Julien Meyer
email: julien.meyer@gipsa-lab.fr

Fanny Meunier
email: fanny.meunier@unice.fr

Laure Dentel
email: ldentel.lab@gmail.com

Noelia Do

Carmo Blanco

Frédéric Sèbe
email: frederic.sebe@univ-st-etienne.fr

Loud and Shouted Speech Perception at Variable Distances in a Forest

Keywords: word recognition, Lombard speech, shouted speech, speech adaptation, vowel and consonant recognition

To increase the range of modal speech in natural ambient noise, individuals increase their vocal effort and may pass into the 'shouted speech' register. To date, most studies concerning the influence of distance on spoken communication in outdoor natural environments have focused on the 'productive side' of the human ability to tacitly adjust vocal output to compensate for acoustic losses due to sound propagation. Our study takes a slightly different path as it is based on an adaptive speech production/perception experiment. The setting was an outdoor natural soundscape (a plane forest in altitude). The stimuli were produced live during the interaction: each speaker adapted speech to transmit French disyllabic words in isolation to an interlocutor/listener who was situated at variable distances in the course of the experiment (30m, 60m, 90m). Speech recognition was explored by evaluating the ability of 16 normal-hearing French listeners to recognize these words and their constituent vowels and consonants. Results showed that in such conditions, speech adaptation was rather efficient as word recognition remained around 95% at 30m, 85% at 60m and 75% at 90m. We also observed striking differences in patterns of answers along several lines: different distances, speech registers, vowels and consonants.

Introduction

Articulating words with strong vocal efforts such as in shouting is developed from early age, in general without any specific learning. Yet, an efficient speech emission in such conditions relies on a homogeneous, powerful, relaxed and precise control of both the airflow and the physiological constraints imposed by over articulations. Compared to modal speech, 'raised', 'loud' or 'shouted' speech forms increase muscle tension in the vocal tract. These tensions reinforce the concentrations of energy in the signal; with the aim to carry oral sounds across distances and/or over noise to ensure good communication. As a sound source, the shouted voice uses the vocal cords and a vocal tract often modified by a low and large pharynx. The resulting signals bear complex frequency spectra characteristic of the human voice. To increase the range of ordinary speech or to overcome noise, individuals adjust their voices by raising amplitude levels in a quasisubconscious way. During this vocal effort, called the "Lombard effect" [START_REF] Lombard | Le signe de l'élévation de la voix[END_REF], the spoken voice progressively passes into the register of the shouted voice. Effort is also intensified with the tendency to lengthen syllables, to reduce the flow of speech and to increase the fundamental frequency. There is a large body of literature on this phenomenon for speech under noisy conditions; e.g., [START_REF] Hanley | Effect of level of distracting noise upon speaking rate, duration and intensity[END_REF][START_REF] Dreher | Effects of ambient noise on speaker intelligibility for words and phrases[END_REF][START_REF] Anglade | Acoustic-phonetic study of Lombard speech in the case of isolated-words[END_REF][START_REF] Olsen | Average Speech Levels and Spectra in Various Speaking/Listening Conditions: A Summary of the Pearson, Bennett, & Fidell (1977) Report[END_REF][START_REF] Garnier | Speaking in noise: How does the Lombard effect improve acoustic contrasts between speech and ambient noise?[END_REF]. However, there are far fewer studies on variations in talker-to-listener distance in natural outdoor conditions [START_REF] Zahorik | Accurate vocal compensation for sound intensity loss with increasing distance in natural environments[END_REF][START_REF] Meyer | Acoustic strategy and typology of whistled languages; phonetic comparison and perceptual cues of whistled vowels[END_REF][START_REF] Cheyne | Talker-to-listener distance effects on speech production and perception[END_REF][START_REF] Fux | Vers un système indiquant la distance d'un locuteur par transformation de sa voix[END_REF].

To date, most studies concerning the influence of distance on spoken communication in outdoor natural environments have focused on the 'productive side' of the human ability to tacitly adjust vocal output to compensate for acoustic losses due to sound propagation. When the perceptive side was taken into consideration, it was done without measuring precisely intelligibility levels, just by checking that the communication was effective. Our study takes a different path as it is based on an adaptive speech production/perception experiment, and its original protocol is presented here for the first time. The setting was an outdoor natural soundscape (a plane forest in altitude). The stimuli were produced live during the interaction: each speaker adapted speech to transmit -in a semi-spontaneous task -French disyllabic words presented in isolation to an interlocutor/listener who was situated at variable distances in the course of the experiment (30m, 60m, 90m). Speech recognition was explored by evaluating the ability of 16 normal-hearing French listeners to recognize these words and their constituent vowels and consonants.

Methods

Participants

The 16 participants were 20 to 25 year-old French native speakers (3 men and 13 women). They were all voluntary Master 2 students of the Ethology and Ecology Master Program of Jean Monnet University. They all knew eachothers for more than 1 year. None of them reported hearing impairment. The present study was conducted in accordance with the Declaration of Helsinki. After being informed of the details of the experimental procedure, all the participants provided written informed consent.

Stimuli

Stimuli presented to the participants were produced in live during an interaction between speakers and listeners to check for intelligibility. 19 lists were prepared in order to randomize word order. Each list contained 17 French isolated words. Lists were matched for word frequency and for the position of Interspeech 2018 2-6 September 2018, Hyderabad each vowel type and each consonant type. The selected words were nouns regularly used in current French vocabulary. They were disyllabic words of mainly CVCV and CVCVC structures but contained also other types of syllabic structure to avoid learning effects from the participants. For all lists, all participants and all simulated distances, the distribution of the word structures was as follows: CVCV (41,9%), CVCVC (35,9%), CVCCV (9,6%), CVCVCC (4,6%), CCVC (3,8%), CVCCVC (2,4%), and CVVC (1,8%).

Design and procedure

General conditions, Experimental field

The experiment took place in winter (December) in a forest situated on a flat land near the summit of the Pilat mountain in France at 1000 meters of altitude. The forest was a mix of resinous and lobed-leaved trees which had lost their leaves at this season. An inventory of trunk sizes was made to further control the reverberation effect in the future. The ground was covered by 10 cm of light fresh snow, which guaranteed quasiideal conditions of ground absorption (ground effect minimized). Meteorological conditions were controlled and the experiment took place in quasi-stationary meteorological conditions (wind speed <1 m/s throughout the session, degree of atmospheric humidity between 45% and 75%, temperature between 7°C and 0°C, measured on a portable meteorological station Geos Skywatch). The recording precautions enabled us to measure a relatively stationary background noise (standard deviation of 1.2 dB) in low level conditions (mean value of 35.4 dBA) measured with a sound level meter Rion NL42. The experiment was stopped only twice due to the presence of a group of birds near the listener and twice due to aircraft noises passing above far in the sky. No other mechanic artificial noise occurred in this isolated area.

Procedure of the test

Participants to the interactive task of this experiment formed pairs. Each participant had to emit aloud to his partner a list of words at each of the three distances of the test: 30, 60 and 90 meters. The test phase began with a list of 17 words to transmit at 30 m, and that after a training phase of 5 to ensure that participants had understood the task. Once all the pairs had performed the task at this distance, the experimenters set up the next step which was to replicate the task at 60 m, and next, at 90 m. For each participant, a different list was presented at each of the 3 distances tested. Each participant had also the simple task of listening to each stimulus said by their partner and trying to recognize the target isolated word, in an open response format. Listeners were asked to speak loud the perceived sounds, even if they did not correspond to a French word, and this answer was audio recorded. To remain in ecological conditions of an interactive communication, listeners could ask for a repetition to their interlocutor. The repeated instances were not analyzed in the present paper. Once the listener had spoken the perceived sounds or in the absence of answer after two repetitions, the speaker moved on to the following word of the list. The participants did not receive any feedback on their performance before the end of the test.

Distances and associated speech registers

Speakers and listeners were alternatively situated at distances of 30 m, 60 m and 90 m, which still permitted visual contact between them. This condition guaranteed that they could adapt their productions and listening by having a visual feedback on the distance to cover to reach the interlocutor. The three distances chosen enabled us to follow the progressive adaptation of speakers and listeners to the constraints imposed by the ecological milieu to word transmission in the distance. The distances of 60m and 90m were chosen to correspond to different levels of the shouted speech register, whereas the distance of 30m was chosen to correspond to spoken speech (but not yet shouted). Simultaneous audio and sound level recordings were made at 1 meter of the speaker and at the distance at which his interlocutor/listener was situated. The two audio-recorders were pointing at the speaker and not at the listener (but still recorded the answers of the listener). Table 1 presents an example -on a sample of /a/ vowels (10 instances per distance in V1 position for female speakers) -of how the adaptation of the speakers affected the main acoustic parameters which are the most often measured for shouted speech (F0, Amplitude, Duration). These measures show a progressive increase of F0, of Amplitude max level and of vowel lengthening as distance increased, typical of the presence of a 'Lombard effect'. The objective here was not to provide a detailed analysis of these productive aspects but to verify that the tendencies found from the stimuli driving our perceptual study were in coherence with the ones commonly observed in the literature on Lombard or shouted speech [START_REF] Hanley | Effect of level of distracting noise upon speaking rate, duration and intensity[END_REF][START_REF] Dreher | Effects of ambient noise on speaker intelligibility for words and phrases[END_REF][START_REF] Anglade | Acoustic-phonetic study of Lombard speech in the case of isolated-words[END_REF][START_REF] Olsen | Average Speech Levels and Spectra in Various Speaking/Listening Conditions: A Summary of the Pearson, Bennett, & Fidell (1977) Report[END_REF][START_REF] Garnier | Speaking in noise: How does the Lombard effect improve acoustic contrasts between speech and ambient noise?[END_REF][START_REF] Zahorik | Accurate vocal compensation for sound intensity loss with increasing distance in natural environments[END_REF][START_REF] Meyer | Acoustic strategy and typology of whistled languages; phonetic comparison and perceptual cues of whistled vowels[END_REF][START_REF] Cheyne | Talker-to-listener distance effects on speech production and perception[END_REF]. Observations and measures also confirmed that speakers' productions corresponded to 'loud' speech at 30 m (below 70 dBA), shouted speech of different intensity at 60 m and 90 m.

Results

General results are based on recognition percentage scores. Altogether, a total of 816 words -272 per distance -were heard by the 16 participants. First, we analyzed word recognition. Next, we analyzed the recognition performance for vowels and consonants separately and as a function of the type of error (confusion, insertion, deletion) when they were mistaken. To assess the influence of each factor, the phoneme error rate was computed separately for each condition and each distance.

Word recognition

We found a mean word recognition rate of 95.2% (SD=3%) at 30 m, 84.6% (SD=7.1%) at 60 m, and 76.4% (SD=8.5%) at 90 m (Figure 1). This means that the intelligibility remained high even if the transmission was less efficient as distance increased. The task increased in difficulty as attested by the increased inter-individual variability with distance -rendered by the standard deviation-with the strongest change between 30 and 60 m. An analysis of variance (ANOVA) was performed on correct answers with 'Distance' as a within factor. It confirmed that the scores varied significantly depending on distance (F(2,30)=13.02, p<0.001). Moreover, Post hoc multiple t-tests with Bonferroni correction (p<0.05) showed that words were significantly less well recognized at 60m and 90m than at 30m, but that no clear significant difference was found between 60 m and 90 m.

Vowel and consonant recognition

Correct answers

Correct answers on vowels differed significantly as a function of distance (F(2,30)=8.23, p=0.01). Mean vowel recognition rates were very high: almost perfect 99.4% (SD=0,8%) at 30m, 95.7% (SD=2,1%) at 60m, and 95,0% (SD=2,5%) at 90m) (Figure 2). Correct answers on consonants also differed significantly as a function of distance (F(2,30)=16.03, p<0.001) and mean correct scores on consonants decreased from 97,4% (SD=1,8%) at 30m, to 91,7% (SD=3,4%) at 60m and 84,0% (SD=6,1%) at 90m)(Figure 3).

Word recognition was correlated to consonant recognition (Rc=0.9882, p<0.1) but not to vowel recognition (Rv=0.9856, n.s.) but this result must be taken with caution because these values close and because sample sizes are different. Moreover, when consonants were all recognized in a word, this almost always led to the identification of the word (except in 1 or 2 cases at each distance), whereas when the vowels were all recognized, we found several errors on consonants at every distance (11 cases at 30m, 27 cases at 60 m, 45 cases at 90 meters).

Two Post hoc multiple t-tests with Bonferroni corrections (p<0.05) -one on vowels and one on consonants -showed that vowels were significantly less well recognized at 60m and 90m than at 30m, but that no clear significant difference appeared between 60 and 90m, whereas consonants were significantly less well recognized at 90m than 60m and 30m, but that no clear significant difference appeared between 30 and 60m.

Mistakes: Confusions, Insertions, Deletions

To further investigate the difference found between vowels and consonants, we dissociated errors on phoneme recognition by studying vowels and consonants along several lines. First, we compared errors on both types of phonemes and found that, overall, errors on consonant dominated over errors due to vowels at every distance. Next, we explored separately confusions (phonemes mistaken for another in the responded word), deletions (suppression of a phoneme in the responded word), and insertions (addition of a phoneme in the responded word). Insertions turned out to be extremely rare for both vowels (Figure 4) and consonants (Figure 5). Moreover, deletions were the most frequent errors for vowels even if they remained below 20 instances at 90m. For consonants, confusions and deletions followed the same tendency of increase between 30m and 60 m with confusions remaining the most frequent errors at any distance. However, between 60 and 90m the consonant confusion rate boosted whereas the consonant deletion rate increased much more slowly.

Discussion and conclusions

The effect of speech-adaptation to distance in ecological conditions was measured through the recognition scores of listeners during an original interactive word production/perception experiment which took place in a forested area. The experimental design aimed at approaching maximally ecological conditions of word transmission between interlocutors at different distances (30m, 60m and 90 m). It was set up to include a spoken speech condition (loud speech but not yet shouted) and two different shouted speech conditions. The focus of the analysis presented here was on perceptual results rather than production or propagation aspects and several interesting results emerged from this approach.

First, we found that word recognition scores remained relatively high as it remained around 95% at 30m, 85% at 60m and 75% at 90m, showing that the adaptation made by the speakers in production was rather efficient for the objective to transmit a word to an interlocutor situated in the distance in this ecological middle.

Next, the recognition performances for vowels and consonants revealed various interesting differences between these two kinds of phonemes. Vowels were, in general, better recognized than consonants at any distance. Therefore, there was a higher stability of vowels over consonants in the speech conditions we used. This is in accordance with the literature on speech in noise perception involving white noise, speechshaped noise, or 'natural quiet' background noise; e.g., [START_REF] Meyer | Speech Recognition in Natural Background Noise[END_REF][START_REF] Benki | Analysis of English Nonsense Syllable Recognition in Noise[END_REF][START_REF] Meyer | Human phoneme recognition depending on speech-intrinsic variability[END_REF]. The second striking difference was about errors on vowels and consonants that showed overall different profiles, a difference which really increased at 90 m. At 90 m, we found almost 4 times more errors and 8 times more confusions on consonants than on vowels. These proportions are unlikely to be specifically due to distributional properties because the CVCV and CVCVC syllabic structures (80% of our corpus) present less than two times as many opportunities to produce similar sounding lexical neighbors due to consonants rather than vowels. Interestingly, similar proportions were found in modal speech perception in natural quiet background noise [START_REF] Varnet | Phoneme resistance during speech-in-speech comprehension[END_REF]. Moreover, the results show that word recognition is significantly correlated to consonant recognition but not to vowel recognition. The fact that words are nearly all recognized at any distance when all consonants are well identified but not when all vowels are well identified illustrates this correlation. Thus, our results seem to confirm the special functional role of consonants during lexical identification, but this time in conditions with the shouted speech register. Overall, these results strongly reinforce the idea that vowels and consonants play different roles in speech processing, even in shouted speech. The literature concerning the respective roles of vowels and consonants in speech recognition is very prolific and is in line with our findings [START_REF] Fletcher | Speech and Hearing[END_REF][START_REF] Fogerty | Perceptual contributions to monosyllabic word intelligibility: segmental, lexical, and noise replacement factors[END_REF][START_REF] Hochmann | Vowels and Consonants in Early Language Acquisition[END_REF]. Yet, our results on this aspect have still to be taken with caution because the lack of correlation between word identification and vowel recognition could be a result of the small dynamic range of vowel recognition and the difference with consonants in this res^pect might be partly due to a difference of sample size with consonants.

Finally, the perceptual results emphasize the difference between the shouted speech conditions (listened to at distances of 60 and 90 m) and Lombard speech condition (30 m). The very high recognition scores at 30 m in the Lombard speech condition contrasted statistically with the two shouted speech conditions. A close look at errors on vowels and consonants at each distance shows that this contrast with Lombard speech was different in nature for the two shouted speech conditions. The mistakes made in Lombard speech condition were very few and balanced between consonant confusions on one side and other types of errors on both vowels and consonants on the other side. At 60m, in the first shouted speech condition, the situation was different because errors due to consonant confusions had a less important role which was almost equivalent to either consonant deletions or to errors on vowels. However, at 90 meters, the contribution of consonant confusions was overwhelmingly dominant. Overall, these results show a non-linearity in sources of errors on word recognition scores between the three conditions corresponding to the different distances of the test. Due to their short duration and low energy, consonants are more rapidly altered in production and more easily masked than vowels [START_REF] Meyer | Speech Recognition in Natural Background Noise[END_REF][START_REF] Phatak | Consonant and vowel confusions in speech-weighted noise[END_REF][START_REF] Phatak | Consonant confusions in white noise[END_REF] These conclusions open exciting perspectives justifying to further explore two lines of research: (i) on one hand the analysis of the main different acoustic characteristics of phonemes in the three conditions which represent different adaptations of speech production. Indeed, speakers adapted their vocal effort to the different distances of the test and thus transformed the phonetic aspects of speech under the Lombard effect. (ii) On another hand, it will be interesting to analyze the scattering due to acoustic propagation at each distance in order to understand better how the noise, the reverberation and the spherical spreading interfered with word recognition and may explain some aspects of the results found here. Finally, one limit of our results is that we do not separate male from female speakers/listeners because we had only two male participants, but this might be interesting to do in order to take into account previously documented differences between male and female productions in Lombard and shouted speech [eg, [START_REF] Olsen | Average Speech Levels and Spectra in Various Speaking/Listening Conditions: A Summary of the Pearson, Bennett, & Fidell (1977) Report[END_REF][START_REF] Junqua | The Lombard reflex and its role on human listeners and automatic speech recognizers[END_REF].

These perspectives are realistic because the original experimental protocol exposed here for the first time was designed to collect all the data necessary for such explorations. A new campaign of data collection on the same experiment was made recently, doubling the number of subject and balancing male and female participants.

Figure 1 :

 1 Figure 1: Word recognition performance as a function of distances and across listeners (plot boxes showing median, 25 to 75 % of participants and SD).

Figure 2 :

 2 Figure 2: Vowel recognition scores as a function of distances and across listeners.

Figure 3 :

 3 Figure 3: Consonant recognition scores as a function of distances and across listeners.

Figure 4 :

 4 Figure 4: Insertions, deletions and confusion in vowels as a function of distances for all listeners.

Figure 5 :

 5 Figure 5: Insertions, deletions and confusions in consonants as a function of distances for all listeners.

Table 1 :

 1 Mean values of fundamental frequency (F0), of the maximum amplitude level (AmpMax) and of duration for Vowel /a/ in the female voice (10 samples per distance in V1 positions). Standard deviations areshown between brackets.

	Distance	F0	AmpMax	Duration
	m	Hz	dBA	s
	30	295.8 (22.6)	62.53 (2.87)	0.131 (0.02)
		348 (20.8)	68.06 (2.58)	0.147 (0.02)
	90	387 (26.7)	73.83 (3.14)	0.156 (0.03)

10.21437/Interspeech.2018-2089

Acknowledgements

The authors would like to thank T. Saint-Germain and the participants for their collaboration during testing. This research was financially supported by the 7th research program of the European Community FP7 2007-2013 (Marie Curie IIF Grant 630076).


Philippe Heinrich

Jonas Kahn

STRONG IDENTIFIABILITY AND OPTIMAL MINIMAX RATES FOR FINITE MIXTURE ESTIMATION

Keywords: MSC 2010 subject classifications: Primary 62G05; secondary 62G20 Local asymptotic normality, convergence of experiments, maximum likelihood estimate, Wasserstein metric, mixing distribution, mixture model, rate of convergence, strong identifiability, pointwise rate, superefficiency

We study the rates of estimation of finite mixing distributions, that is, the parameters of the mixture. We prove that under some regularity and strong identifiability conditions, around a given mixing distribution with m0 components, the optimal local minimax rate of estimation of a mixing distribution with m components is n -1/(4(m-m 0 )+2) . This corrects a previous paper by Chen (1995) in The Annals of Statistics.

By contrast, it turns out that there are estimators with a (nonuniform) pointwise rate of estimation of n -1/2 for all mixing distributions with a finite number of components.

1. Introduction. Finite mixture models have been applied since [START_REF] Pearson | Contributions to the theory of mathematical evolution[END_REF] in various fields including astronomy, biology, genetics, economy, social sciences and engineering [START_REF] Mclachlan | Finite mixture models[END_REF].

Finite mixtures and their estimation naturally arise mostly in three cases. One is model-based clustering. Here the aim is to divide the data into k clusters and assign (new) data to a cluster. A possible approach is to consider that data point from each cluster is generated according to a probability distribution, so that the whole data is generated by mixture with k components [START_REF] Mclachlan | Finite mixture models[END_REF][START_REF] Teh | Dirichlet Process[END_REF].

The second, more traditional case, is the statistical description of possibly heterogeneous data where the underlying mixing distribution has no particular meaning. In that case, mixtures are a tool to describe efficiently the "true" probability distribution and control the convergence rate of mixture estimators to it (van [START_REF] Van De Geer | Rates of convergence for the maximum likelihood estimator in mixture models[END_REF][START_REF] Ghosal | Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities[END_REF][START_REF] Genovese | Rates of convergence for the Gaussian mixture sieve[END_REF].

In the third case, the goal is the mixing distribution itself: its support points and proportions are the parameters we want to estimate. They typically correspond to the phenomenon that is studied. This is the case we are interested in, on the basis of observations drawn from the mixture.

Some works try to bridge the gap between the estimation of the mixture and the one of the mixing distribution, usually at least through estimation of the number of components -the order -in the finite mixture. In particular, [START_REF] Rousseau | Asymptotic behaviour of the posterior distribution in overfitted mixture models[END_REF] have proved that their Bayesian estimator of an overfitted mixture tends to empty the extra components, and [START_REF] Gassiat | Consistent order estimation and minimal penalties[END_REF] have given the minimal penalty on the maximumlikelihood estimator of the order that yields strong consistency.

One could expect that a good estimator for the mixture would be a good estimator for the mixing model. However, this is not so clear. The situation is reminiscent of the difference between estimation and identification in model selection, where [START_REF] Yang | Can the strengths of AIC and BIC be shared? A conflict between model indentification and regression estimation[END_REF] has proved that no procedure can be optimal for both. Moreover, rates of convergence can be very different, as illustrated in an infinite-dimensional case by [START_REF] Bontemps | Bayesian methods for the shape invariant model[END_REF].

Optimal rates are a key information in estimating the mixture parameters. These were unknown (see e.g. [START_REF] Titterington | Statistical analysis of finite mixture distributions[END_REF] till the work of [START_REF] Chen | Optimal Rate of Convergence for Finite Mixture Models[END_REF], who established a n -1/4 local minimax rate, under reasonable identifiability conditions, for one-dimensional-parameter mixtures. This result is somewhat surprising since the rate does not depend on the number of components. It turns out to be erroneous, because of its Lemma 2.

Our article aims at giving correct statements and proofs and its consequences.

The main part consists in finding the correct exponent in the local minimax rate; we do so in Theorem 3.2 and Theorem 3.3. The rate gets worse with more components, which is consistent with the behaviour when there are infinitely many components, such as deconvolution: [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF] had proved that the L 2 -convergence rate was polylogarithmic in general, and [START_REF] Caillerie | Deconvolution for the Wasserstein metric and geometric inference[END_REF] and [START_REF] Dedecker | Minimax rates of convergence for Wasserstein deconvolution with supersmooth errors in any dimension[END_REF] have generalized this kind of rates to the more relevant (for us) Wasserstein metrics. The most original technical tool we shall use is a coarse-graining tree on the parameter indices.

In addition, the optimal local minimax rate and the optimal pointwise rate of estimation everywhere are not the same. This discrepancy is unusual in statistics, and probably the reason why the n -1/4 rate went unchallenged for twenty years. Specifically, if instead of comparing all pairs of mixtures in a ball, we allow only one mixture in it, we get (21) which corrects Lemma 2 of Chen. As a consequence, Theorem 2 of Chen is valid by dropping uniformity: for any fixed mixing distribution say G, the estimator considered there will converge at rate n -1/4 , but with a multiplicative constant that depends on G. It then becomes a statement on the optimal pointwise rate of estimation everywhere, and can even be strengthened to n -1/2 as we show in Theorem 4.1.

The paper by [START_REF] Chen | Optimal Rate of Convergence for Finite Mixture Models[END_REF] has been widely cited and used. Apart from applied papers citing it that may have relied on the theoretical guarantees (see e.g. [START_REF] Kuhn | The Spatial Structure of Young Stellar Clusters. I. Subclusters[END_REF][START_REF] Liu | Unrestricted Mixture Models for Class Identification in Growth Mixture Modeling[END_REF], there are essentially two ways it could play a role. Firstly, when it is used as part of a proof, secondly when it is used as a benchmark.

The first case covers papers that generalize Chen's result in other settings, and re-use its theorems and proofs. For example, [START_REF] Ishwaran | Bayesian model selection in finite mixtures by marginal density decompositions[END_REF] propose a Bayesian estimator that achieves the n -1/4 frequentist rate, and use Chen (1995, Lemma 2) in their analysis. More recently, [START_REF] Nguyen | Convergence of latent mixing measures in finite and infinite mixture models[END_REF] generalizes those results to mixtures with an abstract parameter space and indefinite number of components. But Nguyen (2013, Theorem 1) generalizes Chen (1995, Lemma 2) while transposing the proof with the mistake. The main results of both these articles hold however: they do not need the full strength of Chen (1995, Lemma 2), but merely the weaker version (21). Ho and Nguyen (2016) prove such a sufficient version.

These two papers also use [START_REF] Chen | Optimal Rate of Convergence for Finite Mixture Models[END_REF] article as a benchmark. However, the optimal pointwise rate everywhere would probably be a better reference point in their case, as in many others. In particular, it seems likely that a Bayesian estimator could converge pointwise at speed n -1/2 everywhere. We have not checked whether the proof by [START_REF] Ishwaran | Bayesian model selection in finite mixtures by marginal density decompositions[END_REF] can be improved, or if another prior is necessary.

This use as a benchmark is very usual, as expected for this kind of optimality result (see e.g. Zhu andZhang, 2006, 2004). Let us point in particular to a result by [START_REF] Martin | Convergence rate for predictive recursion estimation of finite mixtures[END_REF]. He achieves almost n -1/2 rate for the predictive recursion algorithm, and tries to explain the discrepancy with [START_REF] Chen | Optimal Rate of Convergence for Finite Mixture Models[END_REF] by the fact that the parameters are constrained to live in a finite space for his algorithm. In fact, since his rate is pointwise, it fits with the continuous case.

Since early versions of this article have been made available, there have been interesting new developments: Ho et al. (2016); [START_REF] Ho | Identifiability and optimal rates of convergence for parameters of multiple types in finite mixtures[END_REF] have made explicit the system of equations underlying the minimax argument, for any finite number of parameters, and solved important special cases. The strong identifiability conditions we shall use in this article ensure that the system of equations is generic.

In Section 2, we give the notations and define and discuss the regularity assumptions we use. In Section 3, we state and discuss the main theorem, giving the optimal local minimax rate. In Section 4, pointwise rate everywhere is investigated. We try to give some intuition in both of these sections. In Section 5, we also dwell on the interpretation and practical consequences of having different rates, and conclude with open questions. In Section 6, we give and explain the meaning of the key intermediate results and prove the main theorems from here. In Section 7, we prove those key intermediate results. In particular, we introduce the most original tool of our proofs: the coarse-graining tree that allows to patch the mistake in the article by [START_REF] Chen | Optimal Rate of Convergence for Finite Mixture Models[END_REF].

Some auxiliary and technical results are detailed in appendices grouped in a supplemental part [START_REF] Heinrich | Supplement to 'Minimax rates for finite mixture estimation[END_REF].

Notations and regularity conditions.

2.1. Basic notations. Throughout the paper, the parameter set Θ, of diameter Diam Θ, is always assumed to be a compact subset of R with nonempty interior. Let G m (resp. G m ) be the set of (resp. at most) m-mixing or m-point support distributions G on Θ. We set also G <∞ = ∪ 1 G .

As usual, we compare two mixing distributions G and G using transportation distances, or L q -Wasserstein metrics, with q ≥ 1. They completely bypass identifiability issues that would arise with the square error on parameters. The definition is:

(1) W q (G, G ) = inf Π Θ 2 |θ -θ | q dΠ(θ, θ ) 1/q
where the infimum is taken over probability measures Π on Θ × Θ with marginals G and G . By Jensen's inequality, W q ≥ W q if q > q and moreover, W q q ≤ W q q (Diam Θ) q-q . We will usually work with the strongest available Wasserstein metric for our results. Endowed with the metric W q , the space G m is compact.

In the special case of W 1 , we will also use its dual representation, where |f | Lip stands for the Lipschitz seminorm of f (e.g. Dudley, 2002, section 11.8):

(2)

W 1 (G, G ) = sup |f | Lip 1 Θ f (θ)d(G -G )(θ).
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Given G = m j=1 π j δ θ j ∈ G m and a family {f (x, θ)} θ∈Θ of probability densities on R w.r.t. some σ-finite measure λ, we can define a finite mixture with m components:

(3) f (x, G) = Θ f (x, θ)dG(θ) = m j=1 π j f (x, θ j ).
To compare mixture distribution functions F (x, G) and F (x, G ), we will use the Kolmogorov metric

F (•, G) -F (•, G ) ∞ .
Here of course we have by definition F (x, θ) =

x -∞ f (y, θ)dλ(y), which extends to F (x, G) by linearity provided that f (y, δ θ ) = f (y, θ) where δ θ denotes the Dirac measure at θ.

Regularity conditions.

(p, α)-smoothness. Hereafter, f (p) (x, θ) or f (p) (•, θ) denote the p-th derivative of f always taken w.r.t. the variable θ.

Definition 2.1. The family {f (•, θ), θ ∈ Θ} w.r.t. some σ-finite measure λ is (p, α)-smooth if E p,α θ, θ , θ = R f (p) (x, θ ) f (x, θ ) α f (x, θ) dλ(x) (4)
is a well-defined [0, ∞]-valued continuous function on Θ 3 , and if there exists

ε > 0 such that |θ -θ | < ε =⇒ ∀θ ∈ Θ, E p,α (θ, θ , θ ) < ∞. (5)
These smoothness conditions are easy to check in practice, and general enough. For example, all exponential families satisfy them, as shown in the supplemental part (Heinrich and Kahn, 2015, E.2). They will be useful for proving local asymptotic normality [START_REF] Cam | Asymptotic Methods in Statistical Decision Theory[END_REF] of relevant families. k-strong identifiability (k ∈ N). Chen (1995) introduced a notion of strong identifiability. We will need a slightly more general version.

Definition 2.2. The family {F (•, θ), θ ∈ Θ} of distribution functions is k-strongly identifiable if for any finite set of say m distinct points

θ j ∈ Θ, k p=0 m j=1 a p,j F (p) (•, θ j ) ∞ = 0 =⇒ a = max p,j |a p,j | = 0,
where • ∞ denotes the supremum norm with respect to the variable x.

Chen's strong identifiability corresponds to 2-strong identifiability. Let us show why this notion is useful. Consider

G n = 1 2 (δ n -1 + δ -n -1 ) in G 2 . We see that F (•, G n ) = F (•, 0)+n -2 F (2) (•, 0)/2+o(n -2 ), provided we can expand around θ = 0. Then 2-strong identifiability ensures that F (•, 0) -F (•, G n ) ∞ is of order n -2 ,
as shown in Proposition 2.3 below, whereas simple (1-strong) identifiability would say nothing. We will need k-strong identifiability with a higher k if more terms cancel.

Proposition 2.3. Let {F (•, θ), θ ∈ Θ} be k-strongly identifiable family of distribution functions with

F (k) (x, θ) continuous in θ. Set for ε > 0 Θ m ε = (θ j ) 1 j m ⊂ Θ : min j =j |θ j -θ j | ε .
Then for all a = (a p,j ) 0 p k

1 j m , inf Θ m ε k p=0 m j=1 a p,j F (p) (•, θ j ) ∞ ε,k,m a , where ε,k,m
means "more than", up to some constant C(ε, k, m) > 0.

Proof. The function (a, (θ j )

1 j m ) → k p=0 m j=1 a p,j F (p) (•, θ j )
∞ is lower semi-continuous on the compact set {a : a = 1} × Θ m ε , so that it admits a minimum. By k-strong identifiability, it is nonzero.

We expect the strong identifiability to be rather generic, and hence the statements of this paper often meaningful. In particular, Chen (1995, Theorem 3) has proved that location and scale families with smooth densities are 2-strongly identifiable. The theorem and the proof straightforwardly generalize to our setting. We merely state the result.

Theorem 2.4. Let k 1. Let f be a probability density w.r.t. the Lebesgue measure on R. Assume that f is k -1 times differentiable with

lim x→±∞ f (p) (x) = 0 for p ∈ [[0, k -1]]. Consider f (x, θ) = f (x -θ), with θ ∈ Θ ⊂ R. Then the corresponding distributions family {F (•, θ), θ ∈ Θ} is k-strongly identifiable. If Θ ⊂ (0, ∞), the result stays true with f (x, θ) = 1 θ f x θ .
For more general conditions see the article by [START_REF] Holzmann | Identifiability of finite mixtureswith applications to circular distributions[END_REF], that also generalize well to k-strong identifiability.

d 0 = m -m 0 .
Lower bounds on local asymptotic minimax rates. 1,4]], the family of densities {f (•, θ)} θ∈Θ is (p, α)-smooth and satisfies, for some point θ 0 in the interior of Θ,

Assumption A(k, θ 0 ). For all (p, α) ∈ [[1, 2k + 2]] × [[
|f (2k+1) (•, θ 0 )|dλ > 0.
Typically, k will be d 0 and θ 0 a support point of G 0 . These conditions allow to prove local asymptotic normality (Le [START_REF] Cam | Asymptotic Methods in Statistical Decision Theory[END_REF] for relevant families. This will give some insight on the reason why the lower bound on the rate holds, and on how the mixtures behave when we change the parameters in the least sensitive direction. The condition on the support point guarantees identifiability locally for the families, and we need more derivatives than usual, since there will be cancellations in the first terms. In what follows, E G will denote the expectation w.r.t. the measure dP G = f (•, G)dλ. And all the mixing distribution estimators denoted by G n below will be based on i.i.d. n-samples.

Theorem 3.2. Recall (6) and set ε n = n -1/(4d 0 +2)+κ for some κ > 0. Let θ 0 be a support point of G 0 . Under Assumption A(d 0 , θ 0 ), for any sequence of estimators G n , we have

sup G∈Gm W 1 (G,G 0 )<εn E G W 1 (G, G n ) n -1/(4d 0 +2) .
Let us give some intuition. The data we have access to is the empirical distribution F n where n is the sample size, and which gets closer to the true mixture F (•, G) at rate n -1/2 . Hence two mixing distributions G and G can be told apart only if

F (•, G) -F (•, G ) ∞ is at least of order n -1/2 .
As an example, let G 0 = δ 0 and consider two-component mixing distributions around, say

G n = 1 2 δ -2n -1/6 + δ 2n -1/6 and G n = 4 5 δ -n -1/6 + 1 5 δ 4n -1/6
. Both have 0 as first moment, and 4n -1/3 as second moment but the third moment is zero for G n and 12n -1/2 for G n . A Taylor expansion in θ = 0 up to the third order gives then

F (•, G n ) = o(n -1/6 ) and F (•, G n ) = o(n -1/6 ).
So that no test can reliably tell G n from G n with an n-sample. On the other hand, we clearly have W 1 (G n , G n ) = n -1/6 for all n. So that the minimax rate for two-mixing distributions cannot be better than n -1/6 . This moment matching argument can be made rigorous and precise with two tools. One is Lindsay's Hankel trick (Lindsay, 1989, Theorem 2A), also used by [START_REF] Dacunha-Castelle | The estimation of the order of a mixture model[END_REF] to estimate the order of a mixture. The other is local asymptotic normality property (LAN) developed by [START_REF] Cam | Asymptotic Methods in Statistical Decision Theory[END_REF]. Section 6 uses them to build a LAN family with scale factor n 1/(4d 0 +2) which gives Theorem 3.2 via Theorem 6.1.

Upper bounds on local asymptotic minimax rates.

Assumption B(k). The family of densities

{f (•, θ)} θ∈Θ satisfies • For all x, F (x, θ) = x -∞ f (•, θ)dλ is k-differentiable w.r.t. θ, • {F (•, θ), θ ∈ Θ} is k-strongly identifiable,
• There is a uniform continuity modulus ω(•) such that

sup x F (k) (x, θ) -F (k) (x, θ ) ω(θ -θ ) with lim h→0 ω(h) = 0.
The latter condition holds if sup x,θ |F (k+1) (x, θ)| exists and is finite. These differentiability conditions should be compared with the usual parametric case, where differentiability in quadratic mean, or twice differentiability in θ for a less technical condition, is enough to get n -1/2 local minimax rate. We will need B(2m) to prove a global minimax rate of n -1/(4m-2) (see ( 9) in Theorem 3.3), and B(1) for a pointwise rate of n -1/2 everywhere (Theorem 4.1).

Theorem 3.3. Let G n (m) be "the" minimum distance estimator, that is any mixing distribution in G m such that (7) F (•, G n (m)) -F n ∞ = inf G∈G m F (•, G) -F n ∞ .
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Under Assumption B(2m), there is ε > 0 such that, with q = 2d 0 + 1:

sup G∈G m Wq(G,G 0 )<ε E G W q ( G n (m), G) n -1/(2q) , (8)
and more globally, with r = 2m -1:

sup G∈G m E G W r ( G n (m), G) n -1/(2r) . (9) Remark 3.4. Since G → F (•, G) -F n ∞ is lower semi-continuous on the compact metric space (G m , W q ),
the infimum in (7) is attained . The minimum distance estimator is discussed by [START_REF] Deely | Construction of Sequences Estimating the Mixing Distribution[END_REF] and Chen.

Theorem 3.3 is proved it by establishing a uniform control of the ratio

F (•, G) -F (•, G ) ∞ /W q (G
, G ) q in Theorem 6.3. To do so, we consider sequences of couples (G n , G n ) minimizing the relevant ratios, and expand

F (•, G n )-F (•, G n ) as a weighted sum on the relevant derivatives F (p) (•, θ j,n ).
A difficulty arises since distinct support points θ j,n may converge to the same θ j , leading to cancellations in the sums. Forgetting this case was the mistake in the proof of Chen (1995, Lemma 2). We overcome the issue in Section 7: we build clusters of support points whose pairwise distances decrease at a given rate and structured as nodes of a coarse-graining tree. We may then use Taylor expansions on each node and its descendants (Lemma 7.4).

Remarks 3.5. It is worth noticing the following from Theorems 3.2-3.3:

• They together imply that the optimal local asymptotic minimax rate is n -1/(4d 0 +2) for estimating a mixture with at most m components around a mixture with m 0 components, for any transportation distance

W p with p ∈ [[1, 2d 0 + 1]].
• The rate is driven by d 0 , that is, it gets harder to estimate the parameters of a mixture when it is close to a mixture with less components. • The worst case is when m 0 = 1, yielding a global minimax rate of estimation n -1/(4m-2) . The rate gets worse when more components are allowed. So that the nonparametric rates for estimating mixtures with an infinite number of components like in deconvolution appear natural. • On the other hand, when the number of components is known, that is m = m 0 , we have the usual local minimax rate n -1/2 . • The global minimax rate on the mixtures with exactly m components stays at n -1/(4m-2) , because G m is not compact, and Theorem 3.2 still apply in the vicinity of m 0 -component mixtures.

4. On pointwise rate and superefficiency. The slow rate n -1/(4m-2) in ( 9) might be a little surprising when for example some Bayesian estimators have n -1/4 rate of convergence [START_REF] Ishwaran | Bayesian model selection in finite mixtures by marginal density decompositions[END_REF]. However this convergence rate is not the local minimax rate, but is closer to a pointwise rate of convergence, that is the speed at which an estimator converges to a fixed G when n increases. The difference with local minimax may be viewed as the loss of uniformity in G. We study here the optimal pointwise rates everywhere.

One motivation for local minimax results was to make clear how the Hodges' estimator (van der Vaart, 1998, ch.8) and other superefficient estimators could cohabit with Cramér-Rao bound, and how much they could improve on it.

Specifically, a superefficient estimator can have a better pointwise convergence rate than any regular estimator, but not a better local minimax convergence rate [START_REF] Hájek | Local asymptotic minimax and admissibility in estimation[END_REF]. Moreover, it turns out that they can only have a better pointwise rate on a Lebesgue-null set (van der Vaart, 1998, ch.8). Now, the set of parameters (weights and support points) defining G <m is a Lebesgue-null w.r.t. the one defining G m . Hence, we might expect that, by biasing the estimators toward the low numbers of components, we might attain better pointwise rates on G <m , up to n -1/2 , which is the value when the number of components is known. By letting m go to infinity, we would have this pointwise rate for all finite mixing distributions. It turns out this is indeed the case.

Theorem 4.1. Consider for each m 1 the minimum distance estimator G n (m) in G m as defined in Theorem 3.3, with G n (∞) arbitrary. Fix κ ∈ (0, 1/2) and set

(10) m n = min m 1 : F (•, G n (m)) -F n ∞ n -1/2+κ . Under Assumption B(1), for any finite mixing distribution G ∈ G <∞ , E G W 1 ( G n ( m n ), G) G,κ n -1/2 .
Remarks 4.2.

• Since the typical distance between empirical and theoretical distribution functions is n -1/2 , this m n in (10) is the lowest number of components that is not clearly insufficient.

• The rate n -1/2 cannot be improved since it is the rate if the number of components is known beforehand.

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• This is slightly stronger than just checking that we find the right number of components and then applying Theorem 3.3, because we need much less regularity. Only Assumption B(1) is required, instead of B(2m).

That is, we do not need more smoothness when the number of components increases. Under the hood we rely on the bound (20) instead of Theorem 6.3. • The estimation of the number of components m n and the estimation of G within G mn are not associated. For example, we may estimate m n with Equation (10), and then use the maximum likelihood estimator G on G mn . Conversely, we may estimate the number of components using Gassiat and van Handel's (2013) penalized maximum likelihood estimator.

5. Practical consequences and perspectives. Disagreement between local minimax rate and pointwise rate everywhere might be rare enough that it is worth recalling first what it means.

The asymptotic rate of convergence to a given G will be the pointwise rate C G n -1/2 where C G is some positive constant. However, the estimator will enter this asymptotic regime only after a long time. More precisely, it enters this regime when G is not any more in any of the balls used in the local minimax bound. Alternatively, we may view this situation as the constant C G exploding when G is close to specific G 0 .

In our case, imagine we have a mixing distribution with three components, with all support points within distance δ > 0 of some θ 0 . Then about δ -(4(3-1)+2) = δ -10 observations are necessary to get an estimator with an error of δ. In particular, if G and G are two such three-component mixing distributions, chosen to have the same first four moments, and G and G are the same mixing distributions, rescaled to be ten times closer, we will need 10 10 as many data points to tell them apart as for G and G .

As a consequence, if the components of the mixing distribution to be estimated are not far apart one from the other, it is quite often impossible to get enough data points to get an appropriate estimate. An experimentalist with any leeway in what he measures (use of different markers, say) might then wish to ensure that the peaks are far apart, even at the cost of many data points.

We end the section by some thoughts on possible further work. This article contains the proof that the optimal local minimax rate of estimation around a mixing distribution with m 0 components among mixing distributions with m components is n -1/(4(m-m 0 )+2) , when the parameter space Θ is a compact subset of R.

We think that extension to a multivariate Θ should be workable, much like Nguyen (2013) did for the former erroneous result. On the other hand, non-compactness of Θ would probably bring about technical difficulties, and cases where the result would not hold. Stronger forms of identifiability would probably be required in general, to avoid problems with limits. Moreover, for many natural higher-dimensional families, strong identifiability does not hold, so that the results would be different.

Finally, another line of inquiry are the results that might be expected in a Bayesian framework. The most natural equivalent to the convergence rate of the a posteriori distribution to the real parameter is the pointwise rate of convergence. Hence the question: can we build Bayesian estimators where the a posteriori distributions converge at rate n -1/2 everywhere? Of course, the convergence would not be uniform.

6. Key tools and proofs.

6.1. Local asymptotic normality for Theorem 3.2. All the densities considered in the sequel are w.r.t. some given dominating σ-finite measure on R. We call experiment a family E of densities. Theorem 6.1. Let G 0 ∈ G m 0 with a support point θ 0 in the interior of Θ. There is a family {G n (u)} n 0,u∈R in G m with the following properties: a. For all distinct u, u in R, we have together

W 1 (G n (u), G n (u )) u,u n -1/(4d 0 +2) u W 1 (G n (u), G 0 ); b. Assume A(d 0 , θ 0 ) for the family {f (•, θ)} θ∈Θ and set the product density f n,u = ⊗ n i=1 f (•, G n (u)).
There is an increasing real sequence U n → ∞ such that the sequence of experiments E n = (f n,u ) u∈[-Un,Un] is locally asymptotically normal (LAN) : there are random variables Z n , asymptotically N (0, 1), and numbers Γ n > 0 such that for all u ∈ R,

Log f n,u (X) f n,0 (X) -uZ n Γ n + u 2 2 Γ n P ---→ n→∞ 0 (11)
where X is a n-sample of density f n,0 . In addition, we have lim inf n Γ n > 0 and lim sup n Γ n < ∞. Remark 6.2. We want only an example of this slow convergence, and it should be somewhat typical. That is why we have chosen the regularity imsart-aos ver. 2014/10/16 file: papierAOSv18.tex date: July 8, 2017 conditions to make the proof easy, while still being easy to check, in particular for exponential families.

In particular, in Assumption A(d 0 , θ 0 ), it could probably be possible to lower α in (p, α)-smoothness to 2 + ε and still get the uniform bound we use in the law of large numbers below. Similarly, less differentiability might be necessary if we tried to imitate differentiability in quadratic mean.

Conversely, under possibly more stringent regularity conditions, Γ n is ex-

pected to converge to E G 0 f (2d 0 +1) (•,θ 0 ) f (•,G 0 )
2 up to a multiplicative constant.

Proof of Theorem 6.1. Write the mixing distribution G 0 as ( 12)

G 0 = m 0 -1 j=1 π j δ θ j + π 0 δ θ 0
with θ 0 in the interior of Θ. Let u ∈ R and replace the Dirac measure δ θ 0 in (12) with a mixing distribution H n (u):

(13) G n (u) = m 0 -1 j=1 π j δ θ j + π 0 H n (u).
We want to choose H n (u) close to δ θ 0 . To this end, set µ 0 = 1 and µ 2d-1 = u with d = d 0 + 1,. Choose in addition numbers µ 1 , . . . , µ 2d-2 such that the

k × k-Hankel matrices (M k ) i,j = µ i+j-2 satisfy det M k > 0 for k ∈ [[1, d -1]].
Then, by Lindsay's Theorem 2A (1989), there is a unique mixing distribution H(u) = m j=m 0 π j (u)δ h j (u) with exactly d support-points h j (u) and first moments µ k up to order 2d -1 satisfying ( 14)

m j=m 0 π j (u)h j (u) k = µ k , k ∈ [[0, 2d -1]].
Define then H n (u) by shifting and rescaling the support points of H(u):

H n (u) = m j=m 0 π j (u)δ θ 0 +εnh j (u) with ε n = n -1/(4d-2) .
Now, using the dual representation (1) of W 1 , we see that

W 1 (G n (u), G 0 ) = π 0 W 1 (H n (u), δ θ 0 ) = π 0 ε n W 1 (H(u), δ 0 ) and likewise, W 1 (G n (u), G n (u )) equals π 0 ε n W 1 (H(u), H(u )
) so that Theorem 6.1.a follows.

To guarantee that the points θ 0 + ε n h j (u) involved in G n (u) stay inside Θ uniformly in u, let us show that the functions h j (•) are continuous. Consider the map

ϕ(π 1 , . . . , π d , h 1 , . . . , h d ) = d 1 π j , d 1 π j h j , d 1 π j h 2 j , . . . , d 1 π j h 2d-1 j on the set {(π 1 , . . . , π d , h 1 , . . . , h d ) : π 1 > 0, . . . , π d > 0, h 1 < • • • < h d }.
The uniqueness in Theorem 2A by [START_REF] Lindsay | Moment matrices: applications in mixtures[END_REF] implies that ϕ is injective. Moreover, its Jacobian is non-zero, as it can be seen by recurrence on d:

J(ϕ) = (-1) (d-1)d 2 π 1 • • • π d 1 j<k d (h j -h k ) 4 .
Thus the inverse of ϕ is locally continuous, so that, in particular, the h j (u) are all continuous. Set now

(15) h(U ) = max j d max |u| U |h j (u)|
which is finite for any U > 0 and choose a positive sequence (U n ) such that

U n → ∞ and ε n h(U n ) → 0.
We can now prove local asymptotic normality (11). Let X = (X 1,n , . . . , X n,n ) be an i.i.d. sample with density f n,0 . Since we proceed along the lines of Chen (1995), the proof is only sketched here. Write the log-likelihood ratio as

Log f n,u (X) f n,0 (X) = n i=1 Log (1 + Y i,n (u)) with Y i,n (u) = f (X i,n , G n (u)) -f (X i,n , G n (0)) f (X i,n , G n (0)) .
The main steps are as follows, see Heinrich and Kahn (2015, Sections A.1, A.2 and A.3) for the details:

Step 1. Use linearity of G → f (•, G) and Taylor expansions up to the order 2d -1 with remainder on Y i,n (u) at θ 0 to show that the r.v.'s

Z i,n = π 0 f (2d-1) (X i,n , θ 0 ) f (X i,n , G n (0))
are centered under f n,0 .

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Step 2. Define

Γ n = E Gn(0) |Z 1,n | 2 and Z n = n -1/2 Γ -1/2 n n
i=1 Z i,n and prove that Z n is asymptotically N (0, 1) via Lyapunov Theorem for triangular arrays.

Step 3. Show the following convergences for all u:

A n (u) := n i=1 Y i,n (u) -uZ n Γ n L 2 -→ 0, B n (u) := n i=1 Y i,n (u) 2 -u 2 Γ n L 1 -→ 0, C n (u) := n i=1 |Y i,n (u)| 3 L 1 -→ 0.
Then derive the LAN property from the equality

Log f n,u (X) f n,0 (X) -uZ n Γ n + u 2 2 Γ n = A n (u)+ 1 2 B n (u)+O P (C n (u)).
6.2. Proof ofTheorem 3.2. Let us show how Theorem 6.1 entails Theorem 3.2 using just two points and contiguity [START_REF] Cam | Locally asymptotically normal families of distributions[END_REF]. Consider any sequence of estimators G n , and G n (u) for u = 0, 1 as defined in Theorem 6.1. It's enough to show that for large n, (16) sup

G∈{Gn(0),Gn(1)} E G [W 1 (G, G n )] n -1/(4d 0 +2) .
Recall that we set here ε n = n -1/(4d 0 +2)+κ and note that G n (0) and G n (1) are in the ball {G : W 1 (G, G 0 ) < ε n } for large n, by Theorem 6.1.a. Consider the probability measures P n,u with the densities f n,u of Theorem 6.1.b for u = 0, 1 and set

g n = f n,1 (X) f n,0 (X) exp(-Z n √ Γ n + Γ n /2). Then, P n,1 (A) = e -Γn 2 A e Zn √ Γn g n dP n,0 e -Γn 2 2 A∩{Zn>0}∩{gn>1/2}
dP n,0 .

We have g n P n,0 --→ 1 by ( 11) so that P n,0 (g n 1/2) 1/16 for large n. Since Z n is asymptotically N (0, 1), we also have P n,0 (Z n 0) 1/2 + 1/16. Thus P n,0 ({Z n > 0} ∩ {g n > 1/2}) is at least 3/8 for n large enough and ( 17) ) , by Theorem 6.1.a. By the triangle's inequality, the complement A c is included in {W 1 (G n (0), G n ) an -1/(4d 0 +2) }. Now, either we have P n,0 (A c ) 1 4 and, for G = G n (0), we get

P n,0 (A) 3 4 =⇒ P n,1 (A) e -Γn/2 . Now, choose A = {W 1 (G n (1), G n ) an -1/(4d 0 +2) } where a > 0 is such that W 1 (G n (1), G n (0)) 2an -1/(4d 0 +2
E G [W 1 (G, G n )1 A c ] a 4 n -1/(4d 0 +2) ,
or we have P n,0 (A) 3 4 and by using ( 17), for G = G n (1), we get

E G [W 1 (G, G n )1 A ] e -Γn/2 an -1/(4d 0 +2) ,
so that ( 16) is proved since lim sup n Γ n < ∞.

6.3.

Comparison between distances for Theorem 3.3 and Theorem 4.1. The key technical tool is Theorem 6.3.

• Let G 0 ∈ G m 0 . Under Assumption B(2m), there are ε > 0 and δ > 0 such that, with q = 2d 0 + 1: inf

G =G ∈G m Wq(G,G 0 )<ε Wq(G ,G 0 )<ε F (•, G) -F (•, G ) ∞ W q (G, G ) q > δ, (18)
and more globally, with r = 2m -1:

(19) inf G =G ∈G m F (•, G) -F (•, G ) ∞ W r (G, G ) r > δ. • Let G 0 ∈ G m 0 . Under Assumption B(1), there are ε > 0 and δ > 0 such that inf G =G ∈G m 0 W 1 (G,G 0 )<ε W 1 (G ,G 0 )<ε F (•, G) -F (•, G ) ∞ W 1 (G, G ) > δ. (20) • Let now G 0 ∈ G m 0 . Under Assumption B(2), there are ε > 0 and δ > 0 such that inf G∈G m 0 W 1 (G,G 0 )<ε F (•, G) -F (•, G 0 ) ∞ W 1 (G, G 0 ) 2 > δ. ( 21
)
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The proof of Theorem 6.3 is postponed to Section 7 where the novel ingredient, a coarse-graining tree, is constructed to prove ( 18) and ( 19), the most difficult points. These one entail Theorem 3.3. The two related bounds ( 20) and ( 21) hold under weaker differentiability assumptions, but are less general. Bound (20) covers the case where the number of components in the mixture is known, and is used for the proof of Theorem 4.1. Bound ( 21) is the valid weaker version of Lemma 2 by [START_REF] Chen | Optimal Rate of Convergence for Finite Mixture Models[END_REF], which is sufficient for the use other authors have made of it. Here, we only compare mixtures in a ball with the mixture at the center of the ball.

For the proofs of Theorem 3.3 and Theorem 4.1, we need in addition:

Lemma 6.4. Let q, d 1 and G ∈ G m . Assume that the minimum distance estimators G n := G n (m) defined in Theorem 3.3 satisfy for some constant C > 0 and on some event A,

W q ( G n , G) d C F (•, G n ) -F (•, G) ∞ . Then E G W q ( G n , G) 2πC 2 1/2d n -1/2d + Diam(Θ)P G (A c ). Moreover, P G (A c ) is at most 2e -2nz 2 if A is either { F n -F (•, G) ∞ z} or { F (•, G n ) -F (•, G) ∞ 2z}.
Sketch of proof. Bound W q ( G n , G) by Diam(Θ) on A c , use the definition (7) and the triangle's inequality to bound P G (A c )) by applying DKW's inequality [START_REF] Massart | The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality[END_REF].

F (•, G n ) -F (•, G) ∞ by 2 F n -F (•, G) ∞ , then use Jensen's inequality on A and bound E G 1 F n - F (•, G) ∞ (and
6.4. Proof of Theorem 3.3. Let ε, δ > 0 such (18) holds. Set z = 1 2 inf G,G ∈G m Wq(G,G 0 ) ε/2 Wq(G ,G 0 ) ε F (•, G) -F (•, G ) ∞ .
The infimum is taken over a compact set and is thus attained. We have z > 0 by identifiability (coming from Assumption B(2m)).

Consider A = { F (•, G) -F (•, G n (m)) ∞ z}. If G is in G m with W q (G, G 0 ) ε/2 then G n (m) must satisfy W q ( G n (m), G 0 ) < ε on the event A so that by (18), On A, W q ( G n (m), G) q < 1 δ F (•, G n (m)) -F (•, G) ∞ .
Applying Lemma 6.4 with d = q = 2d 0 + 1 and C = 1/δ yields

E G [W q ( G n (m), G)] 2π δ 2 1/2q
n -1/2q + 2 Diam(Θ)e -nz 2 /2 so that bound (8) is proved. Bound ( 9) is obtained likewise from (19).

6.5. Proof of Theorem 4.1. Consider a mixing distribution G 0 ∈ G m 0 . Under Assumption B(1), let ε, δ > 0 such (20) holds. Fix κ ∈ (0, 1 2 ) and set

z = n -1/2+κ ∧ 1 4 inf G∈G m 0 W 1 (G,G 0 ) ε F (•, G) -F (•, G 0 ) ∞ .
By compactness and identifiability, the infimum in z is attained and positive. On the event

A = { F (•, G 0 ) -F n ∞ z}, the minimum distance estimator G n (m 0 ) in G m 0 , defined in Theorem 3.3, satisfies F (•, G n (m 0 )) -F n ∞ F (•, G 0 ) -F n ∞ z n -1/2+κ .
so that mn is at most m 0 by ( 10); thus we have

G n ( mn ) ∈ G m 0 . Next, G n ( mn ) must satisfy W 1 ( G n ( mn ), G 0 ) < ε on A since F (•, G n ( mn )) -F (•, G 0 ) ∞ 2 F (•, G 0 ) -F n ∞ 2z,
by the triangle's inequality. Applying then (20) on A, we get

W 1 ( G n ( mn ), G 0 ) < 1 δ F (•, G n ( mn )) -F (•, G 0 ) ∞ .
Now, apply Lemma 6.4 with q = d = 1, C = 1/δ and A as above, so that

E G 0 [W 1 ( G n ( mn ), G 0 )] 2π δ n -1/2 + 2 Diam(Θ) exp(-2n 2κ ).
7. The coarse-graining tree and the proof of Theorem 6.3.

7.1. Proof of (18): the coarse graining tree. Let G 0 ∈ G m 0 . We have to show that, under Assumption B(2m), there is ε > 0 such that

L := inf G =G ∈G m Wq(G,G 0 )<ε Wq(G ,G 0 )<ε F (•, G) -F (•, G ) ∞ W q (G, G ) q > 0 with q = 2d 0 + 1. (22)
Assume on the contrary that L = 0 and choose mixing distributions

G n and G n in G m with W q (G n , G 0 ) ∨ W q (G n , G 0 ) < 1/n such that for each n 1, the ratios F (•, G n ) -F (•, G n ) ∞ /W q (G n , G n ) q are
less than 1/n. We shall prove, up to selecting subsequences, the following contradiction:

(23) F (•, G n ) -F (•, G n ) ∞ W q (G n , G n ) q .
imsart-aos ver. 2014/10/16 file: papierAOSv18.tex date: July 8, 2017 Some notations. We may and do assume that there are integers m, m at most m such that (G n ) ⊂ G m and (G n ) ⊂ G m . We can then write G n = m j=1 π j,n δ θ j,n and G n = m j=1 π j,n δ θ j,n and set

( j,n , ϑ j,n ) = (π j,n , θ j,n ) if j m (-π j-m,n , θ j-m,n ) if j > m ,
so that the signed measure G n -G n = m+m j=1 j,n δ ϑ j,n has total mass zero.

The discrepancy orders of the ϑ j,n 's. We shall first classify the differences between the ϑ j,n 's in an intrinsic way:

Lemma 7.1. For a suitable subsequence of G n -G n , there is a finite number S of "scaling" sequences

0 ≡ ε 0 (n) < ε 1 (n) < • • • < ε S (n) ≡ 1 with ε s (n) = o ε s+1 (n) , such that for all j, j ∈ [[1, m+m ]] there is a unique s(j, j ) ∈ [[0, S]] satisfying ϑ j,n -ϑ j ,n ε s(j,j ) (n).
The proof is given in Heinrich and Kahn (2015, Appendix C). It follows from the definition of s(j, j ) that s(j, j ) max(s(j, j"), s(j , j")) and thus s(•, •) defines an ultrametric on [[1, m + m ]]. The ultrametric makes any two balls either included one into the other, or disjoint, and allows us to build a coarse-graining tree : Definition 7.2. The coarse-graining tree T is the collection of distinct balls J = {s(•, j) s}, called nodes, when j ranges over [[1, m + m ]] and s over [[0, S]]. Moreover:

• The root of T is J r = [[1, m + m ]],
• The parent J ↑ of a node J is defined by

(J ⊂ I J ↑ , I ∈ T ) =⇒ I = J, • The set of children of a node J is Child(J) = {I ∈ T : I ↑ = J}, • The set of descendants of a node J is Desc(J) = {I ∈ T : I ↑ ⊂ J},
• The diameter of a node J is s(J) = max j,j ∈J s(j, j ).

Let us show how the tree T looks like with a partial representation :

imsart-aos ver. 2014/10/16 file: papierAOSv18.tex date: July 8, 2017

diameter N S s(J r ) s(J r ) s(J) s(K) s(K ) 1 0 • 1 • 2 • • • • j • • • • j • • • • m+m root J r • • • • j • • • • j • • • J |ϑ j,n -ϑ j ,n | ε s(J) (n) • • • • j • • • K • • • • j • • • K • j • • • j • ends of s-diameter zero
Note that the ends are not necessarily singletons since the ultrametric s(•, •) does not separate points. Note also that j and j are in different children K and K so that |ϑ j,n -ϑ j ,n | is actually exactly of order ε s(J) (n).

The Wassertsein distances W q (G n , G n ) through the coarse graining tree T . In what follows n is skipped in the ϑ j 's, j 's and ε s 's. Set for short (24) J = j∈J j and ε J = ε s(J) .

Lemma 7.3. For any q 1, we have

W q G n , G n q max J∈Desc(Jr) | J | ε q J ↑ .
Proof. Consider any coupling Π between G n and G n and set

Π(J, J ) = Π {ϑ j } j∈J∩[[1,m]] × {ϑ j } j∈J ∩[[m+1,m+m ]] ,
w q (J, J ) = Set also π J = j∈J∩ [[1,m]] j and π J = -j∈J∩ [[m+1,m+m ]] j . These define the marginal distributions of Π and we have Π(J, J) π J ∧ π J . Note also that

| J | = π J ∨ π J -π J ∧ π J . With J c = J r \ J, this gives Π(J, J c ) ∨ Π(J c , J) | J |.
Notice moreover that if (j, j ) ∈ J × J c , then |ϑ j -ϑ j | ε J ↑ . Hence the lower bound of Lemma 7.3 follows from w q (J r , J r ) w q (J, J c ) + w q (J c , J)

| J | ε q J ↑ .
Conversely, for the upper bound, we show recursively that for all node J, w q (J, J) max K∈Desc(J)

| K |ε q K ↑ . ( 26
)
This is obviously true if J is an end node, with the value of zero. Assume that (26) holds for children K of a given node J. We may develop J on its children:

w q (J, J) = K∈Child(J) w q (K, K) + K ∈Child(J) K =K w q (K, K ) .
Furthermore, we get w q (K, K ) Π(K, K ) ε q J from (25) and

Π(K, K ) Π(K, K c ) π K -Π(K, K),
and if the coupling Π is chosen (see Heinrich and Kahn (2015, Lemma B.2) for a construction) such that Π(K, K) = π K ∧ π K for all node K, then it follows that

Π(K, K ) | K |,
and thus w q (J, J)

K∈Child(J) w q (K, K) + | K |ε q J .
The recurrence hypothesis on children K yields then (26).

Expanding F (x, G n ) -F (x, G n ) through the coarse graining tree. The dependence on n is skipped in the following notations. Consider the additive set-function J → F (x, J) = j∈J j F (x, ϑ j ) and note that F (x, J r ) is equal to

F (x, G n ) -F (x, G n ).
Lemma 7.4. Choose ϑ J in {ϑ j : j ∈ J} for each node J of T .

There are a vector a J = (a J (p)) 0 p 2m and a remainder R(x, J) such that

(27) F (x, J) = 2m p=0 a J (p)ε p J F (p) (x, ϑ J ) + R(x, J),
where:

(a) a J (0) = J and a J 1, (b) There is an integer p J < |J| such that a J |a J (p J )|, (c) The norm a J is bounded from below by a quantity linked to W q :

a J max K∈Desc(J) | K | ε K ↑ ε J |J|-1 , (d) R(x, J) = o a J ε 2m J uniformly in x.
As a remark, the lower bound on F (x, J r ) will stem from points (c) and (d). Points (a) and (b) are mainly there for transmitting recurrence hypotheses. They control the size of F (x, J), together with point (c). The behaviour of F (x, J) only depends on the first |J| terms in the sum. However, the sum goes to 2m so that it is useful when J = J r .

Proof. The proof uses Taylor expansions at θ K for a given generation of children K together with separation and order properties of the coarsegraining tree T . Recall notation (24).

If K is an end of the tree T , then all the θ j for j ∈ K are equal, and F (x, K) = K F (x, θ K ). Choose a K (p) = K 1 {p=0} and R(x, K) = 0 so that the equality (27) holds for the end node K with all the desired estimates (a), (b), (c) and (d).

Assume now that J has children K, each of them satisfying ( 27) with all the estimates (a), (b), (c) and (d):

(28) F (x, K) = 2m =0 a K ( )ε K F ( ) (x, ϑ K ) + R(x, K),
We want to transmit (28) and the estimates to the parent J. Suppose without loss of generality that ϑ J ϑ K and apply Taylor's formula with remainder imsart-aos ver. 2014/10/16 file: papierAOSv18.tex date: July 8, 2017

to F ( ) (x, ϑ K ) at ϑ J for all ∈ [[0, 2m]]:

F ( ) (x, ϑ K ) - 2m-1 p= (ϑ K -ϑ J ) p- (p -)! F (p) (x, ϑ J ) = ϑ K ϑ J (ϑ K -θ) 2m-1- (2m -1 -)! F (2m) (x, θ)dθ. Substract the term (ϑ K -ϑ J ) 2m- (2m-)!
F (2m) (x, ϑ J ) from either side so that

F ( ) (x, ϑ K )- 2m p= (ϑ K -ϑ J ) p- (p -)! F (p) (x, ϑ J ) = ϑ K ϑ J (ϑ K -θ) 2m-1- (2m -1 -)! F (2m) (x, θ) -F (2m) (x, ϑ J ) dθ = (ϑ K -ϑ J ) 2m-O sup θ∈[ϑ J ,ϑ K ] |F (2m) (x, θ) -F (2m) (x, ϑ J )| .
The modulus of continuity of F (2m) (x, •) from B(2m) then yields

F ( ) (x, ϑ K ) = 2m p= (ϑ K -ϑ J ) p- (p -)! F (p) (x, ϑ J ) + o (ϑ K -ϑ J ) 2m-.
The normalised discrepancies φ K := (ϑ K -ϑ J )/ε J for K ∈ Child(J) are by definition at most of order 1 so that

F ( ) (x, ϑ K ) = 2m p= ε p- J φ p- K (p -)! F (p) (x, ϑ J ) + ε 2m- J o (1) .
Subtitute in (28) and change the order of summation:

F (x, K) = 2m p=0 p =0 a K ( ) ε K ε J φ p- K (p -)! ε p J F (p) (x, ϑ J ) + R(x, K) + ε 2m J max 0 2m a K ( ) ε K ε J o(1).
Add up over the children K of J to obtain (27) for J, that is

F (x, J) = 2m p=0 a J (p)ε p J F (p) (x, ϑ J ) + R(x, J), with a J (p) = K∈Child(J) p =0 a K ( ) ε K ε J φ p- K (p-)! and (29) R(x, J) = K∈Child(J) R(x, K) + ε 2m J max 0 2m a K ( ) ε K ε J o(1) .
We have now to prove the estimates (a), (b), (c) and (d) for the defined coefficients a J (p) and remainder R(x, J). Keep in mind that these estimates are assumed to be true for the children K and set for short

M p,K := max 0 p a K ( ) ε K ε J .
Proof of (a) for J . It is immediate from the definition of a J (p) that

a J (0) = K∈Child(J) a K (0) = K∈Child(J) K = J ,
and, using (a) for K together with ε K ε J , we get

(30) |a J (p)| max K∈Child(J) M p,K 1.
Proof of (b) for J . It's enough to establish (31) max

|J| p 2m |a J (p)| max K∈Child(J) M |K|-1,K max 0 p<|J| |a J (p)|.
To prove the l.h.s. of (31), note from (30) that |a J (p)| max K∈Child(J) M p,K . Moreover, for all p |K|,

M p,K M |K|-1,K + max |K| p a K ( ) ε K ε J M |K|-1,K + a K ε K ε J |K| and we have a K max 0 <|K| |a K ( )| by (b) so that, even for p < |K|, (32)
M p,K 1 + ε K ε J M |K|-1,K M |K|-1,K .
Taking the supremum over K ∈ Child(J) and over p give the l.h.s. of (31).

To prove the r.h.s. of (31), write a J (p) = a

(1)

J (p) + a (2) J (p) with a (1) J (p) = K∈Child(J) |K|-1 =0 a K ( ) ε K ε J φ p- K (p -)! 1 p , a (2)
J (p) = K∈Child(J) p =|K| a K ( ) ε K ε J φ p- K (p -)! 1 p .
imsart-aos ver. 2014/10/16 file: papierAOSv18.tex date: July 8, 2017

Note that {φ

K } K∈Child(J) is ε-separated since |φ K -φ K | = |ϑ K -ϑ K |/ε J 1 for K = K . Set λ K, = a K ( )(ε K /ε J )
and apply Corollary D.2 of [START_REF] Heinrich | Supplement to 'Minimax rates for finite mixture estimation[END_REF] to A({φ K } K∈Child(J) ) and Λ = (λ K, ) K∈Child(J), 0 <|K| :

(33) max

0 p<|J| |a (1) J (p)| max K∈Child(J) 0 <|K| a K ( ) ε K ε J = max K∈Child(J) M |K|-1,K ,
which is the r.h.s. of (31) with a

(1) J (•) instead of a J (•). We show now that the |a (2) J (p)| are in fact negligible so that the r.h.s. of (31) will follows. Indeed, easy bounds on a

(2)

J (p) yield max 0 p<|J| |a (2) J (p)| max K∈Child(J) a K ε K ε J |K| ,
whereas, as a by-product of (33), using a

K max 0 <|K| |a K ( )|, max 0 p<|J| |a (1) J (p)| max K∈Child(J) a K ε K ε J |K|-1
.

Proof of (c) for J . From the r.h.s. of (31), and (a) for K, we deduce (34) a J max K∈Child(J)

a K ε K ε J |K|-1 ∨ max K∈Child(J) | K |.
Here we used Note that ( 31) and (32) give max K∈Child(J) M 2m,K a J ; moreover,

M |K|-1,K M 0,K = | K |. Combining (34) with (c) for children K gives a J max K∈Child(J) max F ∈Desc(K) |π F | ε F ↑ ε J |K|-1 ∨ max K∈Child(J) | K |. Now, bound the exponent |K| by |J| to derive (c) for J. Proof of (d) for J . Split (29) as R(x, J) = R (1) (x, J) + R (2) (x, J) with R (1) (x, J) = K∈Child(J) R(x, K), R (2) (x, J) = ε 2m J K∈Child(J) M 2m,K o(1
R (1) (•, J) ∞ max K∈Child(J) o a K ε 2m K
by assumption (d) for K, so that by triangle inequality,

R(•, J) ∞ ε 2m J max K∈Child(J) o a K ε K ε J 2m + a J o(1) .
By (34), a J dominates a K (ε K /ε J ) 2m and thus (d) follows for J.

Concluding the proof of (23). We shall show that ( 35)

F (•, G n ) -F (•, G n ) ∞ max J∈Desc(Jr) | J |ε 2d 0 +1 J ↑ . Recall that F (x, G n ) -F (x, G n ) = F (x, J r
) and distinguish two cases:

Case ε Jr → 0. All the ϑ j,n 's converge to a single support point of G 0 so that m 0 = 1. Apply directly Lemma 7.4 to the root node J := J r : By (b), the optimal p is at most |J| -1, and since |J| 2m, we get

F (x, J) =
F (•, J) ∞ a J ε |J|-1 J .
Now, the estimate (c) for J yields further

F (•, J) ∞ max K∈Desc(J) | K |ε 2m-1 K ↑ .
But this estimate is nothing else than (35) since m 0 = m -d 0 is one. Case ε Jr ≡ 1. This case means either there are more than one support point in the limit G 0 (m 0 > 1) or there is only one support point for G 0 but with possible sequences θ j,n converging to other points (vanishing weights j,n may exist).

imsart-aos ver. 2014/10/16 file: papierAOSv18.tex date: July 8, 2017

Here, all the ε p Jr 's are of the same order (actually identical), so the scheme used for the case when ε Jr → 0 does not work. It works however for the children J of The optimal p is at most |J| -1 by (b), so that F (•, J r ) ∞ dominates max J∈Child(Jr) a J ε

|J|-1 J

, whereas for p = 0 we get that F (•, J r ) ∞ dominates max J∈Child(Jr) | J |, by (a). Together with ε J ↑ ≡ 1 and (c) for J, we obtain

F (•, J r ) ∞ max J∈Child(Jr) max K∈Desc(J)∪{J} | K |ε |J|-1 K ↑
which is nothing else than (36)

F (•, G n ) -F (•, G n ) ∞ max J∈Desc(Jr) | J |ε |J|-1 J ↑
. Now, note that a descendant J of J r of maximal cardinality must be a child of J r , call it J . Since G n and G n converge to G 0 ∈ G m 0 , the root J r has at least m 0 children, each of them containing at least two points. Thus, we have

|J r | |J | + 2(m 0 -1).
Since |J r | is at most 2m, we deduce further that |J | 2m -2m 0 + 2. From (36), we finally arrive at (35), exacly as in the case ε Jr → 0. Now, recall that q = 2m -2m 0 + 1. Lemma 7.3 together with (35) ensure that, whatever the case, ε Jr → 0 or ε Jr ≡ 1,

F (•, G n ) -F (•, G n ) ∞ W q (G n , G n ) q
which is the stated contradiction (23).

7.2. From local to global: how (18) implies (19). We have to show that, under Assumption B(2m), for r = 2m -1,

L := inf G =G ∈G m F (•, G) -F (•, G ) ∞ W r (G, G ) r > 0.
From the definition of L, we can select mixing distributions G n and G n in

G m such that F (•, G n ) -F (•, G n ) ∞ /W r (G n , G n ) r converges to L. Since the set G m × G m is compact, we can assume that (G n , G n ) converges to some limit (G ∞ , G ∞ ). Set w = W r (G ∞ , G ∞ ).
Case w > 0. This case does not depend on (18). By identifiability, there is

x 0 ∈ R such that ∆ 0 := |F (x 0 , G ∞ ) -F (x 0 , G ∞ )| > 0.
Then, for all n, (37)

F (•, G n ) -F (•, G n ) ∞ W r (G n , G n ) r |F (x 0 , G n ) -F (x 0 , G n )| W r (G n , G n ) r .
The numerator of the r.h.s. of (37) goes to ∆ 0 by the triangle's inequality and since the function θ → F (x 0 , θ) is Lipschitz w.r.t. the metric W 1 and thus also w.r.t. W r . As a consequence, we get L ∆ 0 /w r > 0. Case w = 0. Consider (18) with G 0 = G ∞ . For n larger than some n 0 , all W q (G n , G 0 ) and W q (G n , G 0 ) are less than ε so that by (18),

inf n n 0 F (•, G n ) -F (•, G n ) ∞ W q (G n , G n ) q > δ.
Since we have W q (•, •) q Diam(Θ) r-q W r (•, •) r for r q, we get

inf n n 0 F (•, G n ) -F (•, G n ) ∞ W r (G n , G n ) r > δ Diam(Θ) r-q
which gives L δ/Diam(Θ) r-q in the limit and (19) in that case. 7.3. Completing the proof of Theorem 6.3: the easy cases (20) and (21). For the proof of (20), we can simply make use of Theorem 3.1 of [START_REF] Ho | Identifiability and optimal rates of convergence for parameters of multiple types in finite mixtures[END_REF]. Alternatively, a detailed proof with our notations is available in the supplemental part (Heinrich and Kahn, 2015, B.2).

For the proof of ( 21), we can follow the proof of Chen (1995, Lemma 2) which holds here, because the γ j defined in his paper are all non-negative, and at least one is nonzero.

Remark 3. 1 .

 1 When comparing sequences we will write a n b n or a n = O(b n ) for a n Cb n where C > 0 does not depend on n. We will furthermore use a n b n for b n a n b n . If needed, the dependence of C on other parameters, say u, θ will be stressed by subscripts: a n u,θ b n or a n u,θ b n .

  , {j })|ϑ j -ϑ j | q . (25)imsart-aos ver. 2014/10/16 file: papierAOSv18.tex date: July 8, 2017

  p)ε p J F (p) (x, ϑ J ) + R(x, J),so that by the triangle's inequality, Proposition 2.3 and (d) for J,F (•, J) ∞ max 0 p 2m a J (p)ε p J -o a J ε 2m J .

  the triangle's inequality, Proposition 2.3 and (d) for J,F (•, J r ) ∞ max J∈Child(Jr) max 0 p 2m a J (p)ε p J -max J∈Child(Jr)o a J ε 2m J .

imsart-aos ver. 2014/10/16 file: papierAOSv18.tex date: July 8, 2017

Acknowledgements. We thank Sébastien Gadat for numerous suggestions that have greatly improved the presentation of the paper, and Élisabeth Gassiat for helpful discussions.

This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01).

SUPPLEMENTARY MATERIAL

Auxiliary results and technical details (doi: 10.1214/00-AOASXXXXSUPP; .pdf). This supplemental part gathers some proof details on some assertions given in the paper.