Group Family
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In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, especially as that field is used in statistics, a group family of
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomeno ...
s is a family obtained by subjecting a random variable with a fixed distribution to a suitable family of transformations such as a location-scale family, or otherwise a family of probability distributions acted upon by a group. Consideration of a particular family of distributions as a group family can, in
statistical theory The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statisti ...
, lead to the identification of an ancillary statistic.Cox, D.R. (2006) ''Principles of Statistical Inference'', CUP. (Section 4.4.2)


Types of group families

A group family can be generated by subjecting a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
with a fixed distribution to some suitable transformations. Different types of group families are as follows :


Location Family

This family is obtained by adding a constant to a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
. Let X be a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
and a \in R be a constant. Let Y = X + a . Then F_Y(y) = P(Y\leq y) = P(X+a \leq y) = P(X \leq y-a) = F_X(y-a) For a fixed distribution , as a varies from -\infty to \infty , the distributions that we obtain constitute the location family.


Scale Family

This family is obtained by multiplying a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
with a constant. Let X be a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
and c \in R^+ be a constant. Let Y = cX . ThenF_Y(y) = P(Y\leq y) = P(cX \leq y) = P(X \leq y/c) = F_X(y/c)


Location - Scale Family

This family is obtained by multiplying a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
with a constant and then adding some other constant to it. Let X be a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
, a \in R and c \in R^+be constants. Let Y = cX + a . Then F_Y(y) = P(Y\leq y) = P(cX+a \leq y) = P(X \leq (y-a)/c) = F_X((y-a)/c) Note that it is important that a \in R and c \in R^+ in order to satisfy the properties mentioned in the following section.


Properties of the transformations

The
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Tran ...
applied to the random variable must satisfy the following properties. * Closure under composition * Closure under inversion


References

Parametric statistics Types of probability distributions {{Statistics-stub