*2.1. Measure of Concavity*

The global and localised *simplex statistics* proposed in Abrevaya and Jiang (2005) are employed. These loosen the assumption of normality and homoscedasticity of the error disturbances, which is required for most other concavity measures.

#### **Definition 4** (Global Simplex Statistics)**.** *Assume that*


*Then, for a sample of size n, the global simplex statistic is defined as*

$$dJ\_n = \binom{n}{3}^{-1} \sum\_{1 \le t\_1 \le t\_2 \le t\_3 \le n} \text{sign}(a\_1 y\_{\lfloor t\_1 \rfloor} + a\_2 y\_{\lfloor t\_2 \rfloor} - y\_{\lfloor t\_3 \rfloor})\_\prime \tag{11}$$

*where t*1, *t*2, *t*3 ∈ {1, . . . , *<sup>n</sup>*}*,*

$$\text{sign}(q) = \begin{cases} 1 & \text{if } q > 0, \\ -1 & \text{if } q < 0, \\ 0 & \text{if } q = 0. \end{cases} \tag{12}$$

*and a*1, *a*2 *are non-negative numbers with a*1 + *a*2 = 1*.*

Abrevaya and Jiang (2005) proves the asymptotic normality for the global simplex statistic. An interval is considered to be a concave interval if its standardised global simplex statistics *U* ˜ *n* is less than *Zα*/2, where *Zα*/2 is the (*α*/2)-th percentile of the standard normal distribution, vice versa for a convex interval.

For the localised simplex statistic, calculation starts with dividing the time interval [0, *T*] into *G* = *T*/2*h* subintervals, where 2*h* is the window width. Let *<sup>x</sup>*<sup>∗</sup>1, *x*∗2 , ..., *x*∗*G* denote evaluation points. In this case, they are assumed to be the mid-points of each sub-interval. Then, the sub-population for which the *x* values fall in a local window is defined to be

$$V\_h(\mathbf{x}^\*) = \{(y, \mathbf{x}) : \mathbf{x}^\* - h < \mathbf{x} < \mathbf{x}^\* + h\}.$$

**Definition 5.** *Let ph*(*x*<sup>∗</sup>) *be the number of observations in the set Vh*(*x*<sup>∗</sup>)*. The localised simplex statistic at a given x*<sup>∗</sup> *is then defined as*

$$\mathcal{U}\_{\mathfrak{n},h}(\mathbf{x}^\*) = \left(\begin{array}{c} p\_h(\mathbf{x}^\*)\\ \mathbf{3} \end{array}\right)^{-1} \sum\_{t\_1 < t\_2 < t\_3} \left( \text{sign}(a\_1 y\_{[t\_1]} + a\_2 y\_{[t\_2]} - y\_{[t\_3]}) \times \prod\_{k=1}^3 \mathcal{K}\_h(\mathbf{x}\_{t\_k} - \mathbf{x}^\*)\right),\tag{13}$$

*where a*1 + *a*2 = 1 *and Kh*(*v*) = *<sup>K</sup>*(*v*/*h*) *where the kernel function K is to be specified.*

In this paper, we assume a uniform kernel function since no addition information is given. Asymptotic normality also holds for the localised simplex statistics. A sub-interval is defined to be concave if *Un*,*<sup>h</sup>* < *Zα*/2, vice versa for the convex case. We define a concavity measure (localised) as the difference in the proportions of the concave intervals and the convex intervals:

> *Ln*,*<sup>h</sup>* = proportion of convex intervals − proportion of concave intervals.

Here, *Ln*,*<sup>h</sup>* ∈ [−1, 1] measures the level of concavity (convexity). Convexity is concluded for the overall interval when *Ln*,*<sup>h</sup>* ∈ [0, 1], while we report strict concavity when *Ln*,*<sup>h</sup>* ∈ [−1, <sup>0</sup>).

As a measure of tail index, the classic Hill's estimator is implemented. If *X*1, *X*2, ..., *Xn* is a sequence of independent and identically distributed random variables with distribution function *F* and *n* is the sample size, let *<sup>X</sup>*(*<sup>i</sup>*,*<sup>n</sup>*) be the i-th order statistic of *X*1, *X*2, ..., *Xn*. The Hill's estimator is

$$h^{\text{Hill}}\_{(k(n),n)} = \left(\frac{1}{k(n)} \sum\_{i=n-k(n)+1}^{n} \log(X\_{(i,n)}) - \log(X\_{(n-k(n)+1,n)})\right)^{-1},\tag{14}$$

where *k*(*n*) ∈ {1, 2, ..., *n* − 1} is a sequence to be specified. Under these assumptions, *h*Hill (*k*(*n*),*<sup>n</sup>*) converges in probability to the true tail index, and is asymptotically normal when *k*(*n*) → ∞.
