 -q log M q [f ] (7)
the Rényi entropy. Both entropies reduce to the standard Shannon entropy

H 1 [f ] = -´X f (x) log f (x)dx for q = 1.
We will also note by N q [f ] the qentropy power. It is defined as an exponential of the Rényi entropy H q [f ] as

N q [f ] = M q [f ] 2 n 1 1-q = exp 2 n H q [f ] = ˆΩ f (x) q dx 2 n 1 1-q , ( 8
)
for q = 1. For q = 1, we set

N [f ] = N 1 [f ] = exp 2 n H 1 [f ]
. It is well-known that the maximum of the Rényi-Tsallis entropy, among all distributions with a fixed moment

m α = E g [ X α ] , is obtained for a general- ized q-Gaussian distribution with parameter γ G γ (x) ∝ (1 -(q -1) γ x α ) 1 q-1 + for q = 1 exp (-γ x α ) for q = 1. (9)
which is a generalized (or stretched) Gaussian for q = 1, and a standard Gaussian for q = 1, α = 2. This extends the classical result that the standard Gaussian maximizes Shannon entropy subject to a second order moment constraint. Note that γ -1 α is simply a scale parameter which allows in particular to fix the moment of order α. For instance, in the case of a standard Euclidean norm, γ = ((1 + α/n)q -1) m α . For q > 1, the generalized q-Gaussian distributions exhibit heavy tails, while they have compact support when q ≤ 1. The generalized q-Gaussians are used in nonextensive statistical physics, where they present significant agreement with experimental data. They also appear in problems involving non-linear diffusion equations (as we will see in the sequel) and as extremal functions of Sobolev, log-Sobolev or Gagliardo-Nirenberg inequalities. The fact that these distributions maximize the q-entropies can be seen as a consequence of a moment-entropy inequality

m α [f ] 1 2 N q [f ] ≥ m α [G] 1 2 N q [G] (10)
similar to [START_REF] Agueh | Sharp Gagliardo-Nirenberg inequalities via p-Laplacian type equations[END_REF], and where the lower bound is attained if f is any generalized q-Gaussian G γ as in [START_REF] Bobkov | Entropy power inequality for the Rényi entropy[END_REF]. This general inequality is due to Lutwak, Yang and Zhang [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF]. In [START_REF] Carrillo | Asymptotic L1-decay of solutions of the porous medium equation to self-similarity[END_REF], we dropped the index γ of the generalized q-Gaussian G γ to reflect the fact that the lower bound is scale invariant, i.e. does not depend on γ.

On Fisher information -The Fisher information I 2,1 [f ] involved in the Shannon-Fisher-Gauss setting (1)-( 4) is a measure of information attached to the distribution f. This Fisher information is used in inference and understanding in physics [START_REF] Frieden | from Fisher Information: A Unification[END_REF]. It is also used as a tool for characterizing complex signals or systems, with applications, e.g. [START_REF] Vignat | Analysis of signals in the Fisher-Shannon information plane[END_REF], [START_REF] Romera | The Fisher-Shannon information plane, an electron correlation tool[END_REF], [START_REF] Dehesa | Cramér-Rao information plane of orthogonal hypergeometric polynomials[END_REF]. Actually, the Fisher information has been originally introduced in the wider context of estimation theory. It measures the information about a parameter θ in a distribution, corresponds to the Hessian of the log-likelihood and defines, by the Cramér-Rao inequality, a fundamental lower bound on the variance of any estimator. let f (x; θ) be a probability density function defined over a subset X of R n , θ ∈ Θ ⊂ R, h(θ) a scalar valued function and η(θ) = E[T (X)]. Then, under some regularity conditions:

E |T (X) -h(θ)| 2 I 2,1 [f ; θ] ≥ ∂ ∂θ η(θ) 2 , (11)
with

I 2,1 [f, θ] = ˆX ∂ ln f (x; θ) ∂θ 2 f (x; θ)dx, (12)
and with equality iff

∂ ∂θ ln f (x; θ) = k(θ) (T (x) -h(θ)) .
The indexes in the notation I 2,1 [f, θ] are for possible extensions of the definition. The first index corresponds to the quadratic case. Indeed, the definition, here given as the second order moment of the derivative of the log-likelihood -the score function, can be extended to other moments, leading to a generalized version of the Cramér-Rao inequality. This extension, which does not seem to be well known, can be traced back to Barankin [3,Corollary 5.1] and Vajda [START_REF] Vajda | χ α -divergence and generalized Fisher information[END_REF]. The second index will correspond to an extension associated with q-entropies. In the case of a translation family, i.e. f (x; θ) = f (x -θ), that is if the parameter θ is a location parameter, the Fisher information, expressed at θ = 0, becomes a characteristic of information in the distribution

I 2,1 [f, θ = 0] = ˆX d ln f (x) dx 2 f (x)dx.
In this case, the Cramér-Rao inequality leads to inequality (2) for the moments of the distribution, where the equality is achieved by the Gaussian distribution. A part of this paper will be devoted to extensions of the Cramér-Rao inequality for the estimation of a multivariate parameter, in particular with variations on the way of averaging the estimation error, and on ways for handling the multivariate case. These new Cramér-Rao inequalities may prove useful in their own right. A second objective of this paper is to show that the classical Shannon-Fisher-Gauss setting can be extended to generalized q-entropies, to the generalized q-gaussian distribution, and a to suitable generalized Fisher information, that has an estimation theoretic foundation. These ideas have grown in a series of papers [START_REF] Bercher | On a (β,q)-generalized Fisher information and inequalities involving q-Gaussian distributions[END_REF], [START_REF] Bercher | On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized q-Gaussian distributions[END_REF], [START_REF] Bercher | On generalized Cramér-Rao inequalities, generalized Fisher informations and characterizations of generalized q-Gaussian distributions[END_REF], [START_REF] Bercher | Some properties of generalized Fisher information in the context of nonextensive thermostatistics[END_REF]. Of course, some of these results will be reviewed here. However, many proofs are revisited and the paper also provides several new results, in particular the matrix versions of the multivariate Cramér-Rao inequality and further results on the generalized Fisher information.

The rest of this paper is organized as follows. In section 2, we give two possible extensions of Stam's inequality (3) and of de Bruijn's identity (4) to the q-setting. In both cases, the generalized q-Gaussian (9) plays a key role; namely as a limit case or as the distribution which saturates the inequality. In these two relations, a generalized Fisher information also pops up, playing the same role as the classical Fisher information, but for the extended identity and inequality. In section 3, we turn to the extension of the Cramér-Rao inequality. We first describe the setting where we consider a moment of any order of the estimation error, computed with respect to an arbitrary distribution. In section 3.1, we obtain a matrix version of the Cramér-Rao inequality, in the multivariate case, where the Fisher information matrix is defined implicitly as a function of a generalized score function. Special cases are examined. In section 3.2 we also give other extensions of the Cramér-Rao inequality for general norms of the estimation error. In the special case of a translation parameter and using the notion of escort distributions, these generalized Cramér-Rao inequalities lead to an extension of the Cramér-Rao inequality (2), saturated by generalized q-Gaussian distributions. Finally, in section 4, we discuss the main contributions and describe several directions for future work.

Extensions of de Bruijn's identity and of Stam's inequality

As mentioned above, a fundamental connection between the Shannon entropy, Fisher information, and the Gaussian distribution is given by the de Bruijn identity [START_REF] Stam | Some inequalities satisfied by the quantities of information of Fisher and Shannon[END_REF]. We show here that this important connection can be extended to the q-entropies, a suitable generalized Fisher information and the generalized q-Gaussian distributions. The de Bruijn identity states that if

Y t = X + √ 2tZ
where Z is a standard Gaussian vector and X a random vector of R n , independent of Z, then

d dt H[f Yt ] = I 2,1 [f Yt ] = φ 2,1 [f Yt ], (13)
where f Yt denotes the density of Y t = X + √ 2tZ, and

I 2,1 [f Yt ], φ 2,1 [f Yt ]
both denote the classical Fisher information (the meaning of which will be made clear in the following). Although the de Bruijn identity holds in a wider context, its classical proof uses the fact that if Z is a standard Gaussian vector, then Y t satisfies the well-known heat equation ∂f ∂t = ∆f, where ∆ denotes the Laplace operator.

Nonlinear versions of the heat equation are of interest in a large number of physical situations, including fluid mechanics, nonlinear heat transfer or diffusion. Other applications have been reported in mathematical biology, lubrification, boundary layer theory, etc; see the series of applications presented in [28, chapters 2 and 21] and references therein. The porous medium equation and the fast diffusion equation correspond to the differential equation ∂f ∂t = ∆f m , with m > 1 for the porous medium equation and < 1 for the fast diffusion. These two equations have been exhaustively studied and characterized by J. L. Vazquez, e.g. in [START_REF] Vázquez | The Porous Medium Equation: Mathematical Theory[END_REF][START_REF] Vázquez | Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type[END_REF].

These equations are included as particular cases into the doubly nonlinear equation, which involves a p-Laplacian operator ∆ p f := div |∇f | p-2 ∇f , and the power m of the porous medium or fast diffusion equation. This doubly nonlinear equation takes the form It can be shown, see [29, page 192], that for m(β -1) + (β/n) -1 > 0, (14) has a unique self-similar solution, called a Barenblatt profile B(), whose initial value is the Dirac mass at the origin. This fundamental solution is usually given as a function of m. Here, if we set q = m + 1 -α/β, the solution can be written as a q-Gaussian distribution:

∂ ∂t f = ∆ β f m = div |∇f m | β-2 ∇f m , (14)
f (x, t) = 1 t n δ B x t 1 δ , with B(x) =    (C -k|x| α ) 1 q-1 + for q = 1 1 σ exp -|β-1| β α |x| α for q = 1 (15) with δ = n(β -1)m + β -n > 0, k = m(β-1)-1 β 1 δ 1 β-1 and α = β β-1 .
The constants C and σ are uniquely determined by mass conservation, e.g. ´f (x, t)dx = 1. Of course, we observe that the function B(x) above is analogous to the generalized q-Gaussian (9).

As mentioned above, the doubly nonlinear diffusion equation allows to derive a nice extension of the de Bruijn identity [START_REF] Dehesa | Cramér-Rao information plane of orthogonal hypergeometric polynomials[END_REF], and leads to a possible definition of a generalized Fisher information. This is stated in the next Proposition. The case β = 2 has been proved in [START_REF] Johnson | Some results concerning maximum rényi entropy distributions[END_REF].

Proposition 1. [Extended de Bruijn identity, [START_REF] Bercher | Some properties of generalized Fisher information in the context of nonextensive thermostatistics[END_REF]] Let f (x, t) be a probability distributions defined on a subset X of R n and satisfying the doubly nonlinear equation [START_REF] Dembo | Information theoretic inequalities[END_REF]. Assume that X is independent of t, that f (x, t) is differentiable with respect to t, continuously differentiable over X, and that ∂ ∂t f (x, t) q is absolutely integrable and locally integrable with respect to t. Then, for β > 1, α and β Hölder conjugates of each other,

q = m + 1 -α β , M q [f ] = ´f q and S q [f ] = 1 1-q (M q [f ] -1)
the Tsallis entropy, we have

d dt S q [f ] = q m β-1 φ β,q [f ] = m q β-1 M q [f ] β I β,q [f ] (16)
with

   φ β,q [f ] = ´X f (x) β(q-1)+1 |∇f (x)| f (x) β dx and I β,q [f ] = 1 Mq[f ] β φ β,q [f ]. (17)
In [START_REF] Johnson | Some results concerning maximum rényi entropy distributions[END_REF], φ β,q [f ] and I β,q [f ] are two possible generalization of Fisher information. Of course, the standard Fisher information is recovered in the particular case α = β = 2, and q = m = 1, and so de Bruijn's identity [START_REF] Dehesa | Cramér-Rao information plane of orthogonal hypergeometric polynomials[END_REF]. The proof of this result relies on integration by part (actually using the Green identity) along the solutions of the nonlinear heat equation [START_REF] Dembo | Information theoretic inequalities[END_REF]. See [START_REF] Bercher | Some properties of generalized Fisher information in the context of nonextensive thermostatistics[END_REF] for a proof not repeated here. A variant of the result for β = 2, which considers a freeenergy instead of the entropy above, is well-known in certain commities, see e.g. [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF][START_REF] Carrillo | Asymptotic L1-decay of solutions of the porous medium equation to self-similarity[END_REF]. Even more, by carefully using calculations in [START_REF] Dolbeault | Improved interpolation inequalities, relative entropy and fast diffusion equations[END_REF], one can check that d dt φ 2,q [f ] ≤ 0 for q > 1 -1 n , which means the Tsallis entropy is a monotone increasing concave function along the solutions of [START_REF] Dembo | Information theoretic inequalities[END_REF]. In their recent work [START_REF] Savaré | The concavity of Rényi entropy power[END_REF], Savaré and Toscani have shown that in the case β = 2, m = q, the entropy power, up to a certain exponent, is a concave function of t, thus generalizing the well-known concavity of the (Shannon) entropy power to the case of q-entropies. This allows one to obtain as a by-product a generalized version of the Stam inequality, valid for the solutions of [START_REF] Dembo | Information theoretic inequalities[END_REF].

Actually, this extension of Stam's inequality, which links the generalized Fisher information to the q-entropy power [START_REF] Bercher | Some results on a χ-divergence, an extended Fisher information and generalized Cramér-Rao inequalities[END_REF], holds in a broader context. Proposition 2. [Generalized Stam inequality, cf [START_REF] Bercher | Some properties of generalized Fisher information in the context of nonextensive thermostatistics[END_REF])] Let n ≥ 1, β and α be Hölder conjugates of each other, α > 1, and q > max {(n -1)/n, n/(n + α)}. Then for any continuously differentiable probability density on R n , the following generalized Stam inequality holds

I β,q [f ] 1 β N q [f ] 1 2 ≥ I β,q [G] 1 β N q [G] 1 2 . ( 18
)
with equality if and only if f is a generalized q-Gaussian (9).

The generalized Stam's inequality implies that the generalized q-Gaussians minimize the generalized Fisher information within the set of probability distributions with a fixed q-entropy power; or alternatively minimize the entropy power among the distributions with a given (β, q)-Fisher information. A related, but different, Generalized Stam inequality has been given by Lutwak et al [START_REF] Lutwak | Cramér-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information[END_REF]. Their proof involves optimal transportation of probability measures. The proof below is quite different.

Proof. We sketch the proof in the case q < 1. The inequality follows from a sharp Gagliardo-Nirenberg inequality valid for n > 1, due to Cordero et al [START_REF] Cordero-Erausquin | A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities[END_REF]:

∇u β u 1 θ -1 a(β-1)+1 ≥ K u 1 θ aβ with a > 1 and ∇u β = ´ ∇u β * dx 1 β
, and where K is a sharp constant attained if and only if u is a generalized Gaussian with exponent 1/(1 -a), and

θ = n(a -1)/a(nβ -(aβ + 1 -a)(n -β)).
The idea is to take u = g t , for g a probability density function, with aβt = 1, and to set q = [a(β -1) + 1] t.

With these notations, we get that βt = β(q -1) + 1, and a = 1/βt > 1 implies that q < 1. Rearranging the exponents, (18) follows. Similarly, the case q > 1 follows from the sharp Gagliardo-Nirenberg inequality with a < 1. In the case n = 1, the inequality follows similarly from a sharp Gagliardo-Nirenberg inequality on the real line [1] [23].

Let f (x; θ) be a probability distribution, with x ∈ X ⊆ R n and θ ∈ R k . We will first deal here with the estimation of a scalar function h(θ) of θ, with T (x) the corresponding estimator. We extend here the classical Cramér-Rao inequality in two directions: firstly, we give results for a general moment of the estimation error instead of the second order moment, and secondly we introduce the possibility of computing the moment of this error with respect to a distribution g(x; θ) instead of f (x; θ). In estimation theory, the error is T (X) -h(θ), and the bias can be evaluated as

ˆX (T (x) -h(θ)) f (x; θ) dx = E f [T (X) -h(θ)] = η(θ) -h(θ),
while a general moment of of the error can be computed with respect to another probability distribution g(x; θ), as in

E g |T (X) -h(θ)| β = ˆX |T (x) -h(θ)| β g(x; θ) dx.
The two distributions f (x; θ) and g(x, θ) can be chosen arbitrarily. However, one can also build g(x; θ) as a transformation of f (x; θ) that highlights, or on the contrary scores out, some characteristics of f (x; θ). For instance, g(x; θ) can be a weighted version of f (x; θ), i.e. g(x; θ) = h(x; θ)f (x; θ), or a quantized version g(x; θ) = [f (x; θ)] , where [.] denotes the integer part. Another important case appears when g(x; θ) is defined as the escort distribution of order q of f (x; θ): f (x; θ) = g(x; θ) q ´g(x; θ) q dx and g(x; θ) = f (x; θ)

q ´f (x; θ) q dx , ( 19
)
where q is a positive parameter, q = 1/q, provided of course that all involved integrals are finite. These escort distributions are an essential ingredient in the nonextensive thermostatistics context. When f (x; θ) and g(x; θ) are a pair of escort distributions we will recover the generalized Fisher information [START_REF] Johnson | Some results concerning maximum rényi entropy distributions[END_REF] obtained in the extended de Bruijn identity.

Multivariate generalized Cramér-Rao inequality -matrix version

Previous results on generalized Fisher information can be found in [START_REF] Bercher | On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized q-Gaussian distributions[END_REF][START_REF] Bercher | On generalized Cramér-Rao inequalities, generalized Fisher informations and characterizations of generalized q-Gaussian distributions[END_REF] in the case of the direct estimation of the parameter θ. We propose here a new derivation, introducing in particular a notion of generalized Fisher information matrix, in the case of the estimation of a function of the parameters. Let us first state the result.

Proposition 3. Let f (x; θ) be a multivariate probability density function defined for x ∈ X⊆ R n , with parameter θ ∈ Θ ⊆ R k . Let g(x; θ) denote another probability density function also defined on (X; Θ). Assume that f (x; θ) is a jointly measurable function of x and θ, is integrable with respect to x, absolutely continuous with respect to θ. Suppose that the derivatives with respect to each component of θ are locally integrable. Let T (x) be an estimator of a scalar valued function h(θ) : Θ → R and set η(θ) = E f [T (X)]. Then, for any estimator T (x) of h(θ), we have

E g [|T (X) -h(θ)| α ] 1 α ≥ sup A>0 η(θ) T A η(θ) E g | η(θ) T A ψ g (X; θ)| β 1 β . ( 20
)
Equality in [START_REF] Liese | Statistical Decision Theory: Estimation, Testing, and Selection[END_REF] holds if and only if

T (x) -h(θ) = c(θ)sign( η(θ) T A ψ g (x; θ)) η(θ) T A ψ g (x; θ) β-1 (21)
with c(θ) > 0 and where α -1 + β -1 = 1, α > 1, A is a positive definite matrix and ψ g (x; θ) a score function given with respect to g(x; θ) :

ψ g (x; θ) := ∇ θ f (x; θ) g(x; θ) . ( 22
) Proof. Let η(θ) = E f [T (X)]. Let us first observe that E g [ψ g (x; θ)] = d dθ ˆX f (x; θ) dx = 0. Differentiating η(θ) = E f [T (X)] with respect to each θ i we get η(θ) = ∇ θ η(θ) = ∇ θ ˆX T (x) f (x; θ) dx = ˆX T (x) ∇ θ f (x; θ) g(x; θ) g(x; θ) dx = ˆX (T (x) -h(θ)) ψ g (x; θ) g(x; θ) dx.
For any positive definite matrix A, multiplying on the left by η(θ) T A gives

η(θ) T A η(θ) = ˆX (T (x) -h(θ)) η(θ) T A ψ g (x; θ) g(x; θ) dx,
and by the Hölder inequality

E g [|T (x) -h(θ)| α ] 1 α E g η(θ) T A ψ g (x; θ) β 1 β ≥ η(θ) T A η(θ),
with equality if and only if

(T (x) -h(θ)) η(θ) T A ψ g (x; θ) > 0 and |T (x) -h(θ)| α = k(θ) η(θ) T A ψ g (x; θ) β , k(θ) > 0.
The latter inequality, in turn, provides us with the lower bound [START_REF] Liese | Statistical Decision Theory: Estimation, Testing, and Selection[END_REF] for the moment of order α of the estimation error, computed with respect to g .

The inverse of the matrix A which maximizes the right hand side of ( 20) is the Fisher information matrix of order β. Unfortunately, we do not have a closed-form expression for this matrix in the general case. Nevertheless, two particular cases are of interest.

Corollary 4. [Scalar extended Cramér-Rao inequality]

In the scalar case (or the case of a single component of θ), the following inequality holds

E g [|T (X) -h(θ)| α ] 1 α ≥ | η(θ)| E g |ψ g (X; θ)| β 1 β , ( 23
)
with equality if and only if

T (x) -h(θ) = c(θ)sign(ψ g (x; θ)) |ψ g (x; θ)| β-1 .
In the simple scalar case, we see that A > 0 can be simplified in [START_REF] Liese | Statistical Decision Theory: Estimation, Testing, and Selection[END_REF] and thus that [START_REF] Nagy | Über integralungleichungen zwischen einer funktion und ihrer ableitung[END_REF] follows. Note that for α = 2, the equality case implies that

E g [ψ g ] = 0 = E g [T (X) -h(θ)],
which means that E g [T (X)] = h(θ), i.e. the estimator is unbiased (with respect to g) -but this also mean that it will be generally biased with respect to f . Actually, the inequality (23) recovers at once the generalized Cramér-Rao inequality we presented in the univariate case [START_REF] Bercher | On generalized Cramér-Rao inequalities, generalized Fisher informations and characterizations of generalized q-Gaussian distributions[END_REF]. The denominator plays the role of the Fisher information in the classical case, which corresponds to the case g(x; θ) = f (x; θ), β = 2.

A second interesting case is the multivariate case with α = β = 2. Indeed, in that case, we get an explicit form for the generalized Fisher information matrix and an inequality which looks like the classical one.

Corollary 5. [Multivariate Cramér-Rao inequality with

α = β = 2] For α = β = 2, we have E g |T (X) -h(θ)| 2 ≥ η(θ) T J g (θ) -1 η(θ) (24)
with J g (θ) = E g ψ g (X; θ)ψ g (X; θ) T , and with equality if and only if

|T (X) -h(θ)| = k(θ) η(θ) T J g (θ) -1 ψ g (X; θ) . ( 25
)
Proof. The denominator of ( 20) is a quadratic form and we have

E g |T (X) -h(θ)| 2 ≥ sup A>0 η(θ) T A η(θ) 2 E g | η(θ) T A ψ g (X; θ)| 2 ≥ sup A>0 η(θ) T A η(θ) 2 η(θ) T A E g [ψ g (X; θ)ψ g (X; θ) T ] A T η(θ) . (26)
Let J g (θ) = E g ψ g (X; θ)ψ g (X; θ) T and set z(θ) = A 1 2 η(θ). Then, using the inequality (z T z) 2 ≤ (z T Bz) (z T B -1 z) which holds for any B > 0, we obtain that

η(θ) T J g (θ) -1 η(θ) ≥ sup A>0 z(θ) T z(θ) 2 z(θ) T A 1 2 J g (θ) A 1 2 T z(θ) , with B = A 1 2 J g (θ) A 1 2 T .
Since the upper bound is it readily checked to be attained for A = J g (θ) -1 , we finally get [START_REF] Romera | The Fisher-Shannon information plane, an electron correlation tool[END_REF]. Of course, for g = f, the inequality (24) reduces to a classical multivariate Cramér-Rao inequality.

In the quadratic case, it is also possible to derive an analog of the well known result that the covariance of the estimation error is greater than the inverse of the Fisher information matrix (in the Löwner sense). Obviously, the inequality involves the generalized Fisher information matrix J g (θ). The proof follows the lines in [20, pp. 296-297] and we get the following result. Proposition 6. [Mutivariate Cramér-Rao inequality -covariance version] Consider the quadratic case, with β = 2, and the assumptions of Proposition 3, but with T (x) vector valued being an estimator of a vector valued function h(θ). Set

η(θ) = E f [T (X)], denote ψ g (x; θ) := ∇ θ f (x;θ) g(x;θ)
the generalized score function, and define by J g (θ) = E g ψ g (X; θ)ψ g (X; θ) T the Fisher Fisher information matrix J g (θ). Assume that J g (θ) is invertible. Then

E g (T (X) -h(θ)) (T (X) -h(θ)) T η(θ) T J g (θ) -1 η(θ) (27)
with η(θ) = ∇ θ E f [T (X) T ],
and equality if and only if

(T (X) -h(θ)) = η(θ)J g (θ) -1 ψ g (X). (28)
Of course, (24) appears as a special case of [START_REF] Vajda | χ α -divergence and generalized Fisher information[END_REF].

Proof. The proof follows the main lines of [20, pp. 296-297] with some small adaptations. With η(θ) = E f [T (X)], the transpose of the Jacobian matrix, here denoted by η(θ), is given by η

(θ) = ∇ θ E f [T (X) T ], which can be rewritten as η(θ) = E g [ψ g (X) T (X) T ].
Recall that E g [ψ g (X)] = 0. Therefore, for every vectors u and v of ad hoc dimensions, we have

E g u T (T (X) -h(θ)) ψ g (X) T v = u T η(θ) T v.
By the Schwarz inequality

u T η(θ) T v 2 ≤ u T C g u v T J g v (29)
with [START_REF] Vajda | χ α -divergence and generalized Fisher information[END_REF]. For the case of equality, simply observe that

C g := E g (T (X) -h(θ)) (T (X) -h(θ)) T . If v = J -1 g η(θ)u, then (29) reduces to u T η(θ) T J -1 g η(θ)u ≤ u T C g u , which implies
E g (T (X) -h(θ)) η(θ) T J g ψ g (X) T = η(θ) T J -1 g η(θ),
given that E g [ψ g (X)] = 0. Thus

E g T (X) -h(θ) -η(θ) T J -1 g ψ g (X) T (X) -h(θ) -η(θ) T J -1 g ψ g (X) T = E g (T (X) -h(θ)) (T (X) -h(θ)) T -η(θ) T J -1 g η(θ)
which is equal to zero, meaning that the bound is attained in [START_REF] Vajda | χ α -divergence and generalized Fisher information[END_REF] if and only if η(θ)J g (θ) -1 ψ g (X) = (T (X) -h(θ)).

An important consequence of the results above is obtained for a translation parameter, where the generalized Cramér-Rao inequality induces a new class of inequalities. Let θ ∈ R be a scalar location parameter, x ∈ X ⊆ R n , and define by f (x; θ) the family of density f (x; θ) = f (x -θ1), where 1 is a a vector of ones. In this case, we have ∇ θ f (x; θ) = -1 T ∇ x f (x -θ1) (provided that f is differentiable at x -θ1) and the Fisher information becomes a characteristic of the information in the distribution. If X is a bounded subset, we will assume that f (x) vanishes and is differentiable on the boundary ∂X. Without loss of generality, we can assume that the mean of f (x) is zero. Set h(θ) = θ and take T (X) = 1 T X/n, with of course η(θ) = E[T (X)] = θ and η(θ) = 1. Finally, let us choose the particular value θ = 0. In these conditions, the generalized Cramér-Rao inequality ( 23) becomes

E g 1 T X α 1 α E g 1 T ∇ x f (X) g(X) β 1 β ≥ n, (30)
with equality if and only if

1 T ∇xf (x) g(x) = c(θ)sign(1 T X) 1 T X α-1 .

Multivariate generalized Cramér-Rao inequality -general norms

In the multivariate case, we have another result, which involves an arbitrary norm of the estimation error. Recall that if . is an arbitrary norm, then its dual norm . * is defined by

Y * = sup X ≤1 X.Y, (31)
where X.Y is the standard scalar product. For instance, if . is a L p -norm, then . * is the L q -norm. The multivariate generalized Cramér-Rao inequality relies on an Hölder-type inequality for vector-valued functions, with arbitrary norm. This inequality was proved in [START_REF] Bercher | On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized q-Gaussian distributions[END_REF]. Let E = (R n , . ) be a n-dimensional normed space and denote E * = (R n , . * ) its dual space. If X(t) and Y (t) are two functions taking values respectively in E and E * , and if w(t) is a weight function, then

ˆ X(t) α w(t)dt 1 α ˆ Y (t) β * w(t)dt 1 β ≥ ˆX(t).Y (t) w(t)dt (32) with α -1 + β -1 = 1, α ≥ 1.
The equality is obtained if

Y (t) = K X(t) α-1 ∇ X(t) X(t) , with K > 0. ( 33
)
The condition is also necessary if the dual norm is strictly convex. We can now state the following result.

Proposition 7. [Generalized Cramér-Rao inequality for arbitrary norms, c.f. [START_REF] Bercher | On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized q-Gaussian distributions[END_REF]] Let f (x; θ) be a multivariate probability density function defined for x ∈ X⊆ R n , and θ ∈ Θ ⊆ R k . Let g(x; θ) denote another probability density function also defined on (X; Θ). Assume that f (x; θ) is a jointly measurable function of x and θ, is integrable with respect to x, is absolutely continuous with respect to θ, and that its derivatives with respect to each component of θ are locally integrable. Then, for any estimator θ(X) of θ ∈ R n , and arbitrary norms,

E g θ(X) -θ α 1 α E g ∇ θ f (X; θ) g(X; θ) β * 1 β ≥ ∇ θ .E f [ θ(X)] (34)
with α -1 + β -1 = 1, α ≥ 1, and where

I β [f |g; θ] = ˆX ∇ θ f (x; θ) g(x; θ) β * g(x; θ) dx (35)
is generalized Fisher information of order β on θ taken with respect to g. Equality occurs in (34) if (and only if the dual norm is strictly convex)

∇ θ f (x; θ) g(x; θ) = K θ(x) -θ α-1 ∇ θ(x)-θ θ(x) -θ , with K > 0. (36)
Proof. (Sketch of proof, see [START_REF] Bercher | On multidimensional generalized Cramér-Rao inequalities, uncertainty relations and characterizations of generalized q-Gaussian distributions[END_REF]). (a) evaluate the divergence of η(θ

) = E f [ θ(X)],
i.e. ∇ θ .η(θ); (b) average with respect to g and use the fact that the expectation of the score is zero ∇ θ .η(θ) = ´X ∇ θ f (x;θ) g(x;θ) . θ(x) -θ g(x; θ) dx, (c) apply the Hölder inequality (32), with X(x) = θ(x) -θ, Y (x) = ∇ θ f (x;θ) g(x;θ) , and w(x) = g(x; θ).

Let us consider again the case of a location parameter, but in the multivariate case θ ∈ R n . The translation family is f (x; θ) = f (x -θ). Then ∇ θ f (x; θ) = -∇ x f (x -θ). Let us also assume, without loss of generality, that f (x) has zero mean. In these conditions, the estimator T (X) = θ(X) = X is unbiased. Finally, taking θ = 0, (34) leads to Proposition 8. [Functional Cramér-Rao inequality] For any pair of probability density functions, and under some technical conditions,

ˆX

x α g(x) dx

1 α ˆX ∇ x f (x) g(x) β * g(x) dx 1 β ≥ n, (37)
with equality if (and only if when the dual norm is strictly convex)

∇ x f (x) = -K g(x) x α-1 ∇ x x .
Finally, let f (x) and g(x) be a pair of escort distributions such as in [START_REF] Liese | A Cramer-Rao type inequality for a convex loss function[END_REF]. Then, the generalized Fisher information [START_REF] Johnson | Some results concerning maximum rényi entropy distributions[END_REF] is obtained in the following result, which also yields a new characterization of q-Gaussian distributions.

Corollary 9. [q-Cramér-Rao inequality] Assume that g(x) is a measurable differentiable function of x, which vanishes and is differentiable on the boundary ∂X. Suppose that the involved integrals exist and are finite. Then, for the pair of escort distributions [START_REF] Liese | A Cramer-Rao type inequality for a convex loss function[END_REF], the following q-Cramér-Rao inequality holds

E g [ X α ] 1 α I β,q [g] 1 β ≥ n, (38)
with I β,q [g] = (q/M q [g]) β E g(x) β(q-1) ∇ x g(x) g(x)

β * , with equality if and only if g(x) is a generalized q-Gaussian g(x) ∝ (1 -γ(q -1) x α )

1 q-1 + (39)
Proof. The result follows from (37), or [START_REF] Vignat | Analysis of signals in the Fisher-Shannon information plane[END_REF] with n = 1, and the fact that for escort distributions,

∇ x f (X) g(X) = q M q [g] g(X) q-1 ∇ x g(X) g(X) .
The case of equality is obtained by solving the general equality condition

∇ x f (x) = -K g(x) x α-1 ∇ x x .
As a direct consequence of the q-Cramér-Rao inequality (38), the minimum of the generalized Fisher information among all distributions with a given moment of order α, say m α = E g [ X α ] , is obtained for g a generalized q-Gaussian distribution, with parameter γ such that the distribution has the prescribed moment. This extends the well-known result that the minimum of Fisher information with a fixed variance is attained for the Gaussian distribution. This parallels, and complements the known fact that the q-Gaussians maximize the q-entropies subject to a moment constraint, and yields new variational characterizations of generalized q-Gaussian distributions.

Let us also mention that the inequality (38) is similar, but different, to an inequality given by Lutwak et al [START_REF] Lutwak | Extensions of Fisher information and Stam's inequality[END_REF] which is also saturated by the generalized Gaussians (39).

Finally, for a location parameter, the matrix inequality (27) reduces to

Cov g [X] ≥ (J g ) -1 = E g ψ g ψ t g -1 . (40)
When (f, g) is a pair of escort distributions, the equality condition in Proposition 6 shows that equality occurs if and only if g is a generalized q-Gaussian with covariance matrix (J g ) -1 .

Conclusions

To sum up and emphasize the main results, let us point out that we have exhibited a generalized Fisher information, both as a by-product of a generalization of de Bruijn identity and Stam inequality and as a fundamental measure of information in estimation theory. We have drawn a nice interplay between q-entropies, generalized q-Gaussians and the generalized Fisher information. These interrelations yield the generalized q-Gaussians as minimizers of the (β, q)-Fisher information under adequate constraints, or as minimizers of functionals involving q-entropies, (β, q)-Fisher information and/or moments. This is shown through inequalities and identities involving all quantities and generalizing classical information relations (Cramér-Rao's inequality, Stam's inequality, De Bruijn's identity).

In a recent work [START_REF] Bercher | Some results on a χ-divergence, an extended Fisher information and generalized Cramér-Rao inequalities[END_REF], we introduced a variant of a χ α -divergence, which enables us to interpret the generalized Fisher information as a limit case of this divergence. Furthermore, this leads to a new derivation of the generalized Cramér-Rao inequality. More interestingly, this also enables us to derive a generalization of the Fisher Information Inequality (FII) -an inequality that characterizes the behavior of the Fisher information under convolution. The equality case, as usual, occurs for generalized q-Gaussian distributions. Details will come in a future paper.

The entropy power inequality (EPI) is a remaining important inequality of information theory, whose proof is difficult. An extension of the EPI to the q-entropy power N q (X + Y ) ≥ c q (N q (X) + N q (Y )) (q ≥ 1) has been proved very recently in the case q ≥ 1 [START_REF] Bobkov | Entropy power inequality for the Rényi entropy[END_REF], and the subject is very hot [START_REF] Wang | Beyond the entropy power inequality, via rearrangements[END_REF]. In the usual case, the EPI can be derived using de Bruijn's identity and the FII. Thus it would be very interesting to try to derive it from the results presented in this paper and the aforementioned FII.

We have obtained general Cramér-Rao inequalities, in the multivariate case, for a pair of arbitrary densities (f, g). This led us to the definition of a Fisher information I β [f |g; θ] about a parameter θ in f, taken with respect to g. In the case of a pair of escort distributions, this Fisher information reduces to the (β, q)-Fisher information I β,q [f ; θ], which is associated to the q-entropies and generalized q-Gaussians. Hence, a natural question would be to search for possible generalized entropies associated with the generalized Fisher information I β [f |g; θ], and perhaps with other families of maximum entropy distributions.

The extensions presented here rely on the idea of modifying the averaging distribution in the computation of the moment of the estimation error. A further step would be to consider general loss functions instead of a simple power of the estimation error. [START_REF] Kozek | Efficiency and Cramér-Rao type inequalities for convex loss functions[END_REF], [START_REF] Liese | A Cramer-Rao type inequality for a convex loss function[END_REF] and a work in progress bring interesting results in this direction. However, a remaining issue is again to associate some entropy with the new Cramér-Rao inequalities.

Finally, it would certainly be of interest to go further into estimation and look for estimators efficient with respect to the new Cramér-Rao bounds.

  where we use p = β for convenience and coherence with notation in the present paper. The β-Laplacian typically appears in the minimization of a Dirichlet energy such as ´|∇f | β dx which leads to the Euler-Lagrange equation. As we can see, the doubly nonlinear equation includes the standard heat equation (β = 2, m = 1), the β-Laplace equation (β = 2, m = 1), the porous medium equation (β = 2, m > 1) and the fast diffusion equation (β = 2, m < 1).