The Determination of Likelihood and the Transformed Regression Model

Abstract

The traditional model of statistics is a class of probability measures for a response variable. Under reasonable continuity this can be given as a class $C$ of probability density functions relative to an atom-free measure. With a realized value of the response variable, the model $C$ gives the possible probabilities for that realized value--it gives the likelihood function. The likelihood function can be accepted alone or in conjunction with the distribution of possible likelihood functions. In a variety of applications, the variation in a response variable can be traced to a well-defined source having a known probability distribution. The model then is not a class of probability measures but is a single probability measure and a class of random variables. Under moderate conditions this can be given as a probability density function and a class $C_2$ of transformations from the variation space to the response space. And if the distribution for variation is not completely known, the model becomes a class $C_1$ of probability density functions and a class $C_2$ of transformations from the variation space to the response space. With an observed response value, the component $C_2$ identifies a set, the set of possible values for the realized variation. If $C_2$ is a transformation group, then $C_2$ identifies a set--in a partition on the variation space. Standard probability argument using $C_1$ then gives the probability of what has been "observed," and the conditional distribution of what has not been "observed": it gives the likelihood function from the identified set, and the conditional density within the identified set. The likelihood function alone or with its distribution gives the information concerning the parameter of $C_1$; and for any assumed value of that parameter the conditional density gives the information concerning possible values for the realized variation, and accordingly gives the information concerning the parameter of $C_2$, it being what stands between the realized variation and the observed response. The probability of what is identified as having occurred--the likelihood function--is a fundamental output of a model involving density functions. The determination of this probability can however involve certain complexities as soon as the class $C_2$ of random variables is no longer effectively a group. Certainly the class $C_2$ identifies a set on the variation space. But in moderately general cases the range of alternatives can be a partition on the variation space depends on the element of $C_2$. Thus an `event' is identified but the range of possible `events' depends on the parameter of $C_2$. For two kinds of generalized model $(C_1, C_2)$ this paper explores the determination of the probability of what is identified as having occurred--it explores the determination of the likelihood function. In Section 1 the notation and results are summarized for the special model $(C_1, C_2)$ with $C_2$ a transformation group. Two generalizations are examined in Section 2: first, the class $C_2$ is a group but its application as a transformation group has an additional parameter; second, the class $C_2$ is a class of expression transformations $L$ applied to a group of transformations $G$, i.e. $C_2 = LG$. These two generalizations are not as distinct as they may at first appear but they are quite distinct in contexts. The transformed regression model is the central example. Several formulas for volume change in subspaces are recorded in Section 3 and used in Section 4 to make four determinations of likelihood for the generalized model $(C_1, C_2)$. These are applied to the transformed regression model in Section 5 and compared by means of examples in Section 6. The effects of initial variable on the likelihood functions is examined in Section 7 and two compensating routes for analysis are proposed. The class $L$ of expression transformations is examined in Section 8 and shown to be a group under mild consistency conditions. A corresponding invariant likelihood is determined in Section 9, and a transit likelihood in Section 10; the power-transformed regression model is examined in Section 11. In Section 12 the transit likelihood is shown to be the natural likelihood when the semi-direct product $LG$ is itself a group.