變數變換及數學歸納法於機率與數理統計之應用

The Application of Transformation and Mathematical Induction Method for the Probability and Mathematical Statistics

李雅苓(Ya-Ling Li)

指導教授 ： 林協宗

國立臺北大學/商學院/統計學系/碩士(2008年)

摘要

Schwarz和Samanta(1991)基於初學者對於數學歸納法與變數變換法較為簡單，較易瞭解，也較容易為初學者接受，首先利用數學歸納法(The Method of Mathematical Induction)與變數變換法(The Method of Transformation)證明Inverse Gaussian Distribution之加法性(The additive property)。雖意圖證明其母數(Parameters) 與 最大概似估計式(Maximal Likelihood Estimators) 與 之獨立性(Independence)，但事實僅證明 與 之獨立性，並證明 與 之獨立性，又證明 服從Inverse Gaussian Distribution即 與 服從自由度為n-1之卡方分配(Chi-squared Distribution)即 。此外，過去的文獻未曾使用Basu’s定理證明Inverse Gaussian Distribution母數 與 最大概似估計式 與 之獨立性，因此，本文試用Basu’s定理證明Inverse Gaussian Distribution估計式 與 之獨立性。
過去的文獻採用許多方法證明常態分配平均數 與變異數 最大概似估計式 與 之獨立性，惟未曾使用數學歸納法與變數變換法證明常態分配平均數 與變異數 最大概似估計式 與 之獨立性，因此，本文亦基於初學者對於數學歸納法與變數變換法較為簡單，較易瞭解，也較容易為初學者接受，引用數學歸納法與變數變換法證明常態分配(Normal Distribution)之加法性，並證明其平均數 與變異數 之估計式 與 的獨立性及平均數 與變異數 最大概似估計式 與 之獨立性。但本文依Casella和Berger(2002) “Statistical Inference” 之fiducial Inference Theory，平均數 之fiducial scaling revision function定義為

且平均數 的fiducial pdf定義為

致發現常態分配平均數 的fiducial scaling revision function應為

而平均數 的fiducial pdf為

且確為 之常態分配。惟Casella和Berger(2002)依fiducial Inference Theory於Statistical Inferfence P.292發現常態分配平均數 的fiducial scaling revision function為

顯然與本文發現之fiducial scaling revision function不同，致平均數 的fiducial pdf為

斷非 之常態分配，致發現Casella和Berger(2002)的推論過程顯然發生錯誤。此外，我們試用Basu’s定理證明常態分配平均數 與變異數 估計式 與 之獨立性。
最後，我們探討自然界昆蟲卵生子孫(egg’s offspring)繁殖過程之存活(survival)情形(存活率 )，若昆蟲卵生子孫 存活情形服從白努力分配(Bernoulli Distrbution)，則其存活總數(total survival number)為 服從二項分配(Binomial Distribution) 首先利用數學歸納法與變數變換法證明白努力分配和二項分配之加法性，並發現樣本數 之最大概似估值為 ，因此，我們發現當 時樣本數 之最大概似估值並不唯一，而當 則其最大概似估值才為唯一，顯與Casella和Berger(2002)於Statistical Inference P.319?斷言樣本數「 之最大概似估值為唯一」的結論不同。

關鍵字

常態分佈 ； 數學歸納法 ； 變數變換 ； 獨立性 ； 加法性 ； 可信機率密度函數

並列摘要

Since the Method of Mathematical Induction and the Method of Transformation is easier for beginners, Schwarz and Samanta firstly presented the method of transformation and mathematical induction to intend verifying the independence of the Maximal Likelihood Estimators (MLEs) and which means the independence of the sampling distributions for the MLEs and of the parameters and in an inverse Gaussian distributions in their paper (1991). In fact, they had shown that the independence of the estimators and which means the independence of the Sampling Distributions for the estimators and and the independence of the estimators and which means the independence of the Sampling Distributions for the estimators and of the parameters in an inverse Gaussian distribution by the method of transformation and mathematical induction. Next, they can establish the Additive Property that the addition of inverse Gaussian distribution is also inverse Gaussian distribution. Besides, we find that the former documents never presented the Basu’s Theorem to show the independence for the sampling distributions of the estimators and for the parameters of an inverse Gaussian distributions. Hence, this paper takes the Basu’s Theorem to verify the independence for the sampling distributions of the estimators and for the parameters and of an inverse Gaussian distribution.
Many different methods presented in the former documents completely established the independence of the MLEs and which means the independence for the sampling distributions of the MLEs and for the Parameters and of normal distribution. But we have never seen that the former documents take the mathematical induction method and transformation method to prove that. Because the Mathematical Induction Method only requires the multivariable transformations presented in ordinary probability theory and mathematical statistics, we present a mathematical induction method and transformation method to prove the additive property of normal distribution and the independence of the estimators and which means the independence for the Sampling Distributions of the estimators and as well as the independence of the MLEs and which means the independence for the sampling distributions of the MLEs and for the parameters and of normal distribution in the paper. Further, according to the Fiducial Inference Theory in Statistical Inference of Casella and Berger, the fiducial scaling revision function is define as

and the fiducial probability density function (pdf) of is defined as

so that we found that the fiducial scaling revision function of should be

during the process of discussing maximum likelihood estimate for the parameters and of Normal Distribution, then we can obtain that the fiducial probability density function (pdf) of is

Obviously, is normal distribution Unfortunately, Casella and Berger(2002) found that the fiducial scaling revision function in P.292 of their book “Statistical Inference” was

which is significantly different from the previous finding of the fiducial scaling revision function. Accordingly, their fiducial pdf of becomes

which is not a pdf of Normal Distribution Therefore we find that the Statistical Inference process of Casella and Berger's conclusion had obviously made a mistake.
Finally, we intend to consider the survival number of egg’s offspring with the surviving probability for the eggs laid by a mother insect form a large number of mother insects. At first, we used mathematical induction method and transformation method to prove that the additive property of Bernoulli distribution and binomial distribution as the iid observations of a random sample sampled form the Bernoulli population. Next, if the iid observations of a sampled egg’s offspring sampled form the Bernoulli population, we found that the total survival number of sampled egg’s offspring is a probability mass function (pmf ) of binomial distribution and the maximal likelihood estimator of the sample size is . Besides, the maximal likelihood estimate for the number n of the sampled egg’s offspring is not unique as and is unique as which are significantly different from the assertion that “the maximal likelihood estimate for the number is unique” mentioned by the book “Statistical Inference” in P.319?of Casella and Berger (2002)
.

並列關鍵字

Normal Distribution ； mathematical Induction Method ； transformation ； independence ； additive property ； fiducial pdf.

參考文獻

Ross, Sheldon (2006), “A First Course in Probability”, 7th edition, New Jersey︰Pearson Prentice Hall, Pearson Education Inc.

Chhikara, R. S. and Folks, L. S.(1989), the Inverse Gauusian Distribution: Theory, Methodology and Application, New York: Marcel Dekker.

Schwarz, Carl J. and Samanta, M.(1991), “An Inductive Proof of the Sampling Distributions for the MLEs of the Parameters in an Inverse Gaussian Distributions ”, The American Statistican, 45, 223-225.

Shuster, J., (1973), “A Simple Method of Teaching the Independence of and ”, The American Statistican, 27, 29-30.

Stiger, S. M., (1984), “Kruskal’s Proof of the Joint Distribution of and ”, The American Statistican, 38, 134-135.