Monotone trends in the distribution of climate extremes

Abstract

The generalized Pareto distribution (GPD) is often used in the statistical analysis of climate extremes. For a sample of independent and identically distributed observations, the parameters of the GPD can be estimated by the maximum likelihood (ML) method. In this paper, we drop the assumption of identically distributed random variables. We consider independent observations from GPD distributions having a common shape parameter but possibly an increasing trend in the scale parameter. Such a model, with increasing scale parameter, can be used to describe a trend in the observed extremes as time progresses. Estimating an increasing trend in a distribution parameter is common in the field of isotonic regression. We use ideas and tools from that area to compute ML estimates of the GPD parameters. In a simulation experiment, we show that the iterative convex minorant (ICM) algorithm is much faster than the projected gradient (PG) algorithm. We apply the approach to the daily maxima of the central England temperature (CET) data. A clear positive trend in the GPD scale parameter is found, leading to an increase in the 100-year return level from about 31º in the 1880s to 34º in 2015.

Appendix

Lemma

For eachξ > − 0.5, thereexists aσ_ξ ∈ Csuchthat

$$\ell(\boldsymbol{\sigma}_{\xi},\xi)\ge \ell(\boldsymbol{\sigma},\xi) \ \text{ for all} \ \boldsymbol{\sigma}\in C. $$

Consequently, ℓ_p given in Eq. 6 is well defined.

Proof

Fix ξ > 0 and note that σ↦ℓ(ξ,σ) is continuous on C. Moreover, note that by Eq. 5, for y > 0 fixed and σ↓ 0,

$$\ln\left[g_{\xi, \sigma}(y)\right] \sim \frac1\xi \ln(\sigma) \rightarrow - \infty $$

and for σ →∞,

$$\ln\left[g_{\xi, \sigma}(y)\right] \sim -\ln(\sigma) \rightarrow - \infty. $$

Therefore, in maximizing σ↦ℓ(ξ,σ) over C, attention can be restricted to a compact subset of C, namely σ ∈ C for which δ ≤ σ₁ ≤ σ_n ≤ 1/δ for some small δ > 0. This ensures the existence of σ_ξ.

For ξ = 0, ln [g_0,σ(y)] = − ln(σ) − y/σ, leading to the same conclusion. In the case ξ ∈ (− 0.5, 0), the restriction y ≤−σ/ξ implies that σ ≥−ξy. For σ↓−ξy, we obtain

$$\ln\left[g_{\xi, \sigma}(y)\right] \sim \left( -\frac{1}{\xi} - 1\right) \ln(\sigma + \xi y) \rightarrow -\infty, $$

due to the fact that $(-\frac {1}{\xi } - 1) > 0$ for ξ ∈ (− 0.5, 0). For σ →∞, we obtain as before

$$\ln\left[g_{\xi, \sigma}(y)\right] \sim - \ln(\sigma) \rightarrow -\infty. $$

Thus, attention can be restricted again to a compact subset of C, namely for some small δ > 0

$$\cup_{i = 1}^{n} \{\boldsymbol{\sigma} \in C\,:\, -\xi y_{i} + \delta \le \sigma_{i} \le 1 / \delta \}. $$

On this set, σ↦ℓ(ξ,σ) is continuous and hence ℓ_p(ξ) is well defined. □

Consider the first (partial) derivative

$$\frac{\partial \ln g_{\xi, \sigma}(y)}{\partial \sigma} = \frac{y - \sigma}{\sigma (\sigma + \xi y)}. $$

This shows that σ↦ ln g_ξ,σ(y) is unimodal with maximum σ = y for fixed ξ. The second derivative is given by

$$\frac{\partial^{2} \ln g_{\xi, \sigma}(y)}{\partial \sigma^{2}} =\frac{(\sigma - y)^{2} - (\xi + 1)y^{2}}{\sigma^{2} (\sigma + \xi y)^{2}}. $$

It follows that

$$\frac{\partial^{2} \ln g_{\xi, \sigma}(y)}{\partial \sigma^{2}} = 0 \iff \sigma=y(1\pm\sqrt{1+\xi}). $$

This shows that the second derivative exhibits in general at least one change of sign. Thus, the log likelihood is not concave for ξ≠ 0.