A Gamma Model Based On Box-Cox Transformation

During statistical analysis process, establishing a mathematical model is an important research task. For example, meteorologists establish a complicated model consisting of daily rainfall, atmospheric pressure and wind speed to forecast weather; To make a contribution for human security and social development, seismologists are more responsible and eager to set up a mathematical model containing population, pollution, recorded time. We usually apply the method of regression for data sets with many index to establish a model. However, according to much uncertainty of univariate or a few of index, the dataset should be fitted to a probability distribution for statistical inference. For instance, many kinds of counting random variables in the nature field and engineering field follow approximately poisson distribution; the gamma distribution offers a good fit to time of the i-th happened earthquake under conditions of the earthquake sequence with orderliness, homogeneity and counting with independent and stationary incremental features. During modeling, we often deal data with linear or non-linear transformations.This paper conducts a statistical analysis about the different dream levels of students. Based on the frequency table, we make a Box-Cox transformation to the age variable and then employ a gamma distribution to model the transformed variable. Chapter 2 gives two kinds of methods about how to seek for maximum likelihood estimates, such as two-stage estimate and the method of distributing points uniformly. Both supplement each other and can solve almost all loglikelihood functions. Chapter 3 discusses consistency and asymptotic normality, and gives theory proof. Chapter 4 shows that parameters estimates are worked out through the above methods from chapter 2. Besides, we can also get the estimates of three parameters by using regression in 4.3. We record the results of the three methods together and confirm that a gamma model is a perfect fit to data in table 1 based on Box-Cox transformation.