| Summary: | 碩士 === 國立臺北大學 === 統計學系 === 96 === 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 distribution. 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 estimate 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).
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