Prediction in partial duration series with generalized pareto‐distributed exceedances

Abstract

As a generalization of the common assumption of exponential distribution of the exceedances in partial duration series the generalized Pareto distribution has been adopted. Estimators for the parameters are presented using estimation by both method of moments and probability‐weighted moments. The corresponding estimators for the T‐year event are given and approximate expressions for bias and variance of the estimators are derived in both cases. Using the mean square error of the T‐year event estimator as a performance index it is shown that the method of moments is preferable to the probability‐weighted moments. Maintaining the generalized Pareto distribution as the parent exceedance distribution the T‐year event is estimated assuming the exceedances to be exponentially distributed. For moderately long‐tailed exceedance distributions and small to moderate sample sizes it is found, by comparing mean square errors of the T‐year event estimators, that the exponential distribution is preferable to the correct generalized Pareto distribution despite the introduced model error and despite a possible rejection of the exponential hypothesis by a test of significance. For moderately short‐tailed exceedance distributions (with physically justified upper limit) the correct exceedance distribution should be applied despite a possible acceptance of the exponential assumption by a test of significance.

Modeling Properties of Influenza-Like Illness Peak Events with Crossing Theory

“…Results have also been extended to non-Gaussian processes [18] and can be applied to estimate the characteristics of ILI events. The magnitudes of events over the threshold and their durations can be approximated by an exponential-like distribution, such as the generalized Pareto distribution (GPD) [19,20], which can represent such data exhibiting both greater and lesser skew than the exponential itself. In combination with the Poisson assumption, it implies that the annual maximum or -peak week‖ [8] follows a generalized extreme value (GEV) distribution [19], the properties of which can be estimated from this approach, if desired.…”

“…The magnitudes of events over the threshold and their durations can be approximated by an exponential-like distribution, such as the generalized Pareto distribution (GPD) [19,20], which can represent such data exhibiting both greater and lesser skew than the exponential itself. In combination with the Poisson assumption, it implies that the annual maximum or -peak week‖ [8] follows a generalized extreme value (GEV) distribution [19], the properties of which can be estimated from this approach, if desired. In general, the criteria for adopting a specific distribution are; the goodness-of-fit, a strong theoretical basis and the relative ease of computation and interpretation.…”

“…In general, the criteria for adopting a specific distribution are; the goodness-of-fit, a strong theoretical basis and the relative ease of computation and interpretation. Although few influenza-related studies have focused on the statistical properties of peak events, the proposed approach has been extensively used in studies modeling extremes in various fields, including floods, stock market returns and daily maximum temperatures [19][20][21].…”

“…Occasions when weekly cases dropped marginally below the threshold level only to exceed it again in the next week are probably the result of the same event. Parallel considerations in the definition of flood and heat wave events can be found in Rosbjerg et al [19] and Keellings and Waylen [20]. -Events‖ failing to meet this independence criterion are combined and included in subsequent analysis as if they constitute a single event.…”

Comparative Study between POT and AMS Flood Frequency Analysis Models under Changing Environment

“…超定量(POT 或 PDS)抽样两种方法获得 [3] 。其中基于 AMS 抽样的洪水频率分析较常用， 样本长度与实测资 料年数相等。基于 POT 序列的洪水频率分析(POT 模 型)， 是以大于指定门限值的洪峰流量为样本进行分析 [2,4] 。在利用实测资料描述洪水特征上，POT 抽样既反 映洪水量级又反映洪水发生过程，比 AMS 抽样具有 更多物理相关性 [5] 。 关于超定量发生次数和样本分布。超定量的年发 生次数为随机数，在洪水频率分析应用中多假设该序 列服从泊松分布 [6][7][8][9] ；此外还有学者对比分析二项分 布、负二项分布和泊松分布，并给出利用分散指数检 验超定量发生次数服从何种分布的方法 [10,11] 。在拟合 POT 样本方面，近十几年来多采用广义 Pareto(GP)分 布 [12,13] ，作为 POT 模型经典分布的指数分布为 GP 分 布在形状参数为 0 时的特殊情况。 POT 样本选取的关键在于洪峰独立性判别和门 限值的选取 [5] 。POT 模型优点众多，但却未得到广泛 应用，原因可能是 POT 样本选取尚无统一标准 [3] 。目 前样本独立性判别标准主要有美国水资源协会标准、 Cunnane 标准和王善序提出的判别标准 [4] 。门限值选 取的主要方法包括年均超定量发生次数 n 法 [14,15] 、超 定量样本均值法、分散指数法和 Rosbjerg 提出的门限 值选取方法 [5] 。Lang 等 [5] 概括了上述方法并提出门限 值选择建议：通过超定量样本均值法和分散指数法确 定门限值区间，并选择满足 n > 2 或 3 的较大门限值， 该方法综合考虑了门限值选取需注意的问题。年均超 定量发生次数有不同的建议值，Cunnane [14] 认为用指 数分布时 n 应大于 1.65， 我国学者在进行 POT 模型分 析的时候，多通过试算将 n 控制在 2~3 [2,8,9,12] 。董爱红 [13] 分析了用 GP 分布拟合 POT 序列时， 设计值和分布 …”