M Ould Haye

A Philippe
email: anne.philippe@univ-nantes.fr

A Frequency approach for detecting nonstationarity in dependent data

Keywords: Long memory, dependence, stationarity, frenquency

Distinguishing long memory behaviour from nonstationarity can be very difficult as in both cases the sample autocovariance function decays very slowly. Available stationarity tests either do not include long memory or fare poorly in terms of empirical size, especially near the boundary between long memory and nonstationarity. We propose a testing procedure based on evaluating periodograms at different epochs.

Limiting distributions established here are easily tractable as sum of weighted independent χ 2 random variables. Moreover, numerical studies are provided to show that the proposed approach seems to outperform existing methods.

Introduction

Let us consider a stationary process (X t ) with a spectral density of some semi-parametric form:

f (λ) = |λ| -2d f * (λ) (1)
with -1/2 < d < 1/2 and f * is an even, positive, continuous function on [-π, π]. Hereafter, we will refer to processes satisfying condition (1) as satisfying the null hypothesis of stationarity H 0 . It is common to denote this class of processes I(d), -1/2 < d < 1/2.

It is worth noticing that spectral densities of the form (1) encompasse a large variety of stationary processes such as the very popular fractionally autoregressive integrated moving average processes FARIMA(p, d, q)

Φ(B)(1 -B) d X t = Θ(B) t ,
where t is a white noise with variance σ 2 , B is the lag operator BX t = X t-1 , and where

Φ(B) = 1 -φ 1 B -• • • -φ p B p , Θ(B) = 1 + θ 1 B + • • • + θ q B q
are the autoregressive and moving average polynomials. The FARIMA spectral density is then given by

f (λ) = σ 2 2π Θ(e iλ ) Φ(e iλ ) 2 1 -e iλ -2d = |λ| -2d σ 2 2π sin(λ/2) (λ/2) -2d
Θ(e iλ ) Φ(e iλ ) 2 .

We say that the process (X t ) is of short memory if d = 0 (i.e. the spectral density f = f * in (1) is continuous and positive over [-π, π]), and we say that the process is of long memory if 0 < d < 1/2 (i.e. the spectral density is unbounded at zero). The case -1/2 < d < 0 corresponds to the so-called persistent memory, where f (0) = 0. In time-domain terms, short memory means that the covariance function is summable (i.e. covariance function γ(h) = Cov(X i , X i+h ) goes to zero fast as the lag h increases) and long memory means that the covariance function is not summable (i.e. the covariance function goes to zero slowly as the lag increases). Literature is full of real world examples of both types of processes. Over the past decade, several monographs have been written on statistical and probabilistic aspects of long memory processes. The reader is referred to the recent book by [START_REF] Beran | Long-memory processes[END_REF] as well as the book of [START_REF] Giraitis | Large sample inference for long memory processes[END_REF] and references therein.

Under mild conditions, the switch between time-domain and frequency-domain is easily made via the two representations

f (λ) = 1 2π ∞ h=-∞ e -ihλ γ(λ) = 1 2π γ(0) + 2 ∞ h=1 cos(λh)γ(h) , |λ| ≤ π,
and reciprocally

γ(h) = π -π e ihλ f (λ)dλ = π -π
cos(λh)f (λ)dλ, h = 0, ±1, ±2, . . . .

From an applied point of view, long memory behaviour seems to be often confused with a lack of stationarity, as time series exhibiting such a behaviour tend to have a sample autocorrelation function with large spikes at several lags which is well known to be the signature of non-stationarity for many practitioners. There are many stationarity tests such as [START_REF] Phillips | Testing for a unit root in time series regression[END_REF] tests and the frequency test of [START_REF] Bailey | Spectral approach to parameter-free unit root testing[END_REF], which can be seen as extensions of the augmented Dickey-Fuller tests since their stationarity asumption includes larger class than i.i.d. case. For instance the former allows for mixing and the latter includes short memory linear processes. However, the first statistical test for stationarity under a large umbrella of dependence, i.e. short and long memory dependence versus non stationarity is the so-called V/S test introduced by [START_REF] Giraitis | A test for stationarity versus trends and unit roots for a wide class of dependent errors[END_REF].

We develop a statistical test that uses an estimate of the memory parameter d. Contrary to V/S test, this one detects the stationarity for all d in (-1/2,1/2) unlike V/S which requires d to be in a compact [a, b] ⊂ (-1/2, 1/2). This drawback in V/S results in it having high empirical sizes starting from d = .4. The proposed test does not suffer form this issue and keeps empirical sizes close to the nominal levels even for d close to 1/2.

The simple idea of using an estimate d of d with its confidence interval to test the stationarity poses some fundamental problems even if we rewrite the null and alternative hypotheses with semi-parametric forms: et al. (2007) or [START_REF] Giraitis | Large sample inference for long memory processes[END_REF]. Their asymptotic normality requires an assumption on its bandwidth parameter m n that depends in particular on the second order expansion (determined by a coefficient β ∈ (0, 2]) of the spectral density near zero. An optimal choice of the bandwidth, needed to ensure a well-sized test, requires additional assumptions on the null hypothesis stronger than (1), such as knowing β. For instance taking β = 2 we obtain a class of linear models containing FARIMA models. Note that our approach only requires the use of a consistent estimator of d. Moreover, in the simulations section, we will also see that the sole use of d will not suffice to detect slight structural breaks (see Figure 5). An additional difficulty of writing a stationarity test as one side test H 0 : d < 1/2 is to obtain a testing procedure with asymptotic level α(= .05 for example

X t ∈ I(d), -1/2 < d < 1/2 versus it exists integer p ≥ 1 such (1 -B) p X t ∈ I(d) for certain -1/2 < d < 1/2.
P d d > 1/2 -+ 1.65 2 √ m n = .05.
However, this test excludes the main important part of the stationarity domain 1/2 -< d < 1/2 where non stationarity is most suspected in practice.

From a sample X 1 , . . . , X n of the process (X t ), we are interested in building a testing procedure to discriminate between dependence and non-stationarity.

The proposed statistic is built from periodograms taken at different epochs. More precisely, the procedure is as follows: we split our initial sample X 1 , . . . , X n into m blocks (or epochs), each of size , and we construct the periodogram I n,i on the ith block X (i-1) +1 , . . . , X i .

Let

Q n,m (s, d) = m -2d s j=1 I n (λ j ) 1 m m i=1 I n,i (λ j ) , (2)
with

I n (λ) = 1 2πn n t=1 X t e itλ 2 , I n,h (λ ) = 1 2π h t=(h-1) +1 X t e itλ 2 , (3)
where λ j = 2πj/n, and λ j = 2πj/ , j = 1, . . . , s, for fixed s, are the Fourier frequencies.

The statistic Q n,m (s, d) can be viewed as sum of self normalized periodograms I n (λ j ), j = 1, . . . , s. A particular case of the statistic defined in (2) was studied in [START_REF] Gromykov | A frequency-domain test for long range dependence[END_REF], where the statistic Q n,m (s, 0) was considered to test short memory versus long memory, and we established its asymptotic distribution (i.e. under short memory).

It is worth noticing that establishing limiting distributions under long memory necessitates modifying the short memory statistic, including a normalizing coefficient m -2d , as it appears in (2), as well as two sets of Fourier frequencies λ j and λ j . Of course, this will result in different limiting distributions due to the long memory that will strongly affect the asymptotic behaviour of the periodogram.

We assume that m = m(n), and = (n) → ∞ as n → ∞. It is important to emphasize the fact that m and increase with n and are not constant, and that m = n/ . We are simply using notation m and rather than m(n) and (n) only for the sake of simplicity. The rest of the paper is organized as follows. Section 2 contains main limiting theorems related to the statistic Q n,m (s, d), including its behaviour under a wide range of nonstationary alternatives. In Section 3 we present our test statistic to detect nonstationarity using asymptotic properties obtained in Section 2. Section 4 contains some Monte Carlo study to illustrate the performance of our proposed method. The paper ends with an appendix containing all the proofs.

Main results

Let (X t ) t≥0 be a linear process of the form

X t = ∞ j=0 a j t-j , (4)
where a j are the Fourier coefficients of an L 2 -Hermitian function and a 0 = 1, and where ( j ) are i.i.d. random variables with zero mean and finite fourth moment. Denote σ 2 = E( 2 1 ) and η = E( 4 1 ).

For -1/2 < d < 1/2, we define the following memory condition for stationary processes.

Condition H d The process X t is a linear as defined in (4), and its spectral density (which exists) f d is of the form (1). In addition, for d = 0, we assume that

∞ j=0 |a j | < ∞, (5)
and when -1/2 < d < 0, we will also assume that

a j = c(d) j -1+d (1 + O(j -1 )), j ≥ 1, for some c(d) = 0 and ∞ j=0 a j = 0. (6) Condition (5), imposed when d = 0, implies that ∞ h=-∞ |γ(h)| < ∞.
Note that when 0 < d < 1/2, condition (1) on the spectral density implies that

γ(h) ∼ C(d)h 2d-1 , as h → ∞, for some constant C(d) > 0, (7)
where et al. (2012)). Also when -1/2 < d < 0, condition (6) implies that (7) also still holds.(See Proposition 3.2.1. (iii) of [START_REF] Giraitis | Large sample inference for long memory processes[END_REF]. In what follows, we will focus on the statistic Q n,m (s, d) defined in (2). In the next theorem we give the asymptotic distribution of Q n,m (s, d) under the null hypothesis "the process is stationary". As for its behaviour under the alternative of "the process has a stochastic or deterministic trend", it is given in the Theorem 2. We first introduce a covariance matrix that will appear in the limiting 11). The block diagonal form of the matrix is due to the dependence, inherited from the long memory, among normalized real parts and imaginary parts of the discrete Fourier transforms (DFT). Note that DFT's real and imaginary parts will be asymptotically independent even if they are taken at different Fourier frequencies.

a n ∼ b n means that a n /b n → 1 as n → ∞. (see Proposition 3.1.1. (a) of Giraitis
distribution of Q n,m (s, d) in (
Let Σ(d) be a (2s) × (2s) covariance matrix of the form

Σ(d) =    Σ (1) (d) 0 0 Σ (2) (d)    (8)
where Σ (1) (d) and Σ (2) (d) are defined by andΣ (2)

Σ (1) j,k (d) = 1 2 L j,k (d) + L j,-k (d) L j,j (d)L k,k (d)
j,k (d) = 1 2 L j,k (d) -L j,-k (d) L j,j (d)L k,k (d) , (9)
for j, k = 1, . . . , s, and where for all integers j, k = 0

L j,k (d) = |jk| d π ∞ -∞ sin 2 (λ/2) (2πk -λ)(2πj -λ) λ 2π -2d
dλ.

(10)

The next theorem gives the central result that is used to build the critical regions of the statistical test proposed in Section 3.

Theorem 1. Let X t be a stationary process satisfying H d for certain d ∈ (-1/2, 1/2), and let s ≥ 1 be fixed. Assume that there exist δ, δ >

0 such that = O(n 1-δ ) and m = O(n 1-δ ) (with n = m by construction). Then the statistic Q n,m (s, d) defined in (2) satisfies Q n,m (s, d) D -→ Q(s, d) = 2s i=1 ζ i (d)Q i , (11)
where Q 1 , . . . , Q 2s are i.i.d. Remark 1. The limiting distribution form ( 11) is an elaborated form of the limiting form (17) (see in the proof below). Actually (11) can be seen as a sum of independent nonidentically Gamma random variables and therefore its quantiles can easily be computed with high precision (see for instance [START_REF] Hu | Density and distribution evaluation for convolution of independent gamma variables[END_REF]). Evaluating the limit in (17) requires

Monte Carlo simulations which may yield unstable quantiles.

Proof. The proof is relegated to Section 5.1 of the appendix.

The following propositions provide some insight on the limiting distribution obtained in Theorem 1. Namely we show that this distribution can be continuously extended to the case d = 1/2. Moreover, we show that the quantiles (of fixed order) of the limiting distribution are continuous with respect to the memory parameter d. These two propositions will be useful in the next section when it comes to implementing the stationarity test.

Proposition 1. The limiting distribution Q(s, d) converges in distribution, as d → 1/2, to a non degenerated random variable

Q(s, 1/2) := 2s i=1 ζ i (1/2)Q i where Q 1 , . . . , Q 2s are i.i.d. χ 2 1 random variables and ζ 1 (1/2), . . . ζ 2s (1/2) are the eigenval- ues of the covariance matrix Σ(1/2) = lim d→1/2 Σ(d)
where Σ(d) is defined in ( 8) and ( 9).

Proposition 2. Let q α (s, d) be Q(s, d)'s quantile of order 1 -α where 0 < α < 1. For fixed α, the mapping (-1/2, 1/2] → R, d → q α (s, d) is continuous.
The next theorem gives the asymptotic behaviour of the statistic Q n,m (s, d) for a wide variety of non stationary processes, such as random walk (with dependent increments) as well stationary noise with additive deterministic signals. This will ensure the consistency of the testing procedure described in the next section.

Theorem 2. Let Y t be a stationary process satisfying H d for certain d ∈ (-1/2, 1/2), and consider the following two classes of non stationary processes X t

• Stochastic trend (unit root) :

X t = X t-1 + Y t • Deterministic trend (structural breaks): X t = g n (t) + Y t where g n (t) = n β g(t/n), with
β ≥ 0, and where g is a piece-wise continuous1 satisfying 1 0 g(x)e i2πjx dx = 0, for some j ∈ {1, 2, . . . , s}.

Then, for all δ ≤ 1/2

Q n,m (s, δ) P -→ ∞, (13)
where P -→ denotes the convergence in probability.

Proof. The proof is relegated to Section 5.4 of the appendix.

Remark 2. We note in passing that condition ( 12) is satisfied by most of nonconstant functions. For example, nonnegative step functions (allowing change in the mean), polynomials, etc.

3 Testing procedures.

We want to check the stationarity of a process. More precisely we define the following hypotheses: the null hypothesis H 0 : "the process X t is stationary" in the sense that there exists -1/2 < d < 1/2 such that X t satisfies the condition H d . As such the process X t

satisfies Theorem 1 conditions. The alternative H 1 considered here is "the process X t has a stochastic or deterministic trend" as defined in Theorem 2.

Hereafter, and for simplicity, we will refer to H 0 and H 1 as respectively stationarity and nonstationarity hypotheses.

We now describe different decision rules derived from the statistic Q n,m (s, d).

A

Q n,m (s, d) = e -2( d-d) ln m Q n,m (s, d).
To carry out such test, for fixed α ∈ (0, 1) we consider the rejection region

R n = {(X 1 , . . . , X n ) ∈ R n , Q m,n (s, d) > q α (s, d)} ( 14
)
where q α (s, d) is the quantile of order (1 -α) of Q(s, d). Note that the rejection region R n is still defined even if d = 1/2 since Q(s, 1/2) is non degenerated (see Proposition 1). This test has asymptotic level α thanks to Proposition 2 and is also clearly consistent.

Remark 3. The fact that our limiting distribution can be extended to d = 1/2 gives our test a considerable advantage over V/S. Indeed, as the limiting distribution of V/S is degenerated at d = 1/2, d had to be restricted to a compact [-a, a] ⊂ (-1/2, 1/2) in V/S and its implementation. This leads to a blind spot (a, 1/2] in the stationarity null hypothesis. This explains V/S's high empirical sizes observed when d is close to 1/2 (see simulation results in [START_REF] Giraitis | A test for stationarity versus trends and unit roots for a wide class of dependent errors[END_REF]).

4 Numerical Studies

Simulated models

We investigate two kinds of null hypothesis:

• Long memory processes (d = 0).

We simulate FARIMA(0,d,0) processes

X t = (1 -B) -d t where d ∈ (-1/2, 1/2) and
( t ) t are standard Gaussian innovations.

• Short memory processes (d = 0).

We simulate AR(1) processes: X t = aX t-1 + t where a ∈ (-1, 1) and t are standard Gaussian innovations.

Two alternatives are simulated as follows: starting from a FARIMA(0,d,0) processes Y t with d ∈ (-1/2, 1/2), we take

• unit-root process X t = t i=1 Y i . Note that, as an integrated process, X t can be written as a FARIMA(0,d + 1,0) with d ∈ (-1/2, 1/2) which is not stationary (See [START_REF] Taqqu | Fractional brownian motion and long-range dependence[END_REF] for a precise definition)

• Structural breaks (a change point in the mean in the middle n/2):

X t = µ t + Y t where µ t =        0 if t < n/2, ∆ if t ≥ n/2. (15) Q n,m (s, d) V/S m = n 1/2 m = n 1/3 d s = 1 s = 3 s = 5 s = 1 s = 3 s =

Empirical size and power evaluation

Fundamentally, the statistic Q n,m (s, d) depends on the choice of the bandwidth parameter m and the number of Fourier frequencies s. Our strategy is to calibrate m, s in order to ensure an empirical size that remains reasonably close to the nominal level α(= 5%) set in advance, in particular when d is very close to 1/2. In doing so, and contrary to V/S test which is known to suffer from high empirical size, we are trying to drastically reduce the probability of rejecting the stationarity of processes with strong long memory processes.

Table 1 provides empirical size of our test for different values of s and m taking d close to the stationarity boundary 1/2. We also compare them with V /S test. The results show that m = √ n performs better than m = n 1/3 in recognizing the stationarity of strong long memory processes. Moreover, the statistic seems to be little sensitive to the choice of s. Finally, our procedure outperforms V /S in recognizing the stationarity of strong long memory processes. This also confirms that V/S does suffer from very large empirical sizes in the neighbourhood of d = 1/2. In the subsequent numerical studies, we will focus on the bandwidth choice m = √ n, where n is taking a moderate value of 500.

Figure 1 shows the empirical size as function of d of the tests Q n, √ n (s, d) and V /S test.

The figure also shows that the empirical size performance observed near non stationarity of Q n, √ n (s, d) extends to all possible stationarity values of d.

In Figure 2 we consider stationary AR(1) process with coefficient a ranging from .8 to .99.

For this model with short memory, it is legitimate to include Ducky-Fuller test. The comparison shows that, with s = 1, the performance of our statistics is comparable to that of V/S for stationary AR(1) processes near unit root (a close to 1). We notice that, near unit root, the choice of the number of frequencies s is important and the test performs better for s = 1. For this value, the results are similar to those of V/S. Of course Dickey Fuller's test is better than both V/S and our statistics for testing stationary AR processes against unit root. However our null hypothesis H 0 is much wider, as it contains long memory which cannot be seen as stationary by all sorts of Dickey Fuller's test. Note that, when we stay away from unit root, the stationarity is accepted, with probability one by all the above tests.

Figures 3 and4 illustrate the power function of Q n, √ n (s, d) under the alternatives: stochastic and deterministic trends respectively, and confirm the consistency of our test. The test V/S is artificially more powerful than ours in detecting slightly non stationary processes (d near 1/2 from the right). Indeed, as shown in Figure 1, this is due to its high empirical size (about 20% instead of the nominal level of 5%). Despite such high empirical level, with the right choice of s, our test becomes rapidly more powerful than V/S as d increases.

When it comes to choosing an s (the number of lower frequencies to be used), there does not seem to be a need for a trade-off between a good empirical size and a good power. In fact, as seen before, the empirical size is not affected by the choice of s among the tested values and the empirical power increases with s. The choice s = 5 seems to be satisfactory for both size and power.

From Figure 4, we notice a sharp difference in terms of empirical power behaviour between the stochastic trend and structural breaks. Unlike the stochastic trend, the power curves decrease with s. In this case, we recommend the use of s = 1, as it detects small structural breaks as represented in Figure 5. Moreover, this choice guarantees comparable power as V/S. In the structural breaks case, the non-increasing behaviour of the power function (in d)

is consistent with previous findings in time-domain context (see for instance [START_REF] Giraitis | A test for stationarity versus trends and unit roots for a wide class of dependent errors[END_REF]. This is due to the fact that the marginal variance Var(Y

1 ) = σ 2 Γ(1 -2d)/Γ 2 (1 -d)
increases with d, so that any small to moderate jump would be more difficult to detect for d close 1/2, as the stationary noise Y t will dominate.

Conclusion

We Figure 2: Empirical levels evaluated on AR(1) process x t = ax t-1 + t as function of its coefficient a. We compare the tests based on the statistics V/S (solid line) and Q n, √ n (s, d) (dashed red line) and Dickey-Fuller (dashed green line) with n = 500, s = 1, 3, 5 and with 5000 replications. The nominal level is α = 5%.

Appendix

In what follows and for simplicity, c designates a generic positive constant that may be different from one expression to another.

Proof of Theorem 1

From (2) we get

Q n,m (s, d) = m -2d s j=1 f (λ j ) f (λ j ) I n (λ j )/f (λ j ) 1 m m i=1 I n,i (λ j )/f (λ j ) ∼ s j=1 I n (λ j )/f (λ j ) 1 m m i=1 I n,i (λ j )/f (λ j )
, as n → ∞.

We know from [START_REF] Hurvich | Asymptotics for the low-frequency ordinates of the periodogram of a long-memory time series[END_REF] 2 and [START_REF] Terrin | An asymptotic Wiener-Itô representation for the low frequency ordinates of the periodogram of a long memory time series[END_REF] that

I n (λ 1 ) f (λ 1 ) , • • • , I n (λ s ) f (λ s ) D → [L 1 (d)(Z 2 1 (1) + Z 2 2 (1)), • • • , L s (d)(Z 2 1 (s) + Z 2 2 (s))],
where Z(s, d) = (Z 1 (1), . . . , Z 1 (s), Z 2 (1), . . . , Z 2 (s)) is a zero-mean Gaussian random vector with covariance matrix Σ(d) defined in ( 8) and ( 9), and L j (d) = 2L j,j (d), defined in (10).

Actually, the covariances between Z i (j) and Z i (k), i = 1, 2 and j, k = 1, . . . , s given in the references above can easily be worked out to obtain a simplified form under the covariance matrix Σ(d) expressed only in terms of our coefficients L j,k (d) and L j,-k (d).

Using Slutsky Lemma, it will then be enough to show that, for each fixed j, as n → ∞

(and therefore as , m → ∞),

1 m m i=1 I n,i (λ j ) f (λ j ) P -→ L j (d), (16)
2 We note in passing that two errors have slipped into the proof of Theorem 5 of [START_REF] Hurvich | Asymptotics for the low-frequency ordinates of the periodogram of a long-memory time series[END_REF] on page 471 of the article leading to a coefficient of (-1) j+k+1 in their formulae ( 7) and ( 8) while there should be no such a coefficient.

to obtain

Q n,m (s, d) D → Z 2 1 (1) + Z 2 2 (1) + • • • + Z 2 1 (s) + Z 2 2 (s). (17)
The limiting distribution above is that of the squared norm of Z(s, d) whose covariance matrix can be decomposed as

Σ(d) = Ω (d)Λ(d)Ω(d),
where Λ(d) is the diagonal matrix of Σ(d)'s eigenvalues (denoted ζ i (d), i = 1, . . . , 2s) and Ω(d) is an orthogonal matrix. Therefore, we can write

Z 2 1 (1) + Z 2 2 (1) + • • • + Z 2 1 (s) + Z 2 2 (s) = Z(s, d) 2 = Ω(d)Z(s, d) 2 D = Λ(d) 1/2 W 2 = 2s i=1 ζ i (d)W 2 i , (18) where D
= means equality in distribution and W is a × 1) standard Gaussian vector.

To complete the proof of the Theorem, we now establish (16). Using the the fact that I n,i (λ j )/f (λ j ), i = 1, . . . , m are identically distributed, and then [START_REF] Hurvich | Asymptotics for the low-frequency ordinates of the periodogram of a long-memory time series[END_REF], we have for fixed j,

E 1 m m i=1 I n,i (λ j ) f (λ j ) = E I n,i (λ j ) f (λ j ) → L j (d).
Since f (λ j ) ∼ c 2d , it will then be enough to show that

-4d Var 1 m m i=1 I n,i (λ j ) → 0.
Let us first consider the case 0 ≤ d < 1/2. Similar computations as in [START_REF] Gromykov | A frequency-domain test for long range dependence[END_REF] show that the LHS of the expression above can be asymptotically bounded by

2 -4d m m u=1 1 p=1 q=1 -p+1 h=1+q- T ,u (h, p, q), (19)
where

T ,u (h, p, q) = T (1)
,u (h, p, q) + T

(2)

,u (h, p, q) + T

(3)

,u (h, p, q), with

T (1) ,u (h, p, q) = γ( u -h)γ( u -h + q -p), T (2)
,u (h, p, q) = γ( u -h + q)γ( u -h -p), and T

,u (h, p, q) = (η -3)σ 4 ∞ i=0 a i a i+p a i+h+ u a i+h+ u+q .

Let us handle T

(1)

,u (h, p, q) first: with k = q -p and making u starts from 3,

1 m 4d+1 m u=3 p=1 q=1 -p h=1-+q γ( u -h)γ( u -h -p + q) = 1 m 4d+1 m u=3 -1 k=-+1 -k p=1 -p h=1-+k+p γ( u -h)γ( u -h + k) ≤ 1 m 4d+1 m u=3 -1 k=-+1 -k p=1 h=- |γ( u -h)γ( u -h + k)| ≤ 1 m 4d m u=3 -1 k=-+1 h=- |γ( u -h)γ( u -h + k)| = 1 m 4d m u=3 -1 k=-+1
(x + y) 2d-1 dy x 2d-1 dx ≤ 2c m m 2 x 2d-1 dx = O m 2d-1 → 0. T (2)
,u (h, p, q) treats the same way. For the last term T

(3)

,u (h, p, q), we have h=-

T (3) ,u (h, p, q) = |(η -3)|σ 4 ∞ i=0 a i a i+p i+(u+1) s=i+(u-1) a s a s+q ≤ |(η -3)|σ 4 |γ(p)| ∞ s= |a s a s+q | (21) ≤ |(η -3)|σ 4 |γ(p)| ∞ s= a 2 s 1/2 ∞ s=0 a 2 s 1/2 = |γ(p)||o(1)| → 0, as → ∞,
where o(1) is uniform in u, q and p. Therefore Let us consider the case -1/2 < d < 0. Noticing that quantity in ( 20) is bounded by

1 m 4 ∞ h= |γ(h)| 2 = O(1/m) → 0 since under condition H d , we have |γ(h)| ∼ C(d)h -1+2d
as h → ∞ and therefore, the result obtained above for T

(1)

,u (h, p, q) in case 0 < d < 1/2 still holds when -1/2 < d < 0. Same thing remains true for T

(2) ,u (h, p, q). Concerning T

(3) ,u (h, p, q), using (21) and the summability of γ(p) as well as the expression of a j given in H d , we obtain that

-4d 1 m 1 m u=1 p=1 q=1 h=- |T (3) ,u (h, p, q)| ≤ c -4d-1 ∞ s= |a s | ∞ q=1 |a s+q | ≤ c -4d-1 ∞ s= s -1+2d = c -1-2d → 0 as → ∞.
The proof of Theorem 1 is now complete.

Proof of Proposition 1

The first step consists in establishing the following properties for the entries of Σ(d) (which are proven in the appendix)

1. For all fixed j, k = 1, . . . , s, L j,k (d) (defined in ( 10)) are well defined for d = 1/2.

Actually all the singularities of the integrands in the previous formula are removable and the integrands are asymptotically equivalent to K(d)λ -2-2d for some positive constant K(d) (finite for all d ≤ 1/2). Hence the integrals are finite when d = 1/2 (and even for all d > -1/2).

2. As functions of d, L j,k (d) are continuous at d = 1/2. Indeed,

L j,k (d) = K(d) ∞ -∞ h(λ, d)dλ where h(λ, d) = sin 2 (λ/2) (2πk -λ)(2πj -λ) |λ| -2d F F -1 n (α) = F F -1 n (α) -F n F -1 n (α) + F n F -1 n (α) ≤ F n -F + α → α which means that F -1 n (α) → F -1 (α) since F -1 is continuous.
This completes the proof.

Proof of Theorem 2

As δ ≤ 1/2 appears only in the normalization m -2δ of the statistic Q n.m (s, δ), it is enough to prove the convergence (13) for δ = 1/2, i.e. Q n,m (s, 1/2) P -→ ∞.

For a stochastic or deterministic trend, we want to show that

m -1 s j=1 I n (λ j ) 1 m m i=1 I n,i (λ j ) P → ∞.
It will be enough to show that for fixed j,

m -1 I n (λ j ) 1 m m i=1 I n,i (λ j ) P → ∞. (22)
First, consider the deterministic trend case: X t = g n (t) + Y t . For fixed j = 1, . . . , s, and omitting the coefficient 2π in the denominator, we have from (3), with I n,Y denoting the periodogram built from Y 1 , . . . , Y n ,

I n (λ j ) = I n,Y (λ j ) + n 2β+1 |D n (g)| 2 + 2n β+1/2 Re D n (g)D n (Y ) (23)
where

D n (g) = 1 √ 2π n t=1 g t n e i2πtj/n 1 n , D n (Y ) = 1 √ 2πn n t=1
Y t e i2πtj/n .

Since Y satisfies Theorem 1's assumptions, we have as n → ∞,

E(I n,Y (λ j )) ∼ L j (d)f (λ j ) ∼ L j (d)n 2d .
Under assumption (12), we get by Riemann sum approximation of an integral,

|D n (g)| 2 ∼ c > 0.
Finally, for the cross-term, we have

E|D n (g)D n (Y )| = |D n (g)| E|D n (Y )| ≤ |D n (g)| E 1/2 (I n,Y (λ j )) ≤ cn d . ( 24
) Therefore I n (λ j ) n 2β+1 P → c. (25)
We now study the behaviour of the term

1 m m i=1 I n,i (λ j ) = 1 m m i=1 1 2π i t=(i-1) +1 (Y t + g n (t))e itλ j 2
.

Following the same decomposition as in ( 23), we will have a full stochastic term, a trend and a cross term to examine.

Denote I n,i,Y the periodogram built from the ith epoch of the stationary process Y t . Using (16), we then obtain 1 m

m i=1 I n,i,Y (λ j ) = O P ( 2d ). 1 m m u=1 1 t=1 g n (t + (u -1) )e itλ j 2 = -1 n 2β t=1 s=1 m u=1 g t + (u -1) n g s + (u -1) n 1 m e i(t-s)λ j ∼ -1 n 2β t=1 s=1 1 0 g t n + x g s n + x dx e i(t-s)λ j = -1 n 2β 1 0   t=1 g t n x e itλ j 2   dx. (26)
(In the integrals above we extended g beyond (0,1) with value 0). If g is continuous over

[0,1] then it is uniformly continuous and hence

g t n + x = g(x) + n (t, x)
where n (t, x) → 0 uniformly in t = 1, . . . , and x, and hence, since We now consider a stochastic trend X t = X t-1 + Y t where Y t has density of the form (1). Let λ j be a fixed Fourier frequency . We begin with the numerator in ( 22). With

S t = Y 1 + • • • + Y t ,
we write X t = X 0 + S t and using the well known summation by parts formula n t=1

U t v t = U n V n - n t=1 u t V t ,
where

U k = u 1 + • • • + u k , V k = v 1 + • • • + v k ,
and the fact that, j = 1, 2, . . . , s, According to formula (5) in Theorem 4 of [START_REF] Hurvich | Asymptotics for the low-frequency ordinates of the periodogram of a long-memory time series[END_REF], the last term can be evaluated as follows:

  χ 2 1 random variables and ζ 1 (d), . . . ζ 2s (d) are the eigenvalues of Σ(d) defined in (8) and (9). In (11), D -→ represents the convergence in distribution.

Figure 1 :

 1 Figure1: Empirical levels evaluated on FARIMA(0,d,0) process as function of the memory parameter d ∈ (-1/2, 1/2). We compare the tests based on statistics V/S (solid line) and Q n, √ n (s, d) (dashed red lines) with n = 500, s = 1, 3, 5 and with 5000 replications. The nominal level is α = 5%.

Figure 3 :FigureFigure 5 :

 35 Figure3: Empirical power as function of d, the parameter of nonstationary fractional processes with d ∈ (1/2, 3/2). We compare the tests based on statistics V/S (solid line) and Q n, √ n (s, d) (dashed red lines) with n = 500, s = 1, 3, 5 and with 5000 replications. The nominal level is α = 5%.

  |γ(s)γ(s + k)|. (20) 20 When d = 0 then γ(h) is summable and hence (20) is O(1/m). When 0 < d < 1/2 then γ(h) = O(h 2d-1) and hence (20) is bounded by

  = -2d o( ) → 0, as → ∞.

t=1e

  itλ j = 0, the last expression in (26) is bounded by n 2β o( ).If g is piece-wise continuous on [0,1] then there exists M sub-intervals (A k , B k ), j = 1, . . . , M , such that g is continuous on such intervals with finite limits at A k and B k .The previous result (when g is continuous) remains true for each interval [A k , B k ] Hence we find that (26) is still bounded by M n 2β o( ) = n 2β o( ).For the cross-terms and similarly to (24), we easily obtain that)|dx L j (d)f (λ j ) 1/2 = cn β 1/2+d .Hence in total, the denominator is of order n 2β o P (1). Therefore, using (25), for each fixed frequency λ j , the ratio statisticn -1 I n (λ j )which finally implies that Q n,m (s, 1/2) goes to infinity as n → ∞.

  j )Y t ∼ [(2π) 1-2d j -2d f * (0)] 1 2 (L j,j (d) -L j,-j (d))where L j,±j (d) is defined in (10).Straightforward computations show that L j,j (d) -L j,-j (d) > 0 and therefore I n (λ j ) is at least of order n 2+2d . Now let us handle the denominator in (22). Similarly to the numerator, we can easily write, from the summation by parts formula, -W ,u (λ j ) 2 |1 -e iλ j | -2 ,whereS ,u = t=1 Y t+(u-1) ,andW ,u (λ j ) = t=1 Y t+(u-1) e itλ j .Using the stationarity of S ,u , u = 1, . . . , m and the fact that E(S 2 ,1 ) ∼ c 2d+1 (see Proposition 3.3.1 of[START_REF] Giraitis | Large sample inference for long memory processes[END_REF]) and that E W ,u (λ j )2 ∼ c 1+2d (see Theorem 1 ,u (λ j )| 2 O( 2 ) = O( 2 ) m n O P ( 2d+1 ) = O P ( 2d+2 ).Finally, combining the results on the numerator and denominator, we obtain that asymptotically n -1 I n (λ j )

  In this context, the most common d estimators are local Whittle or fully extended local Whittle estimator as defined in Abadir

Table 1 :

 1

	5

Empirical size of tests based on Q n,m (s, d) and V/S for FARIMA(0, d, 0) processes when d is close to 1/2, n=500 with 5000 replications. The nominal level is α = 5%.

the interval [0,1] can be divided into a finite number of sub-intervals such that the function is continuous on each open sub-interval, with finite limit at each endpoint.

That is h(λ, d) is uniformly bounded (in d) by an integrable function and hence

Then these two points imply that Σ(1/2) is well defined and a covariance matrix. Moreover

) to be a zeromean Gaussian vector with covariance matrix Σ(d). Then the convergence of the covariance matrices ensures that Z(s, d)

and therefore Z(s, 1/2) has at least s non-degenerated components. This ensures that the covariance matrix Σ(1/2) has at least s positive eigenvalues. This guarantees that Q(s, 1/2) is a nondegerated random variable.

Proof of Proposition 2

Let s be fixed and let d n be an arbitrary sequence in (-1/2, 1/2] converging to d ∈


N Dumont

L Martrille

E Albuisson

K Baland-Peltre

E Marchand
email: elodie.marchand@univ-lorraine.fr

Examining men as victims of intimate partner violence in a French forensic department

Keywords:

Objective: Intimate partner violence (IPV) is a particular type of interpersonal violence, in which violence is performed between individuals involved in an intimate relationship. Johnson developed a typology in which he differentiates common couple violence (CCV) from intimate terrorism (IT). We led a descriptive review of the cases of men treated at the victimology unit of the forensic department of Nancy for IPVrelated abuse. Method: We retrospectively investigated our database to extract every file of male patient treated at the victimology unit of the forensic department of Nancy (France) from 2014 to 2019 for IPV-related abuse. We studied several parameters relating to the protagonists and their relationship, the violence endured, and the statements made during the forensic consultation. An attempt to perform Johnson's typology was made. Results: We had a total of 226 consultations. The age of the patients ranged from 18 to 88 years old. Almost all patients experienced physical violence. Psychological violence was noted in 138 files. Concerning Johnson's typology, we identified 92 cases consistent with the definition of CCV and 46 cases compatible with IT. Conclusion: Male victims of IPV are now backed by an ever-growing corpus of studies that sheds an opportune light upon this challenging topic. Sadly, too many remain discredited and unaccounted for. We believe that escaping a gender-opposing conception in favor of a global understanding of this phenomenon would be of benefit to both men and women.

Introduction

Intimate partner violence (IPV) is a particular type of interpersonal violence in which violence is performed between individuals involved in an intimate relationship. It is considered a major health problem in many countries, including France, where the estimated number of individual victims of IPV per year is 302,000, 83,000 (28%) of which are men [START_REF] Guedj | Rapport d'enquête " cadre de vie et sécurité[END_REF]. In 2019, a three-month French national symposium ("Grenelle des violences conjugales") was held, during which numerous stakeholders expressed the government's will to work to eliminate IPV. Various types of violence should be considered with physical, psychological, and sexual being the most significant areas, but others, such as economic, should not be disregarded [START_REF]Understanding and addressing violence against women: intimate partner violence[END_REF][START_REF]Intimate partner violence[END_REF].

The involvement of each gender in IPV is a controversial subject. On the one hand, some authors affirm that women are as likely as men to be violent towards their partner and postulate that this violence should be viewed as a chief component of a more global pattern labelled "family violence" [START_REF] Archer | Sex differences in aggression between heterosexual partners: a metaanalytic review[END_REF][START_REF] Morse | Beyond the conflict tactics scale: assessing gender differences in partner violence[END_REF]. On the other hand, many authors assure us that IPV is overwhelmingly committed by men towards women and is a result of the ruling patriarchy that characterizes most societies; this view is supported by the "violence against women" theorists [START_REF] Walker | Psychology and violence against women[END_REF][START_REF] Dobash | Violences against women[END_REF]. Johnson suggested that considering the population in which the study is conducted is essential to correctly interpreting IPV. He developed a typology through which he differentiates common couple violence (CCV) from intimate terrorism (IT) [START_REF] Johnson | Patriarchal terrorism and common couple violence: two forms of violence against women[END_REF]. The first is defined as an outburst of violence during an https://doi.org/10.1016/j.forsciint.2022.111368 0379-0738/© 2022 Published by Elsevier B.V.

Abbreviations: IPV, Intimate partner violence; CCV, common couple violence; IT, intimate terrorism; VR, violent resistance; MCV, mutual control violence argument caused by a malfunction in the management of conflicts that may be equally initiated by men or women. The second is a situation wherein one of the partners tries to control the other by means of diverse coercive behaviors (e.g., stalking, establishing financial dependency, restraining access to friends and relatives, and, of course, physical violence) and, according to Johnson, is performed in the vast majority of cases by men. He then noted that CCV is a phenomenon observed in the general population and described by the "family violence" theorists, while IT is another phenomenon usually studied by the "violence against women" supporters. Later on, Johnson extended his typology by adding two derivatives of IT: violent resistance (VR) occurs when an individual suffering IT retaliates without the intent to gain control and, as a consequence, would be predominantly endured by men-whereas mutual control violence (MCV) can be portrayed as two intimate terrorists fighting for control over the relationship [START_REF] Johnson | A Typology of Domestic Violence: Intimate Terrorism, Violent Resistance, and Situational Couple Violence[END_REF].

Nonetheless, the IPV literature mainly focuses on women as victims [START_REF] Walker | Psychology and violence against women[END_REF][START_REF] Dobash | Violences against women[END_REF][START_REF] Miller | Intimate Partner Violence, 28 Febr[END_REF], so knowledge about male victims remains scarce. In particular, few studies about men resorting to a forensic medicine department have been conducted [START_REF] Savall | A ten-year experience of physical Intimate partner violence (IPV) in a French forensic unit[END_REF][START_REF] Thureau | Conjugal violence: a comparison of violence against men by women and women by men (Apr)[END_REF][START_REF] Carmo | Men as victims of intimate partner violence[END_REF][START_REF] Wörmann | Males as victims of intimate partner violence -results from a clinical-forensic medical examination centre (Sept)[END_REF]. In France, forensic departments deal with victims of assault, mistreatment, or accidents upon the request of the police or gendarmerie, which are both law enforcement agencies; basically, the police take action within a town area (more than 20,000 inhabitants), while the gendarmery operates in rural areas (fewer than 20,000 inhabitants). Forensic doctors assess the physical and psychological impact of an attack, its consistency with the victim's statement (regarding dating for instance), and its quantification via the "total incapacity to work" (incapacité totale de travail = ITT, a typical French concept reflecting the time during which the victim's ability to execute most daily life activities is heavily hindered due to the sustained violence). This ultimately helps the magistrate better estimate the harm produced and deliver a proper condemnation to the perpetrator. As such, the forensic department is a central operator of the judiciary process and a mandatory step in the care of an IPV victim.

In these circumstances, we led a retrospective, descriptive review focusing on the cases of men examined at the victimology unit of the forensic department of Nancy (France) for IPV-related abuse, with the aim of describing their characteristics and then discussing them in light of those recorded in other European forensic departments.

Material and methods

We retrospectively investigated the database storing the file of every patient treated at the victimology unit of the forensic department of Nancy (France) from 2014 to 2019; this period corresponds to the period from the commissioning of the software managing its activity (MIGALE) to the present day. Such files hold information provided by secretaries, nurses, and forensic doctors-in particular, the medical certificate written at the end of the consultation.

As we focused on male patients who consulted for IPV-related abuse, we recovered the files encoded in MIGALE with the items "male" and "intimate partner violence." Expecting the possibility of the same patient being treated multiple times, we distinguished between "constitutional" data that represent the characteristics of the protagonists and their relationship and "case-related" data that describe the context of the case. "Constitutional" data included the age of the patient, his birth country, his professional status, the sex of the abuser, his marital status with the abuser, the duration of the relationship, the existence of children born within the relationship, a history of previous physical violence, the time of onset of the violence, and a history of alcohol and/or drugs abuse by the assaulter. "Case-related" data included the place and time of the assault (morning: 06:00-12:00, afternoon: 12:00-18:00, evening: 18:00-00:00, night: 00:00-06:00); if the protagonists were living together at the moment of the latest assaults; any physical, psychological, sexual or economic violence mentioned in the file that would have been perpetrated by the abuser; the existence of secondary assaulters and/or witnesses; resorting to a caregiver and the subsequent necessity of treatment, such as surgical acts (including stitches), dressings, plastering, advanced medication, and psychological support but not radiologic investigations and grade-I analgesics; the time interval between the latest aggression and the forensic consultation; the law enforcement agency (police or gendarmerie) in charge of the affair; the injury outcome and its topographic distribution at the moment of the forensic consultation and/ or mentioned in the certificate of earlier caregivers; the psychological impact on the victim assessed by the forensic practitioner; a context of custody during the consultation; and, ultimately, the "total incapacity to work" prescribed.

We also attempted to determine which of Johnson's IPV types the case could be regarded as based on the existence and severity of coercive behaviors, the chronicity of the abuse (not reported here due to a rather vague appraisal), and the overall impression produced after the examination of the consultation file.

Statistical analysis

According to their nature and distribution, data were expressed as frequencies (%) or as medians [IQR]. Normality was analyzed with the Kolmogorov-Smirnov test. Characteristics of the patients and characteristics of the aggressor were presented individually (215 subjects), whilst the circumstances and proceedings of the aggression and information regarding the forensic consultation were presented by the consultations (226 consultations). The Chi-squared test was used to analyze the possible association between the variables.

Results

We recovered a list containing 315 consultations. We dismissed 38 of them as wrongly encoded and another 51 as non-exploitable. In the end, we had a total of 226 consultations accounting for 215 individuals (11 individuals consulted twice). These files represent 11.39% of the IPV files, 2.68% of the male files, and 1.27% of the total files registered.

The period covered by our study ranges from the beginning of 2014 to mid-February 2019. The yearly distribution of the number of consultations shows an overall increase from 46 in 2015-59 in 2018. The mean monthly distribution varies from 15 in May, July, and August to 27 in June, without demonstrating any obvious trend.

The characteristics of the studied population are displayed in Table 1. The age of the patients ranged from 18 to 88 years old, the average being 39.85 years and the median 38. The most sizable age group comprised individuals from 30 to 39 years old. Most patients were born in France (n = 182; 84.7%). Of the 138 files provided, 103 patients (47.9%) were employed.

Four patients (1.9%) were assaulted by a partner of the same sex. The most reported type of relationship was marriage (n = 86; 40%). For 133 couples, most of the relationships lasted for at least one year. Of 148 couples, 103 had children (47.9%). Only 17 consulted after being physically assaulted for the first time (7.9%). A story of excessive alcohol and/or drugs consumption by the aggressor was identified in 23 certificates (10.7%).

Data on the aggressions and their subsequent medical repercussions are displayed in Table 2. The aggressions mostly occurred at home (n = 94; 41.6%)-in particular the common residence (n = 66; 29.2%)-and in the evening (n = 51; 22.6%). Almost all patients experienced physical violence (n = 216; 95.6%); more than half of them (n = 116; 51.3%) reported "blows" using natural weapons, especially punches (n = 70; 31%), whilst scratches (n = 51; 22.6%) and slaps (n = 48; 21.2%) were also often reported. Other mechanisms included biting, pushing, strangling, grabbing, blunt and sharp trauma with an object, throwing an object, tearing off clothes, a necklace, etc., liquid spraying, and spitting. Psychological violence was noted in 138 files (61.1%) and mainly exerted through verbal assault (n = 113; 50%), such as insults and humiliations, shouts, and threats. Coercive behaviors such as harassment, involving children in the conflict, deteriorating goods, economic violence, and confinement were also reported. Only two patients reported sexual coercion. The presence of at least one additional aggressor was mentioned in 19 certificates (8.4%). Exposure of children to the abuse was specified in 43 certificates (19%). Prior to the forensic consultation, 81 patients (35.8%) saw a doctor, usually from an emergency service (n = 46; 20.4%). Medical or psychological care was requested in 33 cases (14.6%).

Data on the forensic examination are displayed in Table 3. In 177 cases (78.3%), the forensic consultation occurred within 10 days of the last aggression. The police intervened slightly more often (n = 132; 58.4%) than the gendarmery (n = 93; 41.2%). Consecutively to the attack, pain was declared in 112 certificates (49.6%). Injuries consisted predominantly of abrasions (n = 108; 47.8%), ecchymosis or hematoma (n = 65; 28.8%), and open wounds (n = 36; 15.9%); edema, functional deficit, fractures, and burns were noted, while 48 certificates (21.2%) stated the absence of lesion or pain. Injuries were predominantly located around the head and neck extremity (n = 162; 71.7%) and the upper limbs (n = 80; 35.4%). The psychological impact of the violence and abuse was assessed during 203 consultations (89.8%): it was present in 151 certificates (66.8%). The most reported symptoms were anxiety and sleep disorders (n = 73; 32.3% each). In 27 cases (11.9%), the consultation occurred while the patient was held in custody. Almost all certificates (n = 220; 97.3%) attributed a "total incapacity to work" lasting fewer than 8 days. Concerning Johnson's typology, we identified 92 cases (40,7%) consistent with the definition of CCV and 46 cases (20,4%) compatible with IT. We could not categorize any case as VR or MCV. Through our statistical analysis, we observed that the search for medical counselling was more associated with the police than gendarmerie procedures (p = 0.033) and not with the presence of a significant psychological impact (p = 0.32). As per Johnson's typology, IT was associated with the absence of injury outcome recorded (p = 0.04) and the presence of a psychological impact (p = 0.001) but not with any law enforcement agencies (p = 0.468 and p = 0.488 for police and gendarmerie respectively) and a situation where the protagonists were living together (p = 0.559). CCV and non-classified consultations were not associated with any of these variables. The results of our analysis are summarized in Table 4.

Discussion

We encountered four studies akin to ours: two from France by Savall [START_REF] Savall | A ten-year experience of physical Intimate partner violence (IPV) in a French forensic unit[END_REF] and Thureau [START_REF] Thureau | Conjugal violence: a comparison of violence against men by women and women by men (Apr)[END_REF], one from Portugal by Carmo [START_REF] Carmo | Men as victims of intimate partner violence[END_REF], and one from Germany by Wörmann [START_REF] Wörmann | Males as victims of intimate partner violence -results from a clinical-forensic medical examination centre (Sept)[END_REF]. We estimated that the number of files identified in this study relating to male victims of IPV might be slightly undervalued due to encoding errors. The total number of IPV files (11.39%) are similar to those recorded by Savall (12.8%), Thureau (13%), and Carmo (11.5%) and, furthermore, to the percentage of victims engaging in procedures with the French forces of order for IPV matters (12%) according to the 34th note of the National Observatory of Delinquency and Penal Responses published in March 2019 [START_REF] Sourd | Elements de mesure des violences au sein du couple en 2017 (Note n°34 de l'ONDRP)[END_REF]. Wörmann had a lower rate (6.2%) while specifying that it was lower that German official crime statistics (18%).

Our patients' mean age was just below 40 years old, which is in line with the results from Savall, Carmo, Wörmann, Ebouat [START_REF] Ebouat | Gromb-Monnoyeur, Étude rétrospective descriptive relative aux hommes victimes de violences intrafamiliales (June)[END_REF], and others [START_REF] Douglas | The helpseeking experiences of men who sustain intimate partner violence: an overlooked population and implications for practice (Aug)[END_REF], who all obtained a mean age of 41 years old, whereas Thureau found a slightly lower value of 36 years old. Unsurprisingly, most males in this study were born in France (n = 182; 84.7%), but individuals born in a foreign country still represented a sizable proportion (n = 32; 14.9%).

Using the police-gendarmerie ratio as an indicator of population density, the predominance of police procedures over gendarmery ones might indicate that our patients were predominantly of urban origin. This is consistent with Machado [START_REF] Machado | Characteristics of intimate partner violence victimization experienced by a sample of Portuguese men, Feb 1[END_REF]; however, Machado found a much larger difference, possibly due to a different sizing of "urbanity." We suggest that this could be the result of facilitated access to the services in urban areas and a more conservative mentality in rural ones-particularly regarding the traditional role assigned to each gender-that would prevent men from seeking help or simply a higher population concentration in urban areas.

Most were employed, with highly diversified profiles; we could not determine reliable data to estimate the sociocultural level. Cases of IPV by a partner of the same sex were not widely represented in our work (1.9%), which is similar to the findings of Savall, who found seven incidences of same-sex IPV among 641 males, but noticeably lower than Wörmann (8.4%).

Just as in Thureau, 40% of our patients were married to the abusive partner (although nearly half were in the process of getting a divorce), and roughly 25% were partners or ex-partners respectively. Carmo's patients were also engaged in lasting relationships (mainly between one and 15 years). Less than half of our couples were cohabiting at the time of the assaults, which is not as much as Thureau (69%) and Carmo (66%); nevertheless, this parameter has not been systematically specified and might therefore be undervalued.

A story of substance abuse by the partner was reported in slightly more than 10% of our cases, mostly in a chronical fashion. Specifically, the alcohol abuse rate in our dataset is close to Carmo's (9.8% vs. 9.3%), while our drug abuse rate is a bit higher (2.8% vs. 0.8%).

Aggressions typically occurred at home in the evening; this pattern aligns with those of Thureau, Ebouat, Wörmann, and others [START_REF] Reijnders | Victims of domestic and public violence; an exploratory study based on the reports of assaulted individuals reporting to the police[END_REF]. As such, additional attention should be taken when special circumstances like quarantine induce higher risks of seclusion with the abuser [START_REF] Van Gelder | COVID-19: reducing the risk of infection might increase the risk of intimate partner violence (Apr)[END_REF].

The predominant injurious mechanism was blunt trauma by means of natural weapons, as recorded by Savall, Thureau, Carmo, and Wörmann. Other mechanisms, such as biting, grabbing-especially at the level of the neck (i.e., strangling)-and shoving, were also regularly mentioned in these works. Trauma by blunt (5.2% in Carmo) objects was also likely to be encountered (n = 34; 15% and n = 25; 11.1% respectively in our study); Thureau observed the use of objects or weapons by women against men in 33% of cases and concluded there is a significant difference between the rate of use of objects or weapons by men towards women (19%). We can argue that women tend to use more often objects than men to compensate for a generally lower physical strength. Almost all weapons were unintended objects (e.g., bottles, chairs, walking sticks, or even cars regarding blunt trauma; knives, forks, or broken glass regarding sharp trauma). Only Wörmann reported one case of use of a firearm against a man. Contrary to what one might think, women might perform physical violence on a similar level and frequency to men [START_REF] Swan | A review of research on women's use of violence with male intimate partners[END_REF].

The cervico-cephalic extremity and the upper limbs (n = 162; 71.7% and n = 80; 35.4% respectively)-i.e., the most easily accessible areas by the attacker's upper limbs-were also recognized as favored targets in Savall (61% and 57.4% respectively), Carmo (43.9% and 35.8% respectively), and Wörmann (40% concerning the upper limbs). In all five works, superficial skin lesions (i.e., abrasions, ecchymosis/hematoma, and edema) constituted the common injury outcome. Other injuries, such as fractures and burns, were also reported in Savall (2.4% and 1%, respectively). The subsequent disabilities essentially affected the musculoskeletal system, but sensory impairment (auditive and ophthalmological) was observed in the case of head trauma. Contrary to Savall, no visceral or significant vascular damage has been noted in our work. When secondary aggressors were involved, they were usually blood related to the assaulter.

In margin or intricated to physical offences, many patients described various forms of psychological violence; some of them even said they never suffered any physical harm. Some declared that they endured coercive behaviors, and 15% considered themselves to be mentally harassed. Because the forensic consultation is not suited to extensively document these abuses (due to a shortage of time or a lack of familiarity of the practitioner in this field), these rates might be underestimated. Carmo also identified a high rate of insults, humiliations, and false accusations (60.3%) and noticed several coercive behaviors, such as "threatening to use a sharp instrument" (9.8%), interpersonal control (7.4%), "stalking or spying" (6.3%), and "destruction of victim's affective objects" (5.7%); Thureau and al, pointed out insults in 20% of their cases.

Although easily overlooked, investigating psychological abuse is of primal importance since 66.8% of our certificates found a significant psychological impact (60% in Thureau and al). Furthermore, it has been suggested to be predictive of later physical violence [START_REF] Schumacher | Husbands' and wives' marital adjustment, verbal aggression, and physical aggression as longitudinal predictors of physical aggression in early marriage[END_REF] and bears its own harmful consequences. While Hines [START_REF] Hines | Symptoms of posttraumatic stress disorder in men who sustain intimate partner violence: a study of helpseeking and community samples (Apr)[END_REF] stated that IPV may cause genuine post-traumatic stress disorder (PTSD), men interviewed by Dobash [START_REF] Dobash | Women's violences to men in intimate relationships[END_REF] reported extremely varied feelings about the aggression but frequently felt unconcerned and unaffected. This is congruent with the fact that almost a quarter of our patients declared that there was no psychological impact, while some developed chronic anxious and/or depressive syndromes significantly impacting their quality of life. It has been hinted that men and women's psychological impact regarding IPV may differ [START_REF] Park | This society ignores our victimization": understanding the experiences of korean male victims of intimate partner violence (Dec)[END_REF][START_REF] Próspero | Mental health symptoms among male victims of partner violence (Dec)[END_REF] and is heavily influenced by the cultural standards in which it takes place [START_REF] Schluter | Prevalence and concordance of interpersonal violence reports from intimate partners: findings from the Pacific Islands families study, July 1[END_REF].

Both cases mentioning sexual violence (one from a man, the other from a woman) were categorized as IT. Two other French studies mention a low rate of sexual aggression by an intimate partner: the National Observatory of Delinquency and Penal Responses reported that 2% of the victims of rapes by intimate partners were men [START_REF] Sourd | Elements de mesure des violences au sein du couple en 2017 (Note n°34 de l'ONDRP)[END_REF]; Hiquet, who studied male patients who consulted exclusively for sexual aggressions, found that 2.7% were assaulted by cohabitees [START_REF] Hiquet | L'homme face aux violences sexuelles. Onze ans d'expérience au centre d'accueil en urgence des victimes d'agression (CAUVA) du CHU de Bordeaux[END_REF], whilst Wörmann stated that 1.2% of their patients reported sexual violence, and Thureau found none.

Generally, sexual violence seems to be more frequently endured by women than by men [START_REF] Thureau | Conjugal violence: a comparison of violence against men by women and women by men (Apr)[END_REF][START_REF] Swan | A review of research on women's use of violence with male intimate partners[END_REF][START_REF] Chen | Characteristics of intimate partner violence in China: gender symmetry, mutuality, and associated factors[END_REF]. Being a particularly taboo topic, rapes are certain to be underreported if not specifically asked for. Plus, victims of IPV might have difficulties recognizing sexual violence since sexuality is a specific feature of intimate relationships.

Although contradictory with the idea that men are reluctant to seek help [START_REF] Addis | Men, masculinity, and the contexts of help seeking[END_REF][START_REF] Tsang | Chinese male survivors of intimate partner violence: a three-pillar approach to analyze men's delayed help-seeking decisions[END_REF], the high proportions of medical care prior to the forensic consultation-Savall (49.9%), Carmo (39.6%), and Thureau (66%) found an even higher rate than we did (35.8%), and their patients consulted an ER service-are consistent with the context of men prone to judiciary assistance. These patients were also significantly associated with the police area (i.e., urban areas that, as we already discussed, offer easier access to health facilities). Hospitalization was needed in 3.1% of our cases vs. 2.1% in Savall and 7% in Thureau.

Consistent with Savall and Thureau, patients usually consulted the forensic department a few days after the last assault. ITT is also an indicator of the most suited jurisdiction to take over the case: in cases of voluntary violence, an infraction causing an ITT value inferior or equal to eight days would be qualified as a lower-grade offence, while superior infractions would be called an intermediategrade offence. However, IPV being considered an aggravating circumstance, any violence committed on an intimate partner would qualify as an intermediate-grade offence. In this respect, the vast majority of our cases (n = 220; 97.3%) had an ITT lower than eight days, of which barely 30% were equal to zero, meaning low damage, but these should nonetheless qualify as intermediate-grade offences. Of note, the highest ITT value prescribed was six weeks. Almost 98% of Thureau's ITT cases were shorter than eight days, and 14% were equal to zero. In fact, ITT is an ambiguous and controversial concept whose assessment methods can be quite different from one center to another and is regularly challenged among forensic practitioners.

When it comes to IPV, a crucial element to be reckoned with is the exposure of children to violence. Through our study, we regularly encountered custody of the children or their management as a motive to the onset of IPV. Since an exposed child is a potential victim of mistreatment and is more likely to perpetrate and/or suffer IPV as an adult [START_REF] Miller | Intimate Partner Violence, 28 Febr[END_REF][START_REF] Tjaden | US Department of Justice: Office ot Justice Programs: National Institute of Justice. Extent, nature, and consequences of intimate partner violence[END_REF], the exposure of children to violence should readily be investigated, and, if found, psychological and social advice should systematically be contemplated.

According to Johnson's typology, men usually endure CCV or VR, but very few would face IT. Hines stated that men being victims of IT is not such an extraordinary situation [START_REF] Hines | Intimate terrorism by women towards men: does it exist? July 6[END_REF]. Based on the data in our possession, we classified 20.4% of our cases as IT, which is obviously considerable. Despite this, this assessment should be cautiously comprehended considering the limited material on which this judgement has been built (nearly 40% of the cases could not be classified due to the dearth of data), as well as its subjectivity. Interestingly enough, one patient who attended two consultations received different classifications for each: the first was classified as CCV, the second as ITT. Aside from illustrating the difficulty of the assessment, this led us to wonder about the potential scalability of the behavior of an abuser, i.e., the conversion from one type to another (most likely in the CCV to IT direction). Exploring the dynamic aspects of Johnson's typology thus deserves further consideration.

A strong association between IT patients and the presence of a psychological impact was expected, but we were at first astonished to also observe a significant association with the absence of injury outcome observed. We argue that physical violence is merely an aspect of the IT phenomenon, and, unlike CCV, IT victims undergo several coercive behaviors that might actually be the major motive for suiting. Moreover, fear of reprisal, personal beliefs, or tight control might delay the victims from seeking external help and therefore increase the chance of missing injuries after healing.

No single case could be classified as VR-certainly not because of their non-existence but rather because a one-sided testimony does not allow for a wholly objective grasp of the situation that would unveil IT-evocative behaviors (no patient would spontaneously admit they perpetrated it). Herein lies one huge bias within this study. On the contrary, some files mentioning mutual violence and abuse have been determined as CCV given the absence of complex controlling patterns. In fact, violence between intimate partners seems to often be bidirectional [START_REF] Swan | A review of research on women's use of violence with male intimate partners[END_REF][START_REF] Schluter | Prevalence and concordance of interpersonal violence reports from intimate partners: findings from the Pacific Islands families study, July 1[END_REF]. Forensic doctors having some rudimentary knowledge about the different types of IPV might prove useful in detecting and unveiling sensitive features relevant to the assessment of one's health state, especially mental.

Another tricky question raised regards men consulting while being held in custody: are they the actual author of the infraction, or is it part of a victimization strategy orchestrated by an abusive partner? Filing a complaint can be used as a defensive option to balance another person's complaint or accusation. Again, having a single point of view and limited investigative elements at our disposal did not always allow us to fully grasp the motives of our patients. Still, since forensic doctors are required to display neutral expertise on objective bodily injuries, so this should fortunately not be a major hindering factor to their mission-much more delicate, however, is the psychological aspect of this same mission.

Finally, we would like to underline that we worked with a particular population-that is, French men examined at the request of law enforcement officers. This limits our spectrum of victims to those who ended up filing a complaint, notably excluding those that could choose to come on their own initiative, as can be seen in other European countries such as Switzerland. Men who succumbed to fatal IPV were also not covered in this study. Therefore, this sample cannot be considered representative of the overall population.

Conclusion

While our study provides further material to try and discuss IPV, much remains to be said about it. Despite many discrepant results amongst studies, it appears that IPV is not as unidirectional as it is commonly depicted. Male victims are now backed by an evergrowing corpus of studies that sheds an opportune light upon this challenging topic. Sadly, too many remain discredited and unaccounted for. We believe that escaping a gender-opposing conception in favor of a global understanding of this phenomenon would benefit both men and women.

We observed some striking points of agreement with other studies conducted in European forensic centers about male victims of IPV, while other parameters remain unclear. In the end, further research should definitely focus on the psychological facets of these patients. Forensic doctors used to focus on the somatic consequences of the assaults but are now urged to conjointly proceed to a careful psychological assessment, which is particularly relevant in situations propitious to the commitment of psychological abuse, such as intimate relationships.

Declarations of interest

None.

  Investigation, Writing -original draft. L. Martrille: Supervision, Writing -review & editing. E. Albuisson: Methodology, Formal analysis. K. Baland-peltre: Writing -review. E. Marchand: Writing -review & editing.

Table 1

 1 Characteristics of our patients and their agressor (N = 215).

	AGE OF THE PATIENT	(n = 215; 100%)
	Mean (years)	39.85
	Median (years) [ IQR]	38.0 [30,0:48.0]
	[18-29 y/o]	50 (23.3%)
	[30-39 y/o]	63 (29.3%)
	[40-49 y/o]	56 (26%)
	[50-59 y/o]	32 (14.9%)
	> 60 y/o	14 (6.5%)
	BIRTH COUNTRY OF THE PATIENT	(n = 214; 99.5%)
	France	182 (84.7%)
	• Including Overseas Department Other countries	4 (1.9%) 32 (14.9%)
	PROFESSIONAL STATUS OF THE PATIENT	(n = 138; 64.2%)
	Employed	103 (47.9%)
	Unemployed	24 (11.2%)
	Student	2 (0.9%)
	Retired	9 (4.2%)
	SEX OF THE AGGRESSOR	(n = 215; 100%)
	Woman	211 (98.1%)
	Man	4 (1.9%)
	PARTNER'S STATUS	(n = 210; 97.7%)
	Married	86 (40%)
	Divorced	12 (5.6%)
	Common-law partner	55 (25.6%)
	Ex-common-law partner	51 (23.7%)
	"Civil union pact" a partnership	1 (0.5%)
	Ex "civil union pact" a partnership	4 (1.9%)
	Adulterous partner	1 (0.5%)
	RELATIONSHIP LENGTH	(n = 133; 61.9%)
	< 1 year	15 (7%)
	[1-10 years]	71 (33%)
	> 10 years	47 (21.9%)
	CHILDREN BORN OF THE RELATIONSHIP	(n = 148; 68.8%)
	Yes	103 (47.9%)
	No	45 (20.9%)
	FIRST PHYSICAL VIOLENCE EPISODE	17 (7.9%)
	ONSET OF THE VIOLENCE	(n = 123; 57.2%)
	Since the beginning	22 (10.2%)
	In the course of the relationship	66 (30.7%)
	After breaking up	35 (16.3%)
	SUBSTANCE USE BY THE AGRESSOR	(n = 23; 10.7%)
	Alcohol	21 (9.8%)
	Drugs	6 (2.8%)

a The "union civil pact" is a legal provision of civil partnership with more flexible clauses compared to marriage. Total might not amount to 100% due to some data leaking.

Table 2

 2 Circumstances and proceedings of the aggression (N = 226).

	PLACE OF THE ASSAULT	(n = 133; 58.8%)
	Common residence	66 (29.2%)
	Victim's residence	20 (8.8%)
	Aggressor's residence	8 (3.5%)
	Public road	31 (13.7%)
	Other private places	8 (3.5%)
	TIME OF THE ASSAULT	(n = 102; 45.1%)
	Morning	17 (7.5%)
	Afternoon	25 (11.1%)
	Evening	51 (22.6%)
	Night	9 (4%)
	LIVING TOGETHER	97 (42.9%)
	PHYSICAL VIOLENCE ALLEGED	(n = 216; 95.6%)
	Blows by natural weapons	116 (51.3%)
	• Including punches • Including kicks • Including headbutts Scratching	70 (31%) 37 (16.4%) 7 (3.1%) 51 (22.6%)
	Slapping	48 (21.2%)
	Blunt-object trauma	34 (15%)
	Sharp-object trauma	25 (11.1%)
	Shoving	22 (9.7%)
	Biting	22 (9.7%)
	Strangling	19 (8.4%)
	Grabbing	17 (7.5%)
	Throwing objects	13 (5.8%)
	Falls	12 (5.3%)
	Liquid spraying	7 (3.1%)
	Spitting	7 (3.1%)
	Tearing off clothes, necklaces, etc.	5 (2.2%)
	None	10 (4.4%)
	PSYCHOLOGICAL ABUSE ALLEGED	(n = 138; 61.1%)
	Verbal abuse	113 (50%)
	Insults and humiliations	62 (27.4%)
	Threats	42 (18.6%)
	• Including threats involving the use of an object • Including death threats Harassment	23 (10.2%) 9 (4%) 34 (15%)
	• Including harassment by messaging Children involvement	8 (3.5%) 24 (10.6%)
	Breaking or deteriorating goods	22 (9.7%)
	Shouts	11 (4.9%)
	Economic violence	10 (4.4%)
	Confinement	5 (2.2%)
	SEXUAL VIOLENCE ALLEGED	2 (0.9%)
	ADDITIONAL AGGRESSORS	(n = 19; 8.4%)
	Family of the aggressor	12 (5.3%)
	Other individual	6 (2.7%)
	Child of the couple	1 (0.4%)
	WITNESSES	43 (19%)
	Children	19 (8.4%)
	Adults
	MEDICAL COUNSELLING	(n = 81; 35.8%)
	Emergency doctor	46 (20.4%)
	General practitioner	35 (15.5%)
	NECESSITY OF MEDICAL CARE	(n = 33; 14.6%)
	Medical care	23 (11.5%)
	• Including surgical acts Psychologic or psychiatric care	17 (7.5%) 7 (3.1%)
	Hospitalization	7 (3.1%)

Table 3

 3 Information regarding the forensic consultation (N = 226).

			Table 4
			Statistical analysis.
	TIME INTERVAL UNTIL THE FORENSIC CONSULTATION			Search for medical	IT cases
	Median (days)	5		counselling
	[0-3 days]	85 (39%)	Police	p = 0.033	p = 0.468
	[4-10 days]	92 (42.2%)	Gendarmerie	-	p = 0.488
	> 10 days	41 (18.8%)	Presence of a psychological	p = 0.32	p = 0.001
	LAW ENFORCEMENT AGENCY	132 (58.4%)	impact
	Police	93 (41.2%)	No injury outcome recorded	-	p = 0.04
	Gendarmery		Living together	-	p = 0.559
	GLOBAL INJURY OUTCOME	112 (49.6%)
	Pain	108 (47.8%)
	Abrasions	65 (28.8%)
	Ecchymoses or hematomas	36 (15.9%)
	Open wounds	14 (6.2%)
	Functional deficits	12 (5.3%)
	Edemas	7 (3.1%)
	Fractures	3 (1.3%)
	Burns	48 (21.2%)
	None
	LESIONAL TOPOGRAPHY	162 (71.7%)
	Head and neck extremity	80 (35.4%)
	Superior limbs	41 (18.1%)
	Anterior trunk	23 (10.2%)
	Posterior trunk	37 (16.4%)
	Inferior limbs	4 (1.8%)
	Genitalia
	PSYCHOLOGICAL IMPACT	(n = 203; 89.8%)
	Present	151 (66.8%)
	None	52 (23%)
	Anxiety	73 (32.3%)
	Sleep disorders	73 (32.3%)
	Uneasiness	23 (10.2%)
	Crying	20 (8.8%)
	Depressive syndrome	20 (8.8%)
	Revivification episodes a	19 (8.4%)
	Appetite disorders	16 (7.1%)
	Sadness	15 (6.6%)
	Ruminations	14 (6.2%)
	Asthenia	13 (5.8%)
	Restriction in everyday life activities	9 (4%)
	Auto-destructive behaviors	4 (1.8%)
	CUSTODY	27 (11.9%)
	TOTAL INCAPACITY TO WORK PRESCRIPTION	1.70
	Mean (days)	67 (29.6%)
	0 days	153 (67.7%)
	≤ 8 days	5 (2.2%)
	> 8 days
	JOHNSON'S TYPOLOGY	(n = 138; 61.1%)
	Common couple violence (CCV)	92 (40.7%)
	Intimate terrorism (IT)	46 (20.4%)

a clinical symptom characterized by the repetition of the traumatic experience, reactivated suddenly by a stimulus/external factor.