生態保育已經是刻不容緩的一項課題，使用多樣性指標與相似度指標來量化群落，有助於了解生態的變化，因此各式各樣的多樣性指標與相似度指標被定義與討論，多樣性指標須具備的條件，相似度指標與多樣性之間的關係，是本文首要討論釐清。了解何謂多樣性指標與相似度指標後，如何精確估計指標是本文的重點，根據Hill數值指標為出發點，針對各個階次做估計，最後推廣到多群落的多樣性與相似度指標。

第一部份為單一群落的多樣性指標估計，首先針對最直觀的物種數進行估計，討論取後歸還與取後不歸還這兩種抽樣模式下的物種數估計量，改善文獻上最常用也最穩定的下界估計量。接著估計Shannon熵指標，此指標符合最多的多樣性假設，是非常重要的生態指標，但由於形式複雜，估計不易，本文改善文獻上Chao and Shen (2003) 所提出的修正方式。同時，當估計量形式複雜時，變異數估計量常使用拔靴法進行估計，但樣本出現機率與母體的真實機率實不相同，修正改善拔靴法的樣本出現機率，可使拔靴法更加精確。最後，Hill數值指標的估計更少人探討，文獻上所使用估計整條曲線的估計量皆有進步空間，本文針對階次大於等於2的整數點提出一個近似不偏的修正方式，可以有效減少偏誤，尤其在接次等於2的情況下表現精確。

第二部份為多群落的多樣性指標與相似度指標估計，其大多沿用一群落的多樣性方法，做推廣及延伸，使指標估計有一致性的結果，故單一群落討論的物種數估計可延伸到多群落的共同種估計，同樣也依抽樣方式的不同，分為隨機歸還與隨機不歸還的模式進行估計。接著推廣Shannon熵指標的估計方法至 多樣性指標與Horn相似度指標，同時利用一群落的修正拔靴方法，推廣到兩群落的拔靴修正，使得形式複雜的多樣性指標及相似度指標的估計量，有精確的變異數估計量。最後針對Morisita相似度指標進行估計，估計方式也可以推廣到 估計量。

每個主題的估計方法，皆以電腦模擬進行討論，比較文獻上與本文提出的方法，有何其優劣，同時附加合適的實例分析，說明實務上的應用方式，更加了解指標估計的重要性。

[1] Arvesen JN. 1969. Jackknifing U-statistics. Annals of Mathematical Statistics 40, 2076-2100.
[2] Bulmer MG. 1974a. On Fitting the Poisson lognormal distribution to species-abundance data. Biometrics 30, 101-110.
[3] Bunge J, Fitzpatrick M. 1993. Estimating the number of species: A review. Journal of the American Statistical Association 88, 364-373.
[4] Burnham KP, Overton WS. 1978. Estimation of the size of a closed population when capture probabilities vary among animals. Biometrika 65, 625-633.
[5] Chao A. 1984. Nonparametric estimation of the number of classes in a population. Scandinavian Journal of Statistics 11, 265-270.
[6] Chao A. 2005. Species estimation and applications. New York:Wiley 7907-7916.
[7] Chao A, Chazdon RL, Colwell RK, Shen T-J. 2005. A new statistical approach for assessing similarity of species composition with incidence and abundance data. Ecology Letters 8, 148–159.
[8] Chao A, Jost L, Chiang S-C, Jiang Y-H, Chazdon R. 2008. A two-stage probabilistic approach to multiple-community similarity indices. Biometrics 64, 1178–1186.
[9] Chao A, Lee S-M. 1992. Estimating the number of classes via sample coverage. Journal of the American Statistical Association 87, 210-217.
[10] Chao A, Lin C-W. 2012. Nonparametric lower bounds for species richness and shared species richness under sampling without replacement. Biometrics.
[11] Chao A, Shen T-J. 2003. Nonparametric estimation of shannon's index of diversity when there are unseen species. Environmental and Ecological Statistics 10, 429–443.
[12]Chao A, Wang YT, Jost L. 2012. Species accumulation curve and entropy: A very accurate estimator of entropy. Manuscript.
[13]Chiu CH, Wang YT, Chao A. 2012. A improved lower bound of species richness using Good-Turing formula. Manuscript.
[14] Colwell RK, Chao A, Gotelli NJ, Lin S-Y, Mao CX, Chazdon RL, 2012. Models and estimators linking individual-based and sample-based rarefaction, extrapolation, and comparison of assemblage. Ecology. 5, 3-21.
[15] Colwell RK, Coddington JA. 1994. Estimating terrestrial biodiversity through extrapolation. Philosophical Transactions of the Royal Society of London B - Biological Sciences 345, 101-118.
[16] Corbet AS. 1941. The distribution of butterflies in the maylay peninsula. Proceedings of the Royal Entomological Society of London Series A, General Entomology 16, 101-116.
[17] Efron B, Thisted R. 1976. Estimating the number of unseen species: How many words did Shakespeare know? Biometrika 63, 435-447.
[18] Good IJ. 1953. The population frequencies of species and the estimation of population parameters. Biometrika 40, 237-264.
[19] Gotelli NJ, Colwell. RK, eds. 2011. Estimating species richness. New York, USA:Oxford University Press.
[20] Haas PJ, Liu Y, Stokes L. 2006. An estimator of number of species from quadrat sampling. Biometrics 62, 135-141.
[21] Haas PJ, Stokes L. 1998. Estimating the number of classes in a finite population. Journal of the American Statistical Association 93, 1475-1487.
[22] Hill M. 1973. Diversity and evenness: A unifying notation and its consequences. Ecology 54, 427–432.
[23] Holst L. 1981. Some asymptotic results for incomplete multinomial or Poisson samples. Scandinavian Journal of Statistics 8, 243-246.
[24] Horn HS. 1966. Measurement of “overlap” in comparative ecological studies. American naturalist 100, 419–424.
[25] Horvitz DG, Thompson DJ. 1952. A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association 47, 663-685.
[26] Hurlbert SH. 1971. The nonconcept of species diversity: A critique and alternative parameters. Ecology 52, 577-586.
[27] Jaccard P. 1908. Nouvelles researchers sur la distribution florale. Bull Soc Vaudoise Sci Nat 44, 223-292.
[28] James W, Stein C. 1961. Estimation with quadratic loss. Proceedings of the fourth Berkeley Symposium on Mathematical Statistics and Probability 1, 311-319.
[29] Janzen DH. 1973. Sweep samples of tropical foliage insects: Description of study sites, with data on species abundances and size distributions. Ecology 54, 659-686.
[30] Johnson BM. 1971. On the admissible estimators for certain fixed sample binomial problems. The Annals of Mathematical Statistics 42, 1579-1587.
[31] Jolly GM. 1965. Explicit estimates from capture-recapture data with both death and immigration – stochastic model. Biometrika 52, 225-247.
[32] Jost L. 2006. Entropy and diversity. Oikos 113, 363–375.
[33] Jost L. 2007. Partitioning diversity into independent alpha and beta components. Ecology 88, 2427–2439.
[34] Ledoit O, Wolf M. 2003. Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance 10, 603-621.
[35] MacArthur RH. 1965. Patterns of species diversity. Biological reviews 40, 510–533.
[36] Magurran AE. 2004. measuring biological diversity. Oxford: Blackwell.
[37] Marshall A, Olkin I. 1979. Inequalities: Theory if majorization and its applications. Academic Press.
[38] Morisita M. 1959. Measuring of interspecific association and similarity between communities. Memoires of the Faculty of Science, Kyushu University, Series E (Biolology) 3, 65–80.
[39] Morris CN. 1983. Parametric empirical bayes inference: Theory and applications. Journal of the American Statistical Association 78, 47-55.
[40] Pan HY, Chao A, Foissner W. 2009. A non-parametric lower bound for the number of species shared by multiple communities. Journal of Agricultural, Biological and Environmental Statistics 14, 452–468.
[41] Patil GP, Taillie C. 1982. Diversity as a concept and its measurement. Journal of the American Statistical Association 77, 548-561.
[42] Peet RK. 1974. The measurement of species diversity. Annual Review of Ecology and Systematics 5, 285-307.
[43] Quenouille M. 1949. Approximate tests of correlation in time series. Journal of the Royal Statistical Society 11, 68-84.
[44] Rényi A. 1961. On measures of entropy and information. Proceedings of the 4th Berkeley symposium on mathematics. Statistics and Probability 1, 547-561.
[45] Routledge R. 1979. Diversity indices: which ones are admissible? Journal of Theoretical Biology 76, 503–515.
[46] Sørensen T. 1948. A method of establishing groups of equal amplitude in plant sociology based on similarity of species content and its application to analyses of the vegetation on Danish commons. Biol Skr 5, 1–34.
[47] Schäfer J, Strimmer K. 2005. The berkeley electronic press.
[48] Schechtman E, Wang S. 2004. Jackknifing two-sample statistics. Journal of statistical planning and inference 119, 329–340.
[49] Shannon CE. 1948. The mathematical theory of communication. Bell System Technical Journal 27, 379-423.
[50] Simpson EH. 1949. Measurement of diversity. Nature 163, 688.
[51] Stein C. 1956. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. Proceedings of the Third Berkeley symposium on Mathematical Statistics and Probability 1, 197-206.
[52] Tsallis C. 1988. Possible generalization of Boltzmann-Gibbs statistics. Journal of statistical physics 52, 480-487.
[53] Tukey J. 1958. Bias and confidence in not quite large samples. Annals of Mathematical Statistics 29, 614.
[54] Whittaker RH. 1972. Evolution and measurement of species diversity. Taxon 12, 213–251.
[55] Whittaker RH. 1960. Vegetation of the siskiyou mountains, Oregan and California. Ecological monographs 30, 279–338.
[56] Wolda H. 1981. Similarity indices, sample size and diversity. Oecologia 50, 296–302.
[57] Zahl S. 1977. Jackknifing an index of diversity. Ecology 58, 907-913.
[58] 江霖 物種稀釋曲線之統計推論 Statistical inference for species rarefaction curve 趙蓮菊指導 新竹市國立清華大學, 民99.
[59] 邱春火 系統演化多樣性指標之建立與統計估計 Establishment and statistical estimation of phylogentic diversity indices趙蓮菊指導 新竹市國立清華大學, 民99.
[60]韓若平 Beta生物多樣性的統計估計 Statistical estimation of beta diversity趙蓮菊指導 新竹市國立清華大學, 民97.
[61] 蘇郁如 生物多樣性相似指標之統計分析 Statistical analysis of bio-diversity similarity indices 趙蓮菊指導 新竹市國立清華大學, 民94.