關於描述兩個觀察地A和B相似的程度而言，生物品種是否相同是其中的一個切入點，因此品種重複（species overlap）便為描述兩觀察地相似度的一種指標。就一般的生物或生態研究而言，較常使用的品種重複指數為以品種數為計算基礎的 Jaccard index，公式為 ，其中 和 分別為觀察地A和B的總品種數，而 則為兩地的共同品種數，這樣的計算方式為Gower(1985) 歸類描述兩單位（unit）的相似度（similarity）中的一種。在我們的研究中，將令依觀察到的品種數及品種重複數所計算出的 Jaccard index 視為估計值，記為 ；若描述相似度時僅以品種為計算單位，而忽略個別品種的數量未免有資訊流失的情形，因此我們延伸 Jaccard index 指數而另立以個別品種數為計算單位的 N 指數，並以無母數最大概似估計法（Nonparametric Maximum Likelihood Estimator, NPMLE）估計 N 指數，記為 。另外，Smith, Solow 和 Preston (1996) 也提出利用 delta-beta-binomial 模型修正 Jaccard index 的低估（underestimate）情形，我們將此模型所推估的品種重複記為 ，因此我們的研究重點便在於以模擬實驗比較 、 和 在估計真正參數時的行為。

在模擬實驗中，根據蒙地卡羅（Monte-Carlo）模擬法則，我們設計6種品種發生機率相等的平衡母體，及12種品種發生機率服從幾何分配的不平衡母體，以500次抽樣所得的平均數及標準差決定估計的好壞。根據研究結果，若在已知母體為平衡母體的情形之下， 和 有不錯的估計；而 則是不管在平衡母體或不平衡母體皆有不錯的估計，但 和 在某些不平衡母體時，卻有極偏差的估計。

除了模擬實驗之外，我們並推導出 的期望值和變異數，並證明其為 N 指數的大樣本不偏估計值（asymptotic unbiased estimator），並以台灣西北部濕地的鳥類記錄為實例，計算出三個估計值，並以跋靴法（Bootstrapping）計算出三個估計量的標準差估計值，發現NPMLE 有最小的變異程度。

In describing the similarity between communities A and B, species overlap is one kind of measure. In ecology and biology, the Jaccard index (Gower, 1985) ,denoted , for species overlap is widely used and is useded as an estimation in our research. However, the Jaccard index is simply the proportion of overlapping species, that is those species appearing in more than one community, to unique species, that is those species appearing in only one community. However, this index ignores species proportion information, assigning equal weight to all species. We propose a new index, N, which includes proportion information and is estimated by a Nonparametric Maximum Likelihood Estimator (NPMLE), denoted . Smith et al. (1996) proposed a delta-beta-binomial model to improve underestimation of the Jaccard index, we denoted this estimator .

In our Monte-Carlo simulations, we design 6 balanced populations in which every species has an equal proportion and 12 unbalanced populations in which species proportions follow a geometric distribution. We found that and are accurate for balanced populations but overestimate or underestimate the true value for some unbalanced populations. However, is robust for both balanced and unbalanced populations.

In addition to simulation results, we also give theoretical results, which prove some asymptotic properties of NPMLE .For example, species abundance of wild birds communications occurred at two locations in north-western Taiwan.Via bootstrapping, has smaller standard error than and .

中文書目
王吉松，民88，以用字分析紅樓夢之作者問題，政治大學統計研究所碩士論文。
趙蓮菊，民84，種類知多少--敬獻給為台灣環保努力的朋友們，數學傳播，19，1-7。
吳育美，民86，大學圖書館工程類西文圖書館藏重複及特性之研究：以台大、清大、交大、成大為例，淡江大學教育資料科學研究所碩士論文。
陳恆安，民83，台中市沿都市郊區梯度上動物分佈與物種豐度，東海大學生物學研究所碩士論文。
彭起嘉，民85，塔塔加火災過後地區小型哺乳類之群聚生態研究，東海大學生物學研究所碩士論文。

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