Taneichi(1993)提出一個新的適合度檢定統計量J²，具有近似卡方分配的性質。然而在小樣本的情形下，計算機模擬結果顯示，它的估計顯著水準大於期望顯著水準。所以本論文的重點之一，就是對J²進行改進，根據不同的準則，來選取一個適當的常數a。我們建議對每一觀測次數加一常數0.32，作為我們修正後的統計量，這個統計量我們記為J₁²。

　　另一探討的重點是在比較皮爾生卡方統計量X²，概似比例統計量G²，Cressie & Read統計量 I(2/3)，J²和J₁²之性質，我們想要了解在小樣本的情形之下，何者較接近於卡方分配，何者具有較強的檢定力。研究結果顯示，X²和I(2/3)較接近卡方分配，但J₁²又較G²及J²好；至於檢定力，我們發現沒有一個統計量在文中所探討的對立假設的情況下，同時都具有最大的檢定力。這些現象都可以用觀測次數對期望次數比值間的關係來解釋。

　　Taneichi(1993) introduces a new goodness-of-fit statisticJ², which has an asymptotic chi-squared distribution. However, the results of simulation indicate that the levels of significance are in general bigger than the nominal levels, which prompts us to device a version of J² statistic which would perform better under small sample size situations. We suggest adding 0.32 to each observed value and find that the adjustment indeed works rearonably well. This version of J^2 statistic is denoted as J(1)^2.

　　Although Pearson chi-square statistic X², likelihood ratio statistic G², Cresse-Read statistic I(2/3), J^2 and J(1) ^2 all have asymptotic chi-squared distributions, their small sample behaviors are not expected to be the same. Comparisons based on simulation studies are then made. The conclusions are as follows : (1) In terms of levels of significance, X² and I(2/3) behave more like a chi-squared distribution. Though J(1) ^2 does not perform as good as X² and I(2/3), it does outperform G² and J². (2) In terms of powers, it does not seem that any of the test statistics has a clear advantage over the others.