在生物多樣性的領域中，估計群落中的物種數相當的重要；在估計一個群落的物種數和兩個群落的共同物種數中，現今已有許多的估計量，包括Chao and Lee (1992) 年提出利用樣本涵蓋率，估計一個群落的物種數估計量；Chao et al. (2000) 將其推廣至兩個群落共同種的估計量；Chao et al. (2006) 利用Laplace 方法估計一個群落物種數估計量和兩個群落共同種估計量。
本文主要是針對Laplace 方法之估計量進一步推廣，將其推廣至三、四及五個群落，並估計三個群落的共同物種數，以及至少為兩群落之共同物種的種類數，分別稱為狹義估計量和廣義估計量；其中廣義估計量，是利用兩個群落估計量和三個群落估計量運算所得；接著使用漸進的手法推導三群落下，兩個估計量的標準差，並且加上電腦模擬來比較這兩個估計量的偏誤情況，而四個群落和五個群落，只模擬其狹義估計量部分，最後同時比較樣本涵蓋率估計量和Laplace法估計量在三群落下的表現。
由於利用Laplace 方法所得狹義估計量，在模擬時當母體群落設定下，同質性的母體比較多時 (例如：三個群落至少有兩個群落為同質性母體)，則估計量容易有高估的情形，所以利用偏誤調整形式 (bias-corrected form) 來修正狹義估計量；因為廣義估計量計算過程中包含了狹義估計量，所以廣義估計量的修正，則是在計算式中代入修正後的狹義估計量。最後，採用世界五大洲之原生動物的資料討論和比較。

[1] Bohannan, B. J. M. & Hughes, J. (2003). New approaches to analyzing microbial biodiversity data. Current Opinion in Microbiology 6, 182-187.

[2] Chao, A. (1987). Estimating the population size for capture-recapture data with unequal catchability. Biometrics, 43, 783-791.

[3] Chao, A. and Lee, S-M. (1992). Estimating the number of classes via sample coverage. Journal of American Statistical Association, 87, 210-217.

[4] Chao, A. and Ma, M-C. and Yang, M. C. K.(1993), Stopping rule and estimation for recapture debugging with unequal detection rates. Biometrika, 80, 193-201.

[5] Chao, A., Hwang, W-H, Chen, Y-C., and Kuo C-Y. (2000). Estimating the number of shared species in two communities. Statistica Sinica, 10, 227-246.

[6] Chao, A., Shen, T.-J. and Hwang, W.-H.(2006). Application of Laplace's boundary-mode approximations to estimate species and shared species richness. To appear in Australian and New Zealand Journal of Statistics.

[7] Colwell, R. K. and Coddington, J. A. (1994). Estimating terrestrial biodiversity through extrapolation. Philosophical Transactions of the Royal Society of London B – Biological Sciences 345, 101-118.

[8] Colwell, R. K. (1997). EstimateS: statistical estimation of species richness and shared species from samples. Version 5. Use's Guide and Application published at http://viceroy.eeb.uconn.edu/estimates.

[9] Erkanli, A. (1994). Laplace approximations for posterior expectations when the mode occurs at the boundary of the parameter space. Journal of the American Statistical Association 89, 250-258.

[10] Erkanli, A. (1997). Boundary-mode approximations for posterior expectations. Journal of Statistical Planning and Inference Volume 58, Issue 2 , 15 March 1997, Pages 217-239.

[11] Good, I. J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika, 40,237--264.

[12] Hughes, J. B., Hellmann, J. J., Ricketts, T. H. and Bohannan, B. J. M. (2001). Counting the uncountable: statistical approaches to estimating microbial diversity. Applied and Environmental Microbiology 67, 4399-4406.

[13] Jolly, G.M.(1965).Explicit Estimates from Capture-Recapture Data with Both Death and Immigration-Stochastic Model. Biometrika, 52, 225-247.

[14] Stach, J. E. M., Maldonado, L. A., Masson, D. G., Ward, A. C., Goodfellow, M. and Bull, A. T. (2003). Statistical approaches for estimating actinobacterial diversity in marine sediments. Applied and Environmental Microbiology 69, 6189-6200.

[15] Walther, B. A. and Morand, S. (1998). Comparative performance of species richness estimation methods. Parasitology 116, 395-405.