Edwin Fourrier-Nicolaı

Michel Lubrano
email: michel.lubrano@univ-amu.fr

Edwin Fourrier-Nicolaï

Bayesian inference for TIP curves: an application to child poverty in Germany

Keywords: Bayesian inference, Mixture model, Survey weights, Zero-inflated model, Child poverty

TIP curves are cumulative poverty gap curves used for representing the three different aspects of poverty: incidence, intensity and inequality. The paper provides Bayesian inference for TIP curves, linking their expression to a parametric representation of the income distribution using a mixture of log-normal densities. We treat specifically the question of zero-inflated income data and survey weights, which are two important issues in survey analysis. The advantage of the Bayesian approach is that it takes into account all the information contained in the sample and that it provides small sample credible intervals and tests for TIP dominance. We apply our methodology to evaluate the evolution of child poverty in Germany after 2002, providing thus an update the portrait of child poverty in Germany given in Corak et al. (Rev. Income Wealth 54(4), 2008).

Introduction

Poverty is usually measured as the proportion of households having an income below a poverty line. This proportion, equivalently called the head-count ratio, the at-risk-of-poverty rate or poverty incidence, is often taken as the unique measure of poverty, ignoring the shape of the income distribution among the poor. Since [START_REF] Sen | Poverty: an ordinal approach to measurement[END_REF] and [START_REF] Shorrocks | Revisiting the Sen poverty index[END_REF], research in poverty measurement has developed measures which take into account the distributional aspects of poverty. The Three I's of Poverty (TIP) curve, which was formally considered in [START_REF] Jenkins | Three I's of poverty curves, with an analysis of UK poverty trends[END_REF], is a cumulative poverty gap curve used for representing the different aspects of poverty usually provided by the first three poverty indices of [START_REF] Foster | A class of decomposable poverty measures[END_REF], measuring Incidence, Intensity and Inequality. This curve is also known in the literature as the Poverty Profile Curve or the Poverty Gap Profile (PGP) curve [START_REF] Barrett | Consistent tests for poverty dominance relations[END_REF]. TIP curves guaranty that poverty comparisons can be done in an unambiguous way (see e.g. Jenkins and Lambert 1998a, b;[START_REF] Davidson | Statistical inference for stochastic dominance and for the measurement of poverty and inequality[END_REF]. This desirable property comes from its close relationship with the generalised Lorenz curve. Specifically, the TIP curve is a transformation of the generalised Lorenz curve, and hence its dominance criterion is equivalent, under some conditions, to restricted second-order stochastic dominance. Consequently poverty ranking obtained by TIP dominance is robust according to the choice of a poverty line and of a set of poverty measures. Despite their attractiveness, TIP curves have only been used as a descriptive tool in few empirical studies, as for example in Del Rio and [START_REF] Del Rio | TIPS for poverty analysis. the case of Spain, 1980-81 to 1990-91[END_REF] and [START_REF] Kuchler | Incidence and intensity of smoothed income poverty in European countries[END_REF]. However, a recent growing attention has been devoted on providing statistical inference for TIP curves. [START_REF] Thuysbaert | Inference for the measurement of poverty in the presence of a stochastic weighting variable[END_REF] proposes statistical inference for TIP curves in the presence of stochastic survey weights. More recently, [START_REF] Barrett | Consistent tests for poverty dominance relations[END_REF] provides a non-parametric test for TIP dominance using influence functions. The aim of the present paper is to provide Bayesian inference for TIP curves (and also incidently for generalised Lorenz curves) and Bayesian characterisation of TIP dominance in the context of survey data.

The advantages of using a Bayesian approach are manyfold. First, the statistical test can be performed directly once inference is obtained. Second, distribution-free approaches might suffer from sensitivity to tails' behaviour and might appear unsatisfactory in small samples (see e.g. [START_REF] Cowell | Statistical methods for distributional analysis[END_REF]. As our main focus is on the poor, the question of tail sensitivity becomes crucial because we are concerned with the left tail of the income distribution. Particularly, a distribution-free estimator of the TIP curve throws away all the observations which are above the poverty line, making the question of sample size even more crucial when we are concerned with a sub-population like children. Bayesian inference for TIP curves at least partially overcomes these difficulties. While a distributionfree approach does not make any assumption about the shape of the income distribution, a Bayesian approach has to rely on a parametric or semi-parametric representation of the income distribution. By considering that the income distribution can be represented by a mixture of parametric distributions, we can have both a rather important flexibility obtained by letting the sample determine the number of components of the mixture and the advantage of a parametric representation using the whole sample so that no information is lost. By using a Bayesian approach, we increase accuracy and precision for inference and tests.

Using mixtures for modelling the income distribution has a rather rich history, mainly using log-normal densities. For instance, [START_REF] Flachaire | Estimation of the income distribution and detection of subpopulations: an explanatory model[END_REF] use a mixture of normal densities for the UK's log-transformed income distribution, [START_REF] Lubrano | Income inequality decomposition using a finite mixture of log-normal distributions: A Bayesian approach[END_REF] propose a decomposition of inequality indices using Bayesian inference for a mixture of log-normals and [START_REF] Anderson | Poverty status probability: a new approach to measuring poverty and the progress of the poor[END_REF] use a mixture of log-normals to identify the poor. Alternative specifications have been also used, for instance [START_REF] Ndoye | Tournaments and superstar models: A mixture of two Pareto distributions[END_REF] propose Bayesian inference for a mixture of Pareto densities for a model of high wage formation using US data. [START_REF] Chotikapanich | Estimating income distributions using a mixture of gamma densities[END_REF] and [START_REF] Lander | Bayesian Assessment of Lorenz and Stochastic Dominance[END_REF] use a mixture of gamma densities for comparing income distributions in Canada and in Indonesia. Although the mixture of log-normal densities is particularly appropriated for capturing the left tail of the distribution, we need to take into account both survey weights and the excess of zero value incomes which are two recurrent issues in survey data. This added flexibility makes the analysis more complex, particularly if we have to invert the cumulative distribution function of the mixture using numerical methods in order to compute quantile functions or a Lorenz curve.

We have chosen to illustrate our methods by discussing child poverty in Germany between 2000 and 2012 using the data of the German Socio-Economic Panel (GSOEP). Child poverty in Europe is an important question that has motivated many papers [START_REF] Jenkins | The dynamics of child poverty: Britain and Germany compared[END_REF][START_REF] Jenkins | Why are child poverty rates higher in Britain than in Germany? A longitudinal perspective[END_REF][START_REF] Corak | A portrait of child poverty in Germany[END_REF][START_REF] Hill | Poverty among British children: chronic or transitory?[END_REF][START_REF] Bradbury | The dynamics of child poverty in seven industrialised nations[END_REF], among others). More recent data than those used in these studies are now available and many events that had surely an impact on poverty have occurred since that period; we think specifically about the recent social and labour market reforms (the wellknown Hartz plan, 2003Hartz plan, -2005) ) and the reforms on family policies after 2005. And of course, although not specific to Germany, the 2008 financial crisis. One purpose of the present paper is to show which kind of new information the use of TIP curves can bring in for understanding the evolution of child poverty in Germany, using recently released data of the GSOEP in order to update the results found in [START_REF] Corak | A portrait of child poverty in Germany[END_REF].

The paper is organised as follows. In Section 2, we introduce TIP curves, their relation to the Lorenz curve and define TIP dominance. Section 3 is devoted to Bayesian inference for mixture of log-normal densities in the case of survey weights and zero inflated incomes. In Section 4, we derive the analytical formulae for TIP curves when the income distribution is modelled using a mixture of log-normals with sampling weights and zero-incomes. We also propose a test for comparing TIP curves. Section 5 analyses the evolution of child poverty in Germany. Section 6 concludes.

Measuring poverty using TIP curves

A formal definition

Let us consider a population where each household is endowed with an income y ∈ R + , the distribution of this income being F (y) and its density f (y). A household is said to be poor if its income is below the poverty line z. Poverty intensity is measured by the relative poverty gap, defined as follows:

max(1 -y/z, 0) = (1 -y/z)1(y ≤ z), ( 1
)
where 1(y ≤ z) is the indicator function which is 1 if y ≤ z and 0 otherwise. Integrating this relative poverty gap with respect to the income distribution f (y) provides the TIP curve of Jenkins and Lambert (1997): 1

T I P (p, z) = F -1 (p) 0 (1 -y/z)1(y ≤ z)f (y)dy, ( 2
)
where F -1 (p) is the quantile function, and p the proportion of individuals.

1 In this paper we consider the relative TIP curve built on the poverty gaps normalised by the poverty line. This is a well-known variant of the absolute TIP curve based on absolute poverty gaps max{zy, 0}. The advantages of the relative curve are that the FGT indices come directly from the curve and this variant is more appropriated for comparisons of TIP curves when there are different poverty lines. For values of p greater than the poverty incidence or head-count ratio given by P 0 = F (z), the TIP curve saturates and becomes horizontal. Its ordinate value is the poverty intensity P 1 or average poverty gap. Finally the curvature of the curve represents the inequality among the poor P 2 . A useful feature of the relative TIP curve is that poverty incidence P 0 , poverty intensity P 1 and poverty inequality P 2 are equivalent to the FGT(α) indices introduced in [START_REF] Foster | A class of decomposable poverty measures[END_REF] when α = 0, 1, 2 respectively. A TIP curve is thus a convenient device for displaying at the same time the three essential aspects of poverty, justifying its name the Three 'I's of Poverty.

In Fig. 1, we have represented the TIP curves of two simulated income distributions, using the same poverty line z. The top blue TIP curve represents a case of severe poverty where 30% of the individuals have zero income and the rest has an income greater or equal to the poverty line. In this case, the average poverty gap P 1 is the highest (according to the poverty incidence) i.e. P 1 = P 0 . This is a straight line reflecting the fact that every poor has the same income, hence all poverty gaps are the same. Inequality among the poor P 2 is maximum (maximum slope of 1) and the Gini among the poor is 1. 2 The green curve represents a situation where 40% of the population has an income drawn from a uniform distribution between zero and z and the rest an income greater than z. The average poverty gap is 0.2. The curvature of the TIP curve is directly related to the inequality among the poor, exhibiting a Gini of 0.33. The closer the income of the poor individuals are to the poverty line, the smaller are the poverty gaps, and the closer the slope is to 0. Figure 1 represents a situation where the ranking of poverty incidence is the opposite of the ranking of poverty intensity and inequality.

TIP curves and generalised Lorenz curves

TIP curves are closely linked to the generalised Lorenz (GL) curve as can be shown now. Decomposing Eq. 2 and substituting q for F -1 (p), we obtain:

T I P (p, z) = q 0 f (y)dy - 1 z q 0 y f (y)dy, for p ≤ F (z). ( 3
)
The first integral is the cumulative distribution function evaluated at q and the second integral is the generalised Lorenz curve GL(p). These integrals can be simplified when an analytical form for the quantile function q = F -1 (p) is available and consequently also for the Lorenz curve. In this case, we have access to the direct formulation given in [START_REF] Davidson | Statistical inference for stochastic dominance and for the measurement of poverty and inequality[END_REF]:

T I P (p, z) = p - 1 z GL(p), for p ≤ F (z). ( 4
)
As an illustration, let us consider the log-normal case with f (y|μ, σ 2 ). 3 The Lorenz curve of a log-normal is equal to L (p) = ( -1 (p)σ ) where is the standard normal cdf. Multiplying L (p) by the mean of the log-normal exp(μ + σ 2 /2) in order to get the generalised Lorenz curve GL (p), we find that the TIP curve associated to log-normal density is:

T I P (p, z) = p - 1 z exp(μ + σ 2 /2) ( -1 (p) -σ ) for p ≤ F (z).
(5)

However, we cannot take advantage of the simplicity of Eq. 4 if we want to model the income distribution using a mixture of log-normals. With mixtures of distributions, the total cdf F (.) is a weighted average of the cdf of each member, but its inverse q = F -1 (p) has no analytical form and has to be solved by a numerical procedure. So, we cannot generalise Eq. 5 easily. Instead of considering (4), we have to work with the initial form of the TIP curve, given in (3).

Stochastic dominance and TIP dominance

The use of a dominance criteria leads to poverty comparisons that are robust to the choice of the poverty line. As TIP curves are related to generalised Lorenz curves, they provide a natural framework for testing restricted second order stochastic dominance. We define a TIP ordering or TIP dominance as follows:4

Definition 1 Let us consider two income distributions corresponding to populations A and B and their corresponding TIP curves. Distribution A TIP dominates distribution B for a given poverty line z if T I P A (p, z) ≤ T I P B (p, z) ∀p ∈ [0, 1], and the strict inequality holds at least for one p.

Since generalised Lorenz dominance is strictly equivalent to second order stochastic dominance [START_REF] Atkinson | On the measurement of poverty[END_REF][START_REF] Foster | Poverty orderings[END_REF], it can be easily shown that TIP dominance with a common poverty line z is equivalent to restricted second order stochastic dominance over the interval [0, z] or equivalently for p ∈ [0, F (z)]. This correspond to Theorem 2 of [START_REF] Jenkins | Three I's of poverty curves and poverty dominance: TIPs for poverty analysis[END_REF] which can be rephrased as:

Theorem 1 If distribution A TIP dominates distribution B for all p ∈ [0, F (z)], then A stochastically dominates B at the second order over the interval [0, z]. And this poverty ordering holds for all common poverty lines below z.

In other words, if T I P A (p, z) is always below T I P B (p, z) with a common z, then the two distributions can be ordered with less poverty intensity and inequality in A than in B for all common poverty lines smaller than or equal to z.

At this stage, some remarks have to be made. Second order stochastic dominance and TIP ordering imply the ordering of FGT indices corresponding to P 1 and P 2 (intensity and inequality) and all similar indices built around a relative poverty gap measure (see [START_REF] Davidson | Statistical inference for stochastic dominance and for the measurement of poverty and inequality[END_REF] and [START_REF] Jenkins | Three I's of poverty curves and poverty dominance: TIPs for poverty analysis[END_REF] among others). However, a robust ordering of poverty incidence is not guaranteed, as, for this, we would have to check for first order stochastic dominance. This situation is illustrated in Fig. 1 as commented before. Finally, and like for Lorenz curves, when TIP curves intersect there is indeterminacy since the poverty ranking can be reversed for some values of p. Various ways to remove this indeterminacy have been proposed: imposing more normative conditions and considering higher-degree dominance criteria as in [START_REF] Sordo | Poverty comparisons when TIP curves intersect[END_REF], Jenkins and Lambert (1998a) or considering a narrower range of values of z as in [START_REF] Atkinson | On the measurement of poverty[END_REF].

Bayesian inference for a mixture of log-normals in survey data

Bayesian inference for mixtures of distributions is fairly standard and was detailed for instance in [START_REF] Lubrano | Income inequality decomposition using a finite mixture of log-normal distributions: A Bayesian approach[END_REF] for log-normals. In this section, after recalling the necessary formulae, we extend the analysis to two important features when modelling survey data: the use of sample weights and the modelling of extra zeros. We compare our solutions to the existing literature [START_REF] Thuysbaert | Inference for the measurement of poverty in the presence of a stochastic weighting variable[END_REF][START_REF] Si | Bayesian nonparametric weighted sampling inference[END_REF][START_REF] Savitsky | Bayesian estimation under informative sampling[END_REF][START_REF] Gunawan | Bayesian Weighted Inference from Surveys[END_REF], for the main references).

Bayesian inference for a mixture of log-normals

A finite mixture f (y|ϑ) of log-normals is a linear combination of K log-normal densities f (y|μ, σ ) such that:

f (y|ϑ) = K k=1 η k f (y|μ k , σ 2 k ), 0 < η k < 1, K k=1 η k = 1, (6)
where the η k are the weights of the mixture and (μ k , σ 2 k ) the parameters of the k th component. ϑ = (η, μ, σ 2 ) represents the collection of all the parameters of the mixture. The pdf of the log-normal is written as:

f (y|μ, σ ) = 1 yσ √ 2π exp -(ln y -μ) 2 2σ 2 .
A mixture model is an incomplete data problem which can be completed by an auxiliary variable z which allocates each observation to a given member of the mixture. Then conditionally on a given sample allocation [z i = k], each component can be analysed separately. Following [START_REF] Lubrano | Income inequality decomposition using a finite mixture of log-normal distributions: A Bayesian approach[END_REF], each component is equipped with a natural conjugate prior: a conditional normal prior on

μ k |σ 2 k ∼ f N (μ k |μ 0 k , σ 2 k /n 0 k ) and an inverted gamma prior on σ 2 k ∼ f iγ (σ 2 k |v 0 k , s 0 k ).
A Dirichlet prior is used for the vector of the weights: η ∼ f D (γ 0 1 , ..., γ 0 K ). The hyperparameters of these priors are the k-vectors v 0 , s 0 , μ 0 , n 0 , γ 0 . The conditional posterior densities belong to the same families and serve to draw a new sample allocation, resulting in a Gibbs sampler algorithm which is described as follows: 5 1. Set K the number of components, m the number of draws, m 0 the number of warming draws and initial values of the parameters ϑ (0) = (μ (0) , σ (0) , η ( 0) ). 2. For j = 1, . . . , m + m 0 :

(a) Generate a classification z (j ) i conditionally on ϑ (j -1) , independently for each observation y i according to a multinomial process with probabilities:

P r(z i = k|y, ϑ (j -1) ) = η (j -1) k f (y i |μ (j -1) k , σ 2(j -1) k ) k η (j -1) k f (y i |μ (j -1) k , σ 2(j -1) k ) . ( 7
) (b) Compute the conditional sufficient statistics n k , ȳk , s 2 k : n k = n i=1 1(z i = k), (8) ȳk = 1 n k n i=1 log(y i )1(z i = k), ( 9
)
s 2 k = 1 n k n i=1 (log(y i ) -ȳk ) 2 1(z i = k). ( 10
)
(c) Given the classification z (j ) , generate the parameters σ (j ) , μ (j ) , η (j ) from the posterior densities (11), ( 12) and ( 13):

-The conditional posterior density of σ 2 k is an inverted gamma:

p(σ 2 k |y, z) = f iγ (σ 2 k |v * k , s * k ), (11)
-The conditional posterior density of μ k |σ 2 k is a conditional normal:

p(μ k |σ 2 k , y, z) = f N (μ k |μ * k , σ 2 k /n * k ), (12)
-The conditional posterior density of η is a Dirichlet:

p(η|y, z) = f D (γ 0 1 + n 1 , ..., γ 0 K + n K ) ∝ K k=1 η γ 0 k +n k -1 k , ( 13
)
5 Gibbs sampler algorithms in the context of mixture models are presented in Fruhwirth-Schnatter ( 2001), [START_REF] Fruhwirth-Schnatter | Finite Mixture and Markov Switching Models[END_REF] where is discussed, in particular, the question of label switching and its tentative solutions. [START_REF] Lubrano | Income inequality decomposition using a finite mixture of log-normal distributions: A Bayesian approach[END_REF] discuss the Gibbs sampler when applied to mixtures of log-normals and, in particular, how to select prior information.

with:

n * k = n 0 k + n k , μ * k = (n 0 k μ 0 k + n k ȳk )/n * k , v * k = v 0 k + n k , s * k = s 0 k + n k s 2 k + n 0 k n k n 0 k + n k (μ 0 k -ȳk ) 2 .
(d) Order σ (j ) such that σ (j )

1 < ... < σ (j )
K and sort μ (j ) , η (j ) and z (j ) accordingly to solve label switching. (e) Increase j by one and return to step (a).

3. Finally discard the first m 0 stored draws and compute posterior moments and marginal densities using the remaining draws.

Introducing survey weights

In population studies, it is common to sample individuals through a complex sample design in which probabilities of inclusion are associated with the variables of interest. This correlation results in observations that are not identically distributed: individuals with different characteristics have different probabilities of being selected. Ignoring the sampling design could bias inference results [START_REF] Pfeffermann | The use of sampling weights for survey data analysis[END_REF]. Survey weights are constructed to correct for discrepancies between the sample and the population. The literature on the use of survey weights is not abundant, although there is a recent growing concern about this issue.

Most recently developed methods in a Bayesian context (see e.g. [START_REF] Si | Bayesian nonparametric weighted sampling inference[END_REF] and Savitsky and Toth (2016) for a survey) propose the joint modelling of the weights and of the variable of interest. In this case, weights are random (this is what is considered for instance in [START_REF] Thuysbaert | Inference for the measurement of poverty in the presence of a stochastic weighting variable[END_REF] for classical inference on TIP curves). Savitsky and Toth (2016) consider exogenous weights which serves to modify the likelihood function for a subsequent Bayesian inference. The case of Bayesian inference for mixture of distributions is rather particular, because inference is rarely based on the complete likelihood function. Introducing weights in this context led to a very scare literature. [START_REF] Gunawan | Bayesian Weighted Inference from Surveys[END_REF] propose a general method based on data augmentation where the representative data are treated as missing and are resampled from the observed sample using the weights. Instead, we propose an approach where the observations are given, the weights are exogenous and are introduced at the level of the conditional likelihood. The conditional likelihood formulated from the completed mixture problem with weights should be:

(y|ϑ, z) = K k=1 i;z i =k η w i k f (y i |μ k , σ 2 k ) w i (14)
This is an adaptation of the method of [START_REF] Savitsky | Bayesian estimation under informative sampling[END_REF] to the case of mixtures of log-normals where the conditional likelihood function is modified by the weights w i , once the sample separation is made. Because f (y i |μ k , σ 2 k ) belongs to the exponential family, this is equivalent to computing sufficient statistics using weights.

More precisely, let us consider n individuals that are sampled from the whole population with sampling weights w i = 1/π i constructed to be the inverse of the inclusion probability π i that individual i belongs to the survey. Let wi be the normalised weights such that wi = nw i / w i , so that they sum to the sample size n. Conditional inference using ( 14) is equivalent to using sampling weights when evaluating the sufficient statistics in (8-10), such that:

n k = n i=1 wi 1(z i = k), (15) ȳk = 1 n k n i=1 wi log(y i )1(z i = k), ( 16
)
s 2 k = n k n 2 k -n i=1 wi 2 1(z i = k) n i=1 wi (log(y i ) -ȳk ) 2 1(z i = k). ( 17
)
Given these weighted sufficient statistics, the η k are drawn using ( 13) together with a new sample separation obtained using (7). So a mixture estimate of the income distribution, representative of the original population, is obtained simply by introducing sampling weights at the level of the conditional sufficient statistics.

Modelling zero-inflated data

Another recurrent feature of survey data is the excess number of zeros (greater than expected under the distributional assumption). Particularly in income studies, zero incomes are numerous when measured before taxes and transfers. Actually, a large part of the population, such as elderly persons or unemployed workers, has no market income. As the log-normal is defined with strictly positive support, we propose to add an extra component for modelling the zero incomes:

f (y|ϑ) = ω if y = 0, (1 -ω) K k=1 η k f (y|θ k ) if y > 0, (18)
where ω = Pr(y = 0) ( i 1(y i = 0)w i )/ w i . This is a zero-inflated mixture model where ω is estimated as the (weighted) proportion of zeros in the sample, while inference on the other parameters is performed on the sample excluding the zeros. The cdf corresponding to this zero-inflated mixture is:

F (y|ϑ) = ω if y = 0, ω + (1 -ω) K k=1 η k F (y|θ k ) if y > 0. ( 19
)
It will be used in Section 4.2 for making Bayesian inference on TIP curves. This modelling of zero-inflated data is simple and particularly useful within a parametric approach, knowing that values close to 0 cause problems for many parametric families. Even more, non-parametric estimators might fail to represent this feature of the sample. Figure 2 represents estimates of the German disposable income distribution for the whole population in 2009 using the GSOEP data. 6 The kernel density estimate smooths out the zero-excess observations. Worse, still the kernel density estimate seems to be less flexible than the 3 component mixture at the beginning of the distribution, which is important for analysing poverty. Even if the proportion of zeros seems negligible in this sample of disposable income with 0.15%, it is still important to take it into account. If we had considered market income, that proportion would have gone up to 5% for the same year.

Bayesian inference for TIP curves

In order to provide inference for TIP curves, we have to express them as a function of the model's parameters. Since the quantile function of a mixture has no closed form, we cannot generalise Eq. 4 to the case of mixtures of log-normals. We have to start from the original definition of the TIP curve given in (3), where the expression of the quantile q is left unspecified.

An alternative TIP curve formula

Assuming that the data generating process of f (y) is a mixture of log-normal as given in (6), we can decompose the general TIP formula, using the linearity property of the integral:

T I P (p, z) = K k=1 η k q 0 f (y|μ k , σ k ) dy - 1 z K k=1 η k q 0 yf (y|μ k , σ k ) dy,
for p ≤ F (z). The first part of the right-hand side is the cdf of the mixture of log-normals:

K k=1 η k F (q|μ k , σ k ) = K k=1 η k ln q -μ k σ k ,
knowing that is the standard normal cdf and that the mixture's cdf is the weighted sum of the components' cdf. In the second part of the right-hand side, we find the weighted sum of components' generalised Lorenz curve for the log-normal:

K k=1 η k q 0 yf (y|μ k , σ k ) dy = K k=1 η k exp(μ k + σ 2 k /2) ln q -μ k -σ 2 k σ k .
As a matter of fact, q cannot be substituted within each component as in equation ( 5), since q here represents the value of the p quantile of the complete mixture and the quantile of a is, of course, not a linear function of the quantile of each component and has to be evaluated numerically. We can write the following expression of the TIP curve for a mixture of log-normal densities:

T I P (p, z) = K k=1 η k ln q -μ k σ k - 1 z exp(μ k + σ 2 k /2) ln q -μ k -σ 2 k σ k , ( 20
)
for p ≤ F (z). It turns out that the TIP curve of a mixture of log-normal densities is the weighted sum of the components' TIP curves, as given in ( 5), evaluated at the mixture's quantiles. As the left-hand side is a function of p, while the right-hand side is a function of q, we have to complete this equation by a relation between p and q, solving numerically in q the equation p = F (q). The expression of F (y) is K k=1 η k F (y|θ k ). We consider a grid of points p s for p ∈ [0, 1] and s = 1, . . . , S. For each point p s and for each value of ϑ, we solve numerically in q the equation:

F (q|ϑ) = p s .
This can be done using the uniroot function in R. This function is based on the [START_REF] Brent | An algorithm with guaranteed convergence for finding a zero of a function[END_REF][Chapter 4]'s method, a root-finding algorithm combining the bisection method, the secant method and the inverse quadratic interpolation. Convergence is always guaranteed provided that the predefined interval contains the solution. The execution time of the Brent's algorithm is negligible. 7Let us now suppose that we have obtained m posterior draws of the parameters ϑ (j ) from the Gibbs sampler. For each point p s of a predefined grid, we can obtain m draws for the TIP curve, applying (20) and solving numerically for q the equation F (q|ϑ (j ) ) = p s . The posterior mean of each point of the TIP curve corresponding to a value of p s is obtained as the mean of all these m posterior draws for each value of p s . The 0.05 and 0.95 quantiles of these m draws provide an evaluation of a 90% credible interval of the TIP curve.

TIP curves for the zero-inflated model

When we take into account the excess of zero incomes, the expression of the cumulative distribution function is changed into that given in equation ( 19). Moreover, we have to solve Eq. 2 separately for 0 ≤ p ≤ ω and then for ω ≤ p ≤ 1. For the range 0 ≤ p ≤ ω, we know that y = 0, so equation (2) becomes:

T I P (p, z) = F -1 (p) 0 f (y)dy = p.
For the range ω ≤ p ≤ 1, we make use of Eq. 19, so the expression of the TIP curve (2) becomes:

T I P (p, z) = F -1 (p) 0 (1 -ω) K k=1 η k f (y|θ k )dy - 1 z F -1 (p) 0 y(1 -ω) K k=1 η k f (y|θ k )dy,
and after integration:

T I P z) = ω + (1 -ω) K k=1 η k ln q -μ k σ k - (1 -ω) z K k=1 η k exp(μ k + σ 2 k /2) ln q -μ k -σ 2 k σ k . ( 21
)
The TIP curve for an inflated-zero model can be written as:

T I P (p, z) = p 0 ≤ p ≤ ω ω + (1 -ω)T I P (p, z) ω ≤ p ≤ 1, (22)
where T I P (p, z) was defined in (20). For evaluating (22), the value of q has to be determined using (19).

Testing for TIP dominance

Let us consider two dominance curves D s A (y) and D s B (y) at the order s for the two states A and B, respectively. We say that A dominates B if D s A (y) ≤ D s B (y). When testing for stochastic dominance (or TIP dominance), we have to compare two curves at a given number of points. Most statistical tests of stochastic dominance consider a null of dominance, see [START_REF] Cowell | Statistical methods for distributional analysis[END_REF] for a survey of the classical approach. This means that the null hypothesis H 0 : D s A (y) ≤ D s B (y) has to be verified for all values of y while the alternative is chosen if it is valid for some values of y. It is worth mentioning that non-rejection of the null does not imply a situation of non-dominance. Davidson andDuclos (2000, 2013) consider a test where the null is non-dominance and the alternative is dominance. In this case, we say that A does not dominate B and the null is H 0 : D s A (y) ≥ D s B (y) which has to be verified for some y. If the null is rejected, the alternative is dominance and is valid for all values of y. Because the test is based on finding the least favourable case, it relies on the minimum over y of the distance between the two curves. Translated into the domain of TIP dominance, this type of test would consider the minimum distance between two TIP curves computed over a grid for p.

Hypothesis testing is the domain where there is the greatest difference between the classical and the Bayesian approach. Essentially in a Bayesian framework there is no privileged hypothesis. The two hypotheses H 0 and H 1 are compared by means of the ratio between their posterior probability, the famous Bayes factor Pr(H 0 |y)/ Pr(H 1 |y). The main question for evaluating a Bayes factor is to find a convenient way to compute the posterior probability of a hypothesis. In our case, the probability we want to compute is:

Pr[δ(x, p|θ) ≤ 0], ∀p ∈ [0, 1],
where δ(x, p|θ) = T I P A (p, z|θ A ) -T I P B (p, z|θ B ), and θ A and θ B are the parameters associated to TIP curves A and B, respectively. The probability that δ(x, p|θ) ≤ 0 is the probability that A TIP dominates B. We evaluate each TIP curve on a grid of fixed points, conditionally on a draw of the parameters, say θ (j ) A and θ (j ) B . The condition δ(x, p|θ) ≤ 0 defines a logical vector of zeros and ones of dimension S, the dimension of the grid. It is equivalent to check any of the three following conditions:

S i=1 1(δ(p i |ϑ) < 0) = 1, max i 1(δ(p i |θ) < 0) = 1, min i 1(-δ(p i |θ) > 0) = 1.
The posterior probability of this event can be evaluated easily once we have obtained m posterior draws of the parameters ϑ (j ) from the Gibbs sampler. More formally:

Pr max p d(p|y) < 0 = ϑ 1 max p d(p|ϑ) < 0 μ(ϑ|y)dϑ 1 m m j =1 1 max p d(p|ϑ (j ) ) < 0 , ( 23
)
where μ(ϑ|y) is the posterior density of the parameters.8 This procedure is general and allows us to obtain the probability of stochastic dominance, restricted stochastic dominance, Lorenz dominance and TIP dominance depending on the grid defined and the poverty lines chosen.

Because TIP dominance is equivalent to restricted second order stochastic dominance, there is no guarantee that in the case of TIP dominance there is also less poverty incidence as shown for instance in Fig. 1. We thus have to complement our TIP dominance test by a test comparing two poverty incidence measures. When evaluating T I P A (p, z|θ (j ) A ) and T I P B (p, z|θ (j ) B ), we have to compute

P 0 A = h(z, θ (j ) A ) and P 0 B = h(z, θ (j )
B ) in order to determine at which point the graph of the TIP curve becomes horizontal. Testing for a lower poverty incidence in A than in B is equivalent to computing the proportion of cases where h(z, θ

(j ) A ) < h(z, θ (j ) B ) for j = 1, . . . , m.
Finally, when can we say that the situation in A is not statistically different from the situation in B? Equality is rejected if, for at least one value of p s , δ(p s |ϑ) is statistically different from zero. This means that we have to compute a credible interval for δ(p s |ϑ) and see if zero is included in this interval. If we find a single p s for which zero does not belong to a say 90% credible interval for δ(p s |ϑ), then we can reject at the 90% level that the two TIP curves are equal.

How do our method is related to the existing literature? [START_REF] Chotikapanich | Bayesian assessment of Lorenz and stochastic dominance in income distributions[END_REF] relate Lorenz dominance and stochastic dominance to parametric restrictions when the income distribution is modelled with either a Dagum or a Singh-Maddala distribution.

On one side, they compute the posterior probability that these parametric restrictions are satisfied. On the other side, they compute the probability that two dominance curves are ordered for all values of y or all values of p when they compare Lorenz curves. In a MCMC framework, these probabilities are evaluated by counting the number of times the inequality condition is satisfied. [START_REF] Lander | Bayesian Assessment of Lorenz and Stochastic Dominance[END_REF] take into account the necessity to have a flexible formulation for modelling the income distribution using a mixture of gamma distributions. Numerous papers have proposed statistical tests based on stochastic dominance in a classical approach, see [START_REF] Xu | A distribution free tests for deprivation dominance[END_REF] and [START_REF] Berihuete | Welfare, inequality and poverty analysis with rtip: An approach based on stochastic dominance[END_REF] among others. [START_REF] Corak | A portrait of child poverty in Germany[END_REF] provides some of the most recent results concerning the evolution of child poverty in Germany, using data from the German Socio-Economic Panel (GSOEP). They provide information on the evolution of poverty incidence both for children and for adults (adults are defined as those in households without children). They analyse the East-West contrast, the impact of family composition and of citizenship status. They provide information on poverty dynamics by examining the average length of poverty spells, computing probabilities of entry into poverty and exit of poverty. Finally, they compare the impact of the redistributive system between Germany and a group of English speaking countries.

A new portrait of child poverty in Germany

However, their reporting period ends in 2004. As new data are now available, we propose to investigate the period 2000-2012 in order to check if child poverty follows the same trends as the ones depicted before or if there has been a structural break. TIP curves will also complement the portrait of poverty incidence given in [START_REF] Corak | A portrait of child poverty in Germany[END_REF] with information about poverty intensity and poverty inequality. They will also provide information on poverty dynamics when applied to smoothed income as in [START_REF] Kuchler | Incidence and intensity of smoothed income poverty in European countries[END_REF] in order to depict chronic poverty, as explained below.

Data and methodological issues

The GSOEP is a socio-economic panel provided by the German Institute for Economic Research in Berlin (DIW). It is a representative sample of households living in Germany since 1983 and including former East German households after the reunification. For the period 2000-2012, each wave covers on average 11 623 households. Among these, on average 3 154 households have children and their average number of children is 1.48 so that on average, the sample contains 4 680 children in each wave.

The survey is a stratified sample with weights. Cross sectional weights are introduced in order to match individual and household profiles to those of the population. Longitudinal weights are computed to account for the probability of a household to stay in the survey next year. New households are regularly added to the panel to compensate for attrition.

Two types of income can be reconstructed: an annual market income which represents labour income, capital income, in fact all incomes coming from a market activity; a disposable income which is the market income minus taxes and plus redistribution including unemployment benefits, social security pensions, family allowances and all remaining forms of social redistribution. We consider the real disposable income obtained by dividing the current income by the Consumer Price Index (2005) provided in the GSOEP. In order to keep coherency with the paper of [START_REF] Corak | A portrait of child poverty in Germany[END_REF], we define the poverty line as 50% of the sample median disposable income (taking into account all households, those having children and those without children). Remark that the disposable income is normalised by the new OECD equivalence scale and expressed in real terms. 9 Computed poverty lines are provided in the left panel of Fig. 3. The average poverty line is 8 455 euros per year and per equivalent adult over the whole period. The annual poverty line is slightly greater than this value before 2006 and slightly below after that date. But the fluctuation is less than 2%, so we have decided to keep the same poverty line over the entire period, which is more convenient to implement tests of TIP dominance.

Analysing child poverty means that we consider households with children having an equivalent disposable income below the poverty line. We define a child as a person under 18 years old. It would be misleading to consider a household as the unit of observation, ignoring the number of children in the household. This would certainly under-estimate the number of poor children in the country. A more realistic approach, followed by [START_REF] Hill | Poverty among British children: chronic or transitory?[END_REF], [START_REF] Jenkins | Why are child poverty rates higher in Britain than in Germany? A longitudinal perspective[END_REF], [START_REF] Corak | A portrait of child poverty in Germany[END_REF] and many others, consists in considering a child (and not a household) as the unit of observation, which means having possibly several observations coming from the same household. The child poverty incidence rate corresponds thus to the ratio between the number of children living in poor households over the total number of children.

We report in the right panel of Fig. 3 the evolution of child poverty incidence, distinguishing between East and West Germany. The child poverty head-count ratio follows a rising trend until 2006. This corroborates the findings of [START_REF] Corak | A portrait of child poverty in Germany[END_REF] for the period 1999-2004. But this rising evolution stops in 2006 as from that date child poverty incidence decreases, except in East Germany after the financial crisis. This change might reflect the delayed effect of the Hartz reforms which introduced large changes in social assistance. 10We conclude that there is a clear break around 2006, which justifies considering separately two periods for dynamic analysis.

Analysing poverty dynamics has a long history starting with [START_REF] Bane | Slipping into and out of poverty: the dynamics of spells[END_REF]. They propose to study the length of poverty spells, a poverty spell; that is, the period in which a household is below the poverty line. [START_REF] Corak | A portrait of child poverty in Germany[END_REF] analyse the length of poverty spells of households with children. They also compute the probability of entering into poverty and the probability of exiting from poverty. However, this approach has been criticised because of the bias introduced by censored spells and also because it considers only one dimension of poverty, namely poverty incidence. [START_REF] Rodgers | Chronic poverty in the United States[END_REF] promoted the idea that households can transfer income from one year to the other, then poverty has to be portrayed with respect to the smoothed or permanent income. It is computed at the household level over a given period, usually taking the mean income over the period as in [START_REF] Hill | Poverty among British children: chronic or transitory?[END_REF]. Then, chronic poverty is when the household smoothed income (as evaluated over a given period) is below the poverty line. [START_REF] Kuchler | Incidence and intensity of smoothed income poverty in European countries[END_REF] pushed the analysis one step further, considering TIP curves using smoothed income to depict chronic poverty.

Analysing the dynamics of poverty requires panel data. We have to be able to follow the same household over a given period, which means that the panel has to be balanced over that period. Because the year 2012 led to huge attrition, we preferred to discard this end-of-sample year. Second, we have seen in Fig. 3 that the dynamics of poverty incidence has changed in 2006. So there is a strong interest in considering two subperiods and thus providing a separate analysis of chronic poverty for each of them. The balanced panel for the subperiod 2007-2011 covers five years and contains 2 236 children. In order to have the same number of years in the first period, we drop 2000 and 2001 so that the period 2002-2006 contains also five years and 2 991 children. We then compute two series of smoothed income obtained as individual weighted means using longitudinal weights.

The evolution of child poverty

We shall first compare TIP curves computed at the beginning of a sub-period and at the end of the same subperiod to measure the evolution of the three aspects of current poverty. If there is TIP dominance, this will give a clear indication on the evolution of at least two dimensions of poverty, intensity and inequality. Investigating the first dimension, incidence, requires a specific formal test.

The left part of Fig. 4 clearly shows that there is a significant increase in the three aspects of child poverty over the first period as a) TIP curves do not intersect, b) their 90% credible intervals do not overlap, c) the 90% credible intervals of poverty incidence also do not overlap. We have a confirmation of the results of [START_REF] Corak | A portrait of child poverty in Germany[END_REF] who concluded to an increase of the incidence of child poverty in Germany between 1999 and 2004. But this result is extended a) to the three dimensions of child poverty and b) till the year 2006, and for all poverty lines lower than 8 455 euros.

The behaviour of child poverty during the second period (2007-2011) is just the opposite. As the graphs have the same scale, visual comparison is made easy. Current poverty has decreased significantly between the beginning and the end of that period. There is a significant reduction of child poverty in all its dimensions as all credible intervals do not overlap.

Formal dominance tests confirm these results. For current child poverty, the last line of Table 1 shows that 2011 dominates all the other reported years and that poverty incidence dominates all the other years with probability 1.000). The ordering of poverty incidence is in accordance, so that we can order simultaneously the three aspects of poverty at similar degree of probability. We have also tested and found that the TIP curves in 2002 and 2007 were not statistically different. The global portrait of child poverty depicted in [START_REF] Corak | A portrait of child poverty in Germany[END_REF] has thus completely changed.

Chronic poverty and the East-West contrast

It is much worse for a household to be in a state of chronic poverty over a long period (here five years) than being temporarily in poverty. So we shall concentrate from now on chronic poverty, computing TIP curves with smoothed income instead of current income, Fig. 5 The West-East contrast of chronic child poverty following the approach of [START_REF] Kuchler | Incidence and intensity of smoothed income poverty in European countries[END_REF], in order to explore the East-West contrast. East and West Germany have been reunified in 1990. However, the convergence between these two regions is slow and the economic differences are still important. [START_REF] Corak | A portrait of child poverty in Germany[END_REF] concluded that child poverty incidence was much more important in the East part of Germany. This is confirmed for child chronic poverty during period I (2002-2006) as seen when comparing the two panels of Fig. 5. However because the credible intervals are large, there is no TIP dominance of the West over the East for the first period (the probability of TIP dominance is 0.212, but incidence is lower with a probability of 1.000 as reported from Table 2).

There is a massive reduction of chronic child poverty during period II (2007II ( -2011)), both in East and West Germany as seen from the two panels of Fig. 5. As a matter of fact, we observe TIP dominance of period II over period I for the two regions (left part of Table 2) and also a significant lower child poverty incidence (right part of Table 2). This massive reduction of chronic child poverty has erased the differences between the two regions concerning chronic child poverty. The two TIP curves were tested not to be statistically different between the two regions (the two red dashed curves in the two panels of Fig. 5). We conclude that the redistributive system has been very efficient during the second period for fighting against chronic child poverty and after the Hartz reforms came into force. We have provided Bayesian inference for TIP curves. Specifically, we have proposed parametric modelling of the income distribution using a mixture of log-normal densities. We also solved two questions raised by the use of survey data: incorporating survey weights and taking into account explicitly zero-income data. Once we have obtained random draws of the income distribution parameters from the posterior distributions, the TIP curves are a (not so) simple transformation of these draws, which means that we have direct access to statistical inference (both credible intervals and testing). In doing so, we propose a solution for computing the quantiles of a mixture model.

Using TIP curves, we gave an empirical portrait of child poverty in Germany, which is complementary to that of [START_REF] Corak | A portrait of child poverty in Germany[END_REF], both for the period considered and for the tools we used. We showed that poverty head-counts rates have to be completed by other dimensions and that TIP curves are very convenient for this issue. In some cases, the poverty incidence ordering is consistent with the ordering in the other dimensions, but in other cases, this is not so clear. We provided an example with adult poverty reported in Table 3 of the online Appendix C.

Child poverty continued to follow the upward trend found in [START_REF] Corak | A portrait of child poverty in Germany[END_REF] and that up to 2006. However, after 2006, the portrait of child poverty in Germany has totally changed with a decrease in both current and chronic poverty and a reduction of the gap between adult and child poverty. The gap between East and West chronic child poverty was also much reduced, becoming not significant. There has thus been a convergence on this point between the two parts of Germany. Meanwhile, adult chronic poverty (as documented in online Appendix C) has increased a lot in East Germany while it was decreasing in the West. We might see there a consequence of the structural changes introduced by the Hartz plan (2003)(2004)(2005) and the related reforms.

Fig. 1

 1 Fig. 1 TIP curves from different income distributions

Fig. 2

 2 Fig.2The distribution of disposable household income in 2011 using weights and modelling zero incomes. Source: authors calculations using the 2011 wave of the GSOEP

Fig. 3

 3 Fig. 3 How to define a poverty line and the corresponding poverty rates. (The left panel displays yearly poverty lines based on 50% of the sample median. The horizontal line is the average poverty line which is equal to 8 455 euros. It is used for computing distribution-free poverty head-counts rates in the right panel.)

  The three I's of child poverty in Germany. (90% credible intervals are represented by dotted lines. In black solid line is represented the TIP curve at the beginning of each sub-period. The red dashed line corresponds to the TIP curve of the end of each subperiod.) is the lowest in 2011. The other lines of Table1show that child poverty has significantly increased between2002 and 2006 (2002 TIP dominates 2006 with probability 0.952) while it decreases after that date (2007 TIP dominates 2006 with probability 0.947 and 2011

					First period							Second period
			2002							2007
		0.020	2006						0.020	2011
	Intensity	0.010 0.015						Intensity	0.010 0.015
		0.005							0.005
		0.000							0.000
		0.00	0.02	0.04	0.06	0.08	0.10	0.12	0 . 0 0	0 . 0 2	0 . 0 4	0 . 0 6	0 . 0 8	0 . 1 0	0 . 1 2
					p							p
	Fig. 4

Table 1

 1 Probability of TIP dominance for current child poverty

		TIP dominance			Lower incidence
	Year	2002	2006	2007	2011	2002	2006	2007	2011
	2002	-	0.952	0.401	0.000	-	0.999	0.481	0.000
	2006	0.000	-	0.001	0.000	0.001	-	0.001	0.000
	2007	0.384	0.947	-	0.000	0.519	0.999	-	0.000
	2011	1.000	1.000	1.000	-	1.000	1.000	1.000	-

Each line represents the probability that there is less poverty in the corresponding year than in the year given in column. The first panel corresponds to TIP dominance (intensity and inequality) while the right panel indicates the probability of lower incidence

Table 2

 2 TIP dominance test for child chronic poverty between West and East GermanyEach column represents the probability that there is less poverty for the category indicated in line. The first panel corresponds to TIP dominance (intensity and inequality) while the right panel indicates the probability of lower incidence. Period I corresponds to2002[START_REF] Fruhwirth-Schnatter | Finite Mixture and Markov Switching Models[END_REF] and Period II to 2007[START_REF] Cowell | Measuring Inequality[END_REF]

			West		East		West		East
			I	I I	I	I I	I	I I	I	I I
	West	I	-	0.001	0.212	0.042	-	0.017	1.000	0.267
		II	0.997	-	0.989	0.668	0.982	-	1.000	0.647
	East	I	0.019	0.000	-	0.003	0.000	0.000	-	0.001
		II	0.917	0.166	0.842	-	0.733	0.353	0.999	-

The Gini is 1 among the poor because all the poor have a zero income while one is supposed to have z which is a typical case of maximum inequality.

See for instance[START_REF] Cowell | Measuring Inequality[END_REF] for the properties of the log-normal density.

TIP dominance according to the definition given in the 4th footnote of[START_REF] Jenkins | Three I's of poverty curves, with an analysis of UK poverty trends[END_REF] implies that there is more poverty in A than in B if A TIP dominates B. This might appear counter-intuitive when confronted to stochastic dominance. Thus, in our context, TIP dominance will mean less poverty as presented in[START_REF] Thuysbaert | Inference for the measurement of poverty in the presence of a stochastic weighting variable[END_REF] or[START_REF] Davidson | Statistical inference for stochastic dominance and for the measurement of poverty and inequality[END_REF].

The GSOEP will be described in Section 5 and disposable as well as market income defined in the same section.

Note also the method proposed in Appendix D of[START_REF] Gunawan | Bayesian Weighted Inference from Surveys[END_REF] which is a stochastic algorithm for computing the quantiles of a Gamma mixture.

Note here that the range of p has to be slightly restricted because all TIP curves are zero at p = 0 and solving numerically for the p = 1 quantile can be troublesome. So the practical range for the test should be something like p ∈ [0.01, 0.99], values adopted in e.g.[START_REF] Davidson | Testing for restricted stochastic dominance[END_REF].

We used the cross-section household weights for computing the median. We eliminated the households which were given a zero cross-section weight in the data set.

The Hartz reforms, started in 2003 and ended at the beginning of 2005, have fundamentally changed the labour market, the social assistance and insurance systems. They have triggered a lot of political and social protests. See the online Appendix B for more details.

Acknowledgements A first version of this paper was presented in May 2016 at the 10th Annual RCEA Bayesian Econometric Workshop in Rimini. Useful comments by participants are gratefully acknowledged. We are especially grateful to Denis de Crombrugghe, Christian Schluter, Luc Bauwens, the editor and an anonymous referee for a careful reading of a previous version of the paper and for their insightful suggestions. Of course usual disclaimers apply. This work has been carried out thanks to the support of the A*MIDEX project (No ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir" French Government program, managed by the French National Research Agency (ANR).

Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


J J Gao

I Thibon

D Laillé

P Castany

T Gloriant
email: thierry.gloriant@insa-rennes.fr

Influence of texture and transformation strain on the superelastic performance of a new Ti-20Zr-3Mo-3Sn alloy

Keywords: titanium alloys, superelasticity, martensitic transformation, texture, transformation strain

In this work, the superelastic behavior of a new metastable β Ti-20Zr-3Mo-3Sn (at.%) biomedical alloy displaying stress-induced α" martensitic transformation (SIM) was investigated by cyclic tensile tests, electron back-scattered diffraction (EBSD) and in situ synchrotron X-ray diffraction (SXRD) for 3 different solution treatment temperatures: 700°C, 800°C and 900°C. The EBSD observations revealed that the Ti-20Zr-3Mo-3Sn alloy showed a strong dependence of the solution treatment temperatures on crystallographic texture. SXRD diffraction profiles acquired under cyclic loading/unloading tensile tests clearly illustrated the reversible SIM α" transformation. In addition, the lattice parameters of β and SIM α" phases were measured for each cycle in order to establish the evolution of lattice parameters during the deformation and to calculate the maximum transformation strain for any crystallographic direction. A high recovery strain of 3.5% was obtained by tensile test after a solution treatment at 700°C for 30 minutes. This high recovery strain is due to the fact that the maximum transformation strain is obtained along the tensile direction displaying the favorable <011> β {100} β recrystallization texture. The performance of the Ti-20Zr-3Mo-3Sn alloy is discussed and compared with other superelastic alloys such as NiTi and (Ti-Nb)-based alloys.

Introduction

Near equiatomic NiTi alloys are commonly used for the manufacture of biomedical devices such as orthodontic arches, orthopedic staples, endovascular stents… due to their high strength, good corrosion resistance and their high and useful superelastic effect which is required for such applications. However, Ni hypersensitivity still remains an issue in terms of biocompatibility [START_REF] Wever | Cytotoxic, allergic and genotoxic activity of a nickel-titanium alloy[END_REF] and superelastic Ni-free β-titanium alloys displaying stress-induced α" martensitic transformation (SIM) are now considered as the right alternative option for biomedical applications.

Consequently, numerous superelastic (Ti-Nb)-based alloys have been widely developed, such as Ti-Nb-Al, Ti-Nb-Zr, Ti-Nb-Ta, Ti-Nb-O, Ti-Nb-N, Ti-Nb-Zr-Sn, Ti-Nb-Zr-Ta, Ti-Nb-Ta-Zr-O [START_REF] Kim | Martensitic transformation, shape memory effect and superelasticity of Ti-Nb binary alloys[END_REF][START_REF] Inamura | Relationship between texture and macroscopic transformation strain in severely cold-rolled Ti-Nb-Al superelastic alloy[END_REF][START_REF] Sun | A thermo-mechanical treatment to improve the superelastic performances of A c c e p t e d M a n u s c r i p t biomedical Ti-26Nb and Ti-20Nb-6Zr (at.%) alloys[END_REF][START_REF] Hosoda | Effects of short time heat treatment on superelastic properties of a Ti-Nb-Al biomedical shape memory alloy[END_REF][START_REF] Kim | Shape memory characteristics of Ti-22Nb-(2-8) Zr (at.%) biomedical alloys[END_REF][START_REF] Kim | Texture and shape memory behavior of Ti-22Nb-6Ta alloy[END_REF][START_REF] Ramarolahy | Microstructure and mechanical behavior of superelastic Ti-24Nb-0.5 O and Ti-24Nb-0.5N biomedical alloys[END_REF][START_REF] Castany | In situ synchrotron X-ray diffraction study of the martensitic transformation in superelastic Ti-24Nb-0.5 N and Ti-24Nb-0.5 O alloys[END_REF][START_REF] Hao | Microstructure and mechanical behavior of a Ti-24Nb-4Zr-8Sn alloy processed by warm swaging and warm rolling[END_REF][START_REF] Hao | Elastic deformation behaviour of Ti-24Nb-4Zr-7.9 Sn for biomedical applications[END_REF][START_REF] Tahara | Effect of nitrogen addition and annealing temperature on superelastic properties of Ti-Nb-Zr-Ta alloys[END_REF][START_REF] Besse | Mechanisms of deformation in gum metal A c c e p t e d M a n u s c r i p t TNTZ-O and TNTZ titanium alloys: A comparative study on the oxygen influence[END_REF][START_REF] Yang | Multiple deformation mechanisms of Ti-22.4Nb-0.73Ta-2.0Zr-1.34O alloy[END_REF]. However, most of (Ti-Nb)-based alloys showed low recovery strain (less than 3.0%) compared to NiTi alloys (more than 8.0%). The poor recovery strain is attributed to the small transformation strain from body-centered cubic β phase (space group N o 229, Im-3m) to C-centered orthorhombic α'' martensitic phase (space group N o 63, Cmcm) [START_REF] Castany | In situ synchrotron X-ray diffraction study of the martensitic transformation in superelastic Ti-24Nb-0.5 N and Ti-24Nb-0.5 O alloys[END_REF]. Recently, it was reported that hafnium [START_REF] Ijaz | Design of a novel superelastic Ti-23Hf-3Mo-4Sn biomedical alloy combining low modulus, high strength and large recovery strain[END_REF] or zirconium [START_REF] Ijaz | Superelastic properties of biomedical (Ti-Zr)-Mo-Sn alloys[END_REF][START_REF] Pavón | Effect of Nb content and heat treatment temperature on superelastic properties of Ti-24Zr-(8-12)Nb-2Sn alloys[END_REF] can be effective alloying elements to improve the superelastic performance of Ni-free titanium-based alloys. Indeed, the substitution of niobium by zirconium allowed the control of the martensitic transformation start temperature in order to promote superelasticity at room temperature and a strain recovery as high as 7.0% could be reached in some optimized chemical compositions [START_REF] Pavón | Effect of Nb content and heat treatment temperature on superelastic properties of Ti-24Zr-(8-12)Nb-2Sn alloys[END_REF].

In this work, a new quaternary superelastic Ti-20Zr-3Mo-3Sn (at.%) alloy was designed with high zirconium content to replace niobium. Molybdenum was selected as beta stabilizing element and tin was used to suppress the ω phase formation [START_REF] Ijaz | Superelastic properties of biomedical (Ti-Zr)-Mo-Sn alloys[END_REF].

Once elaborated, the alloy was investigated by cyclic tensile tests and characterized by electron back-scattered diffraction (EBSD) for 3 different solution treatment temperatures: 700°C, 800°C and 900°C with the objective to evaluate the influence of the recrystallization texture on its superelastic performance. On the other hand, in situ synchrotron X-ray diffraction (SXRD) characterizations on loading were conducted to A c c e p t e d M a n u s c r i p t follow the deformation sequence and the evolution of the lattice parameters of both phases in order to calculate the maximum transformation strain. Indeed, it is well established that the superelastic recovery strain is attributed to the lattice transformation strain capacity between β and α'' phases [START_REF] Kim | Martensitic transformation, shape memory effect and superelasticity of Ti-Nb binary alloys[END_REF][START_REF] Inamura | Relationship between texture and macroscopic transformation strain in severely cold-rolled Ti-Nb-Al superelastic alloy[END_REF][START_REF] Ijaz | Superelastic properties of biomedical (Ti-Zr)-Mo-Sn alloys[END_REF][START_REF] Pavón | Effect of Nb content and heat treatment temperature on superelastic properties of Ti-24Zr-(8-12)Nb-2Sn alloys[END_REF]. Thus, an objective of this study is also to clarify the role of the chemical composition, and particularly the zirconium element, on the superelastic performance. Although a maximum recovery strain is required for efficient superelastic medical devices, minimum values must also be taken into consideration for a good functionality. Indeed, superelastic medical devices often present a complex geometry and it remains difficult to control the texture during their fabrication. As this point is rarely addressed in literature, the minimum recovery strain is also investigated in the present study and inverse pole figures of transformation strains for all crystallographic orientations will be calculated for comparison with other superelastic alloys such as NiTi and (Ti-Nb)-based alloys.

Material and methods

Alloy synthesis and sample preparation

The Ti-20Zr-3Mo-3Sn alloy (at.%) was synthesized by cold crucible levitation melting (CCLM) technique under argon atmosphere. For this synthesis, pure raw metals included 99.95% pure titanium, 99.078% pure zirconium, 99.9% pure molybdenum and 99.99% pure tin were used. After melting, ingot was homogenized at 950°C for 1200 min under high vacuum (10 -7 mbar), followed by water quenching.

Then, the ingot was cold rolled until 95% reduction rate in thickness. Dog-bone tensile specimens were machined from the cold-rolled sheets with normalized shape: 3 mm width, 0.5 mm in thickness and a gage length of 15 mm. In order to restore a fully recrystallized microstructure from the cold-rolled state, the tensile specimens were finally solution-treated under high vacuum (10 -7 mbar) at 3 different temperatures (T=700, 800, and 900°C) for 30 minutes in the β-phase domain followed A c c e p t e d M a n u s c r i p t by a water quenching in order to retain the β-phase in its metastable state.

In the present study, the 3 solution treatment temperatures were chosen after the determination of the β transus temperature of the alloy, which was obtained by measuring the electrical resistivity variation upon temperature. Indeed, electrical resistivity, which is very sensitive to microstructural changes, is an effective thermal analysis technique for detecting such beta transus temperature [START_REF] Gordin | Characterization of Ti-Ta Alloys Synthesized by Cold Crucible Levitation Melting[END_REF][START_REF] Sun | High-strength nanostructured Ti-12Mo alloy from ductile metastable beta state precursor[END_REF]. The resistivity measurements were performed on a lamella sample by using the four-probe method under high vacuum (10 -7 mbar) between room temperature and 950°C and with a heating/cooling rate of 5°C/min. For the measurements, serial electrical circuit set-up composed of the sample (with resistivity ρ), a reference resistor (2 ohm) and a d.c. source was used. If we neglect the thermal expansion of the sample, variation of the electrical resistivity ratio ρ/ρ 0 (ρ 0 : initial resistivity at room temperature) can be plotted as a function of temperature. Fig. 1 shows the typical resistivity curve usually observed from metastable β-Ti alloys, which was obtained between 400°C and 900°C during a heating/cooling cycle in the present study. On this curve, the red part corresponds to the heating and the blue part to the cooling. The detection of the β transus in titanium alloys by resistivity measurements corresponds to the temperature from which the heating and cooling measurements are superimposed [START_REF] Gordin | Characterization of Ti-Ta Alloys Synthesized by Cold Crucible Levitation Melting[END_REF]. As shown on this resistivity curve, the β transus temperature was measured to be 690°C for the studied Ti-20Zr-3Mo-3Sn alloy.

Tensile tests and microstructural analysis methods

Tensile tests with the strain rate of 10 -4 s -1 were performed to evaluate the mechanical properties of the different samples. The tensile direction was chosen parallel to the cold rolling direction (RD). In this study, cyclic loading-unloading tensile tests were particularly employed in order to characterize the superelastic behavior of the Ti-20Zr-3Mo-3Sn alloy after the different solution treatments. For that, tensile stress is applied until the strain reaches 0.5% for the first cycle, and then the load is removed. The measurement is repeated through increasing the strain by 0.5% A c c e p t e d M a n u s c r i p t steps upon loading for the same specimen. An extensometer was used to measure the strain of the specimens during these tests.

To characterize the microstructure after the different solution treatments, electron back-scattered diffraction (EBSD) analyses were conducted with a scanning electron microscope (Jeol JSM 7100F, SEM) equipped with an Oxford HKL EBSD system.

Prior to the EBSD observations, all the specimens were mechanically mirror-polished by using several SiC papers with decreasing grid size and followed by colloidal silica suspension (particles size: 0.05 μm). In this work, the colloidal silica suspension was mixed with H 2 O 2 solution to release stress due to polishing and reveal the microstructure.

In situ synchrotron X-ray diffraction (SXRD) analyses under cyclic tensile tests were conducted in this study. SXRD experiments were carried out at the ID-22 high resolution powder diffraction beamline of the European Synchrotron Radiation Facility (ESRF, Grenoble, France) with a 1 mm² X-ray beam size and a 0.35453630 Å wavelength radiation. The in situ cyclic tensile tests (at the rate of 10 -4 s -1 ) under synchrotron radiation were performed with increments of 0.5% until 5.0% of strain and then increased by 1.0% until 6.0% of strain during each loading and unloading condition. A nine-channel detector was used to collect transmitted diffracted beams and all the SXRD profiles between 2θ=6° and 2θ=20° were obtained for each cycle. behavior performance, the recoverable strain can return back to origin totally. With increasing applied strain, residual strain appears meaning that the inverse martensitic transformation is not complete. In this figure, the cycle showing the maximum recovered strain is marked in bold for each sample. The maximum recovery strain is then measured to be 3.5%, 2.7% and 2.5% for ST-700-30, ST-800-30 and ST-900-30, respectively.

Results

Tensile tests

Electron back-scattered diffraction characterizations (EBSD)

Analyses by EBSD were carried out in this study to characterize the recrystallization texture induced by the different solution treatments. Typical EBSD inverse pole figure (IPF) maps obtained from the alloy subjected to the three different solution treatments (ST-700-30, ST-800-30 and ST-900-30) are shown in Fig. 3.a, 3.b and 3.c, respectively. The color code corresponding to the crystallographic grain orientation is shown in Fig. 3.d (the Rolling Direction is represented). All the samples show a typical equiaxed β-grain microstructure. A predominant green color is observed in the Fig. 3.a, indicating a dominant of {hkl} β <101> β texture when the alloy is solution-treated at 700°C. However, red-colored grains are dominating in Fig. 3.b, demonstrating that {hkl} β <001> β is the preferential orientation when the alloy is solution-treated at 800°C. On the other hand, a random distribution of color is obtained in Fig. 3.c when the alloy is solution-treated at 900°C. Consequently, it is worth noting that the temperature of the solution treatment is on great influence on the crystallographic orientation. This observation is not surprising since it is well known that the cold-rolled rate and the recrystallization temperature greatly influence the texture in many metallic alloys [START_REF] Doherty | Recrystallization and texture[END_REF]. In order to determine more precisely the recrystallized textures, the ODF (orientation distribution function) have been calculated for each sample (available as supplementary materials) and inverse pole figures of textures are presented in Fig. 4 for each solution treatment. In this figure, RD and ND correspond to the rolling direction (parallel to the tensile direction) and the normal direction, respectively. For the ST-700-30 sample (Fig. 4.a), a dominant <011> β {100} β texture is evidenced but a <111> β {213} β texture component is also

In situ synchrotron X-ray diffraction (SXRD)

In order to evidence the SIM transformation occurring in the Ti-20Zr-3Mo-3Sn alloy, in situ SXRD under cyclic loading/unloading tensile test has been conducted on the ST-700-30 specimen. Fig. 5 displays the initial SXRD profile (in black), the SXRD profile under loading maintained at 6.0% of strain (in red) and the SXRD profile after unloading (in blue). Prior to deformation, the initial SXRD profile shows the typical diffraction peaks: (110) β , (002) β , (112) β , (202) β , (013) β of the cubic β phase (black profile). After being deformed to 6.0% strain, only the diffraction peaks related to the orthorhombic α'' martensitic phase are detected (red profile): (020) α'' , (002) α'' , (021) α'' , (022) α'' , (113) α'' , etc.… meaning that the alloy is fully martensitic under this 6.0% of strain. When the load is released, both α'' and β peaks are detected. In order to characterize more precisely the martensitic phase transformation, it is presented in Fig. 6 the evolution of the most intense β phase peak and α'' martensitic phase peaks within the restricted 7.6 -9.4° angle range during the entire cyclic tensile test from 0 to 6.0% strain on loading (Fig. 6.a) and after unloading (Fig. 6.b). One can notice that the (110) β peak is slightly shifted to the left at the beginning of the deformation (up to 1.5% of strain), compared to its initial position. On the other hand, the (002) α'' , (020) α'' , (021) α'' martensitic peaks start to be detected at 1.5% of strain, which is in good agreement with the cyclic tensile curve. When the strain increases, the intensity of detected martensitic peaks increases progressively to the detriment of the β phase (110) β peak. After unloading from the lowest 4.0% of strain (Fig. 6.b), the main detected peak seems to be the (110) β peak of the β phase but in fact, a slight (002) α'' martensitic peak is also observed in superimposition. Consequently, the reversible stress-induced martensitic transformation is thus clearly evidenced by these in situ SXRD experiments. The other peaks of the martensitic phase can also be observed A c c e p t e d M a n u s c r i p t after unloading but only when the alloy was deformed at a strain superior to 4.5%.

For both parent β and martensitic α'' phases, the shift of peaks indicates a variation of the lattice parameters. Thus, the evolution of lattice parameters of β phase (a β ) and α'' martensite phase (a α'' , b α'' , c α'' ) determined from the whole SXRD profiles is plotted in Fig. 7 as a function of the applied strain on loading. Before deformation, the lattice parameter of the β phase is measured to be a β =0.3332 nm. When the strain increases, the β phase is elastically deformed and its a β parameter raises a maximum value of 0.3365 nm at 1.5% of strain and remains constant for higher strain values.

Concerning the stress-induced α'' martensitic phase, which starts to be detected at 1.5% of strain, its a α'' lattice parameter remains more or less constant in the studying strain range but its b α'' and c α'' parameters continually increase. Indeed the b α'' lattice parameter increases from 0.5080 nm (at 2.0% of strain) to a constant value of 0.5104 nm while the c α'' parameter increases from 0.4740 nm to 0.4770 nm at 5.0% of strain, where the plastic deformation begins. Consequently, the deformation sequence of the Ti-20Zr-3Mo-3Sn alloy under tensile load can be divided in three parts as shown in Fig. 7: the β phase is first deformed elastically until 1.5% of applied strain (I), secondly, the SIM transformation occurred from 1.5% to 5.0% of strain together with its elastic deformation (II), and finally, the plastic deformation occurred from 5.0% of strain (III).

Discussion

Recently, new (Ti-Zr)-based superelastic alloys have demonstrated a remarkably recovery strain, which is due to the fact that a strong <101> β type crystallographic texture was observed parallel to the tensile direction [START_REF] Ijaz | Superelastic properties of biomedical (Ti-Zr)-Mo-Sn alloys[END_REF][START_REF] Pavón | Effect of Nb content and heat treatment temperature on superelastic properties of Ti-24Zr-(8-12)Nb-2Sn alloys[END_REF]. Thus, around 7.0% of recovery strain was obtained by tensile test for the Ti-24Zr-10Nb-2Sn alloy composition (at.%) heat treated at 1173 K for 1.8ks. It is worth noting that this remarkable large elastic recovery value is more than three times that of other Ni-free (Ti-Nb)-based superelastic alloys [START_REF] Pavón | Effect of Nb content and heat treatment temperature on superelastic properties of Ti-24Zr-(8-12)Nb-2Sn alloys[END_REF], which constitutes an excellent result since it becomes comparable with those obtained with the commonly superelastic NiTi alloy A c c e p t e d M a n u s c r i p t used for biomedical applications [START_REF] Héraud | In Situ Synchrotron X-ray Diffraction of the Martensitic Transformation in Superelastic Ti-27Nb and NiTi Alloys: A Comparative Study[END_REF]. In superelastic alloys, the crystallographic texture was shown to be a very important parameter influencing the recovery strain.

However, the transformation strain capability between β and α'' phases must also be taken into account in order to optimize the superelastic recovery performance. Indeed, the transformation strain does not only depend on the orientation of the β grains but also on the lattice parameters of both β and α'' phases, which depend on the chemical composition of the alloy.

From the superelastic titanium alloys, there are 6 equivalent correspondence variants that can be formed when a cubic crystal of β phase transforms into an orthorhombic α" martensite crystal with respect of the classical {110} β ∥(001) α'' and <111> β ∥[110] α'' orientation relationships. But when the martensite is stress-induced, a variant selection operates in order to form only the variants that can accommodate the imposed deformation. It is thus commonly assumed that the variant that is activated and growth is the one that gives the maximum transformation strain in each grain.

According to that, the transformation strain induced by the lattice distortion due to the martensitic transformation from the β cubic phase to the orthorhombic α" phase can be calculated for any direction as a function of lattice parameters of both phases [START_REF] Kim | Martensitic transformation, shape memory effect and superelasticity of Ti-Nb binary alloys[END_REF].

As the maximum transformation strains depend on the lattice parameters of each phase, one can easily calculate them for the present Ti-20Zr-3Mo-3Sn alloy for which the lattice parameters for both β and α'' phases were evaluated by SXRD in this study.

Table 1 reports these lattice parameter values for the Ti-20Zr-3Mo-3Sn alloy of the present study and those obtained on a typical (Ti-Nb)-based superelastic alloy (the Ti-24Nb-4Zr-8Sn alloy composition in wt.%) elaborated and characterized by our group with an identical approach [START_REF] Yang | Characterization of the martensitic transformation in the superelastic Ti-24Nb-4Zr-8Sn alloy by in situ synchrotron X-ray diffraction and dynamic mechanical analysis[END_REF][START_REF] Yang | Texture A c c e p t e d M a n u s c r i p t investigation of the superelastic Ti-24Nb-4Zr-8Sn alloy[END_REF] for comparison. For example, the maximum transformation strain values, ε hkl (given in %), that correspond to the three principal crystallographic directions of the body-centered cubic β phase can be calculated as follow [START_REF] Kim | Martensitic transformation, shape memory effect and superelasticity of Ti-Nb binary alloys[END_REF][START_REF] Tahara | Lattice modulation and superelasticity in oxygen-added β-Ti alloys[END_REF][START_REF] Bertrand | Deformation twinning in the full-α ″martensitic Ti-25Ta-20Nb shape memory alloy[END_REF]:

ε 110 = b α" -√2a β √2a β × 100 (1)
A c c e p t e d M a n u s c r i p t

ε 001 = �b α" 2 +c α" 2 -2a β 2a β × 100 (2)
ε 111 = �a α" 2 +b α" 2 -√3a β √3a β × 100 (3)
Where a β corresponds to the lattice parameter of the β phase and a α'' , b α'' , c α'' correspond to the lattice parameters of the α'' phase.

These calculated values for both Ti-20Zr-3Mo-3Sn and Ti-24Nb-4Zr-8Sn alloys are reported in table 1 and it can be seen that the highest transformation strain is always obtained for the <101> β direction. Indeed, the theoretical maximum transformation strain, ε 110 , that can be expected is thus 7.3% and 3.6% for the Ti-20Zr-3Mo-3Sn and Ti-24Nb-4Zr-8Sn alloys, respectively. Consequently, it is not surprising to obtain the best recovery strain when the present alloy is treated at 700°C since the principal texture observed, which is <101> β direction, is aligned to the tensile direction, unlike those obtained for the other treatments (800°C and 900°C).

These calculated values represent a similar tendency with the experimental superelastic recovery strains that were measured for each alloy to be 3.50% and 2.25%, respectively. Consequently, the zirconium element, instead of niobium, seems to play an important role since the b α'' lattice parameter is much higher in the case of the Ti-20Zr-3Mo-3Sn alloy with the consequence that this alloy presents a larger stain recovery compared to the Ti-24Nb-4Zr-8Sn alloy.

As previously mentioned, the calculated maximum transformation strain induced by the lattice distortion between β and α'' can be generalized for all crystallographic directions [START_REF] Kim | Martensitic transformation, shape memory effect and superelasticity of Ti-Nb binary alloys[END_REF][START_REF] Tahara | Lattice modulation and superelasticity in oxygen-added β-Ti alloys[END_REF][START_REF] Bertrand | Deformation twinning in the full-α ″martensitic Ti-25Ta-20Nb shape memory alloy[END_REF]. Indeed, the deformation in any x vector of the β phase is transformed to x ′(Vi) through the martensite α'' transformation by:

x ′(Vi) = T (Vi) . x (4)
Where T (Vi) represents the lattice distortion matrix expressed in the coordinates of the β phase for each variants Vi. In the cubic to orthorhombic transformation, there are 6 equivalent correspondence variants of martensite (i=1 to 6), therefore there are 6 lattice distortion matrices to take into consideration [START_REF] Tahara | Lattice modulation and superelasticity in oxygen-added β-Ti alloys[END_REF].

A c c e p t e d M a n u s c r i p t

Hence the maximum transformation strain, ε hkl (Vi) , due to the lattice distortion for each variant and for any direction can be calculated as follows:

ε hkl (Vi) = �x ′(Vi) �-|x| |x| (5)
From this calculation using the lattice parameters determined by SXRD (Table 1), inverse pole figure distribution of maximum transformation strains (IPF-TS) can be drawn as shown in Fig. 8. This figure presents the IPF-TS for the Ti-20Zr-3Mo-3Sn

(top) and Ti-24Nb-4Zr-8Sn (bottom) alloys, respectively. As shown, the maximum transformation strain is obtained for the <101> β direction and the minimum for the <111> β direction (2.5% and 0.6% for the Ti-20Zr-3Mo-3Sn and the Ti-24Nb-4Zr-8Sn alloys, respectively). Thus, the fact to substitute niobium by a relatively large amount of zirconium is very judicious because this later can modify significantly the lattice parameters implying a large difference between a β and b α'' , which must be obtained together with a pure <011> β {100} β recrystallization texture in order to promote the highest superelastic recovery. Of course, calculated transformation strains and experimental recovery strains cannot be compared directly. First because the pure <011> β {100} β recrystallization texture was not obtained, even in the case of the most favorable thermal treatment (700°C) where a <111> β {213} β component is also obtained, and secondly, because the calculated maximum transformation strains don't take into account the presence of grain boundaries and grain sizes or other defects, such as the formation dislocations that can be on great influence on the real recovery strain [START_REF] Sun | Influence of a short thermal treatment on the superelastic properties of a titanium-based alloy[END_REF]. However, comparable tendency is observed and the higher is the calculated transformation strain, the higher is the experimental recovery strain.

Another point that deserves to be discussed concerns the minimum values obtained. Although a maximum recovery strain is required for efficient superelastic medical devices, minimum values must also be taken into consideration for a good functionality. Indeed, superelastic medical devices often present a complex geometry and it remains difficult to control the texture during their fabrication. Consequently, parts of the device can present unfavorable texture leading to low recovery strain, which represents a weakness for a considered device. It is then demonstrated through this study the advantage of using zirconium instead of niobium as principal alloying element due to the fact that the minimum value of transformation strain obtained for the Ti-20Zr-3Mo-3Sn alloy (2.5%) is 4 times higher than the minimum value of transformation strain obtained for the Ti-24Nb-4Zr-8Sn alloy (0.6%, which is very low). On the other hand, calculated transformation strains for the B2-B19' martensitic transformation in NiTi alloys can be found in literature for comparison [START_REF] Isola | Martensitic transformation strain and stability of Ni 50-x -Ti 50 -Co x (x= 3, 4) strips obtained by twin-roll casting and standard processing techniques[END_REF][START_REF] Mao | EBSD studies of the stress-induced B2-B19′ martensitic transformation in NiTi tubes under uniaxial tension and compression[END_REF].

Although NiTi alloys can display a very high maximum transformation strain value in tension reaching more than 10.0% in the most favorable <355> B2 crystallographic direction, the minimum calculated value is only 2.7% in the most unfavorable <001> B2 direction, which is comparable with the minimum value obtained with the alloy of the present study (2.5%). Consequently, this new Ti-20Zr-3Mo-3Sn alloy composition is potentially very interesting and competitive to replace NiTi alloy for the manufacture of superelastic medical devices as new Ni-free alloy.

Conclusions

As a summary, a newly developed Ti-20Zr-3Mo-3Sn (at.%) superelastic alloy for biomedical applications was designed and elaborated in this study. In order to investigate the superelastic performance of this alloy, cyclic tensile tests, EBSD and in situ SXRD characterizations have been conducted after 3 different solution treatment temperatures: 700°C, 800°C and 900°C.

The EBSD observations revealed that the Ti-20Zr-3Mo-3Sn β-grain microstructure showed a strong dependence of the solution treatment temperature on crystallographic texture and three different textures were obtained. By in situ SXRD, the typical diffraction peaks of the β and SIM α" phases during loading/unloading were clearly observed and the reversible transformation between the β phase and the SIM α" phase was evidenced. On the other hand, the lattice parameters of β and SIM α" phases were measured for each cycle and their evolutions during the deformation were established. From these measurements, the maximum transformation strain for any crystallographic direction was evaluated and inverse pole figures of maximum alloy and for the Ti-24Nb-4Zr-8Sn alloy, which was characterized in a previous work by the same approach. By comparing the IPF-TS of the two alloys, it was showed that the Ti-20Zr-3Mo-3Sn alloy presents much higher maximum transformation strain values than Ti-24Nb-4Zr-8Sn alloy whatever the crystallographic orientation. Finally, the highest recovery strain obtained with the Ti-20Zr-3Mo-3Sn alloy (3.5%) can then be explained by the high zirconium content that modifies significantly the lattice parameters implying a large difference between a β and b α'' , which must be obtained together with a pure <011> β {100} β recrystallization texture in order to promote the highest strain recovery.

Fig. 2

 2 Fig.2displays the engineering stress-strain curves, which were obtained from cyclic tensile tests for specimens subjected to three different solid solution treatments in the β-phase domain: 700°C for 30 min (ST-700-30, Fig.2.a), 800°C for 30 min (ST-800-30, Fig.2.b) and 900°C for 30 min (ST-900-30, Fig.2.c). In these tensile curves, the presence of hysteresis between loading and unloading is due to the superelastic property of the alloy in which reversible SIM transformation between β and α'' phases occurs. Up to four cycles, they all show a very good superelastic

  A c c e p t e d M a n u s c r i p t observed to a lesser extent. For the ST-800-30 specimen, it is shown a strong <100> β {011} β texture (Fig.4.b). For the last one, in the case of the ST-900-30 specimen, a principal <113> β {110} β texture and a slightly <110> β {111} β textures are obtained (Fig.4.c).

A c c e p t e d M a n u s c r i p t

A c c e p t e d M a n u s c r i p t

  transformation strain (IPF-TS) were established for the present Ti-20Zr-3Mo-3Sn

Fig. 1 .

 1 Fig. 1. Electrical resistivity curve of the Ti-20Zr-3Mo-3Sn alloy (heating and cooling cycle).

A c c e p t e d M a n u s c r i p t

Fig. 2 .

 2 Fig. 2. Engineering stress-strain curves of the Ti-20Zr-3Mo-3Sn alloy obtained from cyclic tensile tests at room temperature: ST-700-30 (a); ST-800-30 (b); ST-900-30 (c).

A c c e p t e d M a n u s c r i p t

Fig. 3 .

 3 Fig. 3. EBSD RD (Rolling Direction) IPF maps of solution treated alloy samples: ST-700-30 (a); ST-800-30 (b); ST-900-30 (c); IPF color code (d).

A c c e p t e d M a n u s c r i p tFig. 4 .

 4 Fig. 4. Inverse pole figures of the solution treated alloy samples: ST-700-30 (a); ST-800-30 (b); ST-900-30 (c). RD is parallel to the tensile direction.

A c c e p t e d M a n u s c r i p tFig. 5 .

 5 Fig.5. SXRD profiles for ST-700-30 specimen acquired before deformation (black), under loading at 6.0% of strain (red) and after unloading from 6.0% of strain (blue).

A c c e p t e d M a n u s c r i p tFig. 6 .

 6 Fig. 6. SXRD profiles zoomed within 2θ=7.6°-9.4° obtained during in situ cyclic tensile test on loading (a) and after unloading (b) for the ST-700-30 specimen.

A c c e p t e d M a n u s c r i p tFig. 7 .

 7 Fig. 7. Evolution of lattice parameters of β and α'' phases as a function of the applied strain for the ST-700-30 sample during in situ cyclic tensile test.

A c c e p t e d M a n u s c r i p t

A c c e p t e d M a n u s c r i p t

Table 1 . Lattice parameters of β and α'' phases, calculated maximum transformation strains in the principal crystallographic directions and measured recovery strains for Ti-20Zr-3Mo-3Sn and Ti-24Nb-4Zr-8Sn alloys.

 1

	Alloys	lattice parameter(nm) β phase α'' phase a β a α'' b α''	c α''	ε 110	ε 001	ε 111	recovery strain (%)
	Ti20Zr3Mo3Sn	0.336	0.312 0.510 0.477	7.3	3.8	2.5	3.50
	Ti24Nb4Zr8Sn [22]	0.333	0.315 0.488 0.484	3.6	3.2	0.6	2.25

Acknowledgment J.J. Gao acknowledges the China Scholarship Council (CSC) for his Ph.D financial support (No. 2016-6329). The authors also acknowledge the SCANMAT platform of the University of Rennes for providing access to SEM facilities.