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
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
be a constant. Let
. Then
For a fixed distribution , as
varies from
to
, 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
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
be a constant. Let
. Then
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
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
be constants. Let
. Then
Note that it is important that
and
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
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