Statistical parameter
Encyclopedia
A statistical parameter is a parameter that indexes a family of probability distribution
s. It can be regarded as a numerical characteristic of a population
or a model
.
Among parameterized families
of distributions are the normal distributions, the Poisson distribution
s, the binomial distributions, and the exponential distribution
s. The family of normal distributions has two parameters, the mean
and the variance
: if these are specified, the distribution is known exactly. The family of chi-squared distributions, on the other hand, has only one parameter, the number of degrees of freedom.
In statistical inference
, parameters are sometimes taken to be unobservable, and in this case the statistician's task is to infer what he can about the parameter based on observations of random variables distributed according to the probability distribution in question, or, more concretely stated, based on a random sample taken from the population of interest. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out (for example, the number of degrees of freedom
in a Pearson's chi-squared test
).
Even if a family of distributions is not specified, quantities such as the mean
and variance
can still be regarded as parameters of the distribution of the population from which a sample is drawn. Statistical procedures can still attempt to make inferences about such population parameters. Parameters of this type are given names appropriate to their roles, including:
Where a probability distribution has a domain over a set of objects that are themselves probability distributions, the term concentration parameter
is used for quantities that index how variable the outcomes would be.
Quantities such as regression coefficients, are statistical parameters in the above sense, since they index the family of conditional probability distributions that describe how the dependent variables
are related to the independent variables.
as a statistic
is to a sample.
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s. It can be regarded as a numerical characteristic of a population
Statistical population
A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...
or a model
Statistical model
A statistical model is a formalization of relationships between variables in the form of mathematical equations. A statistical model describes how one or more random variables are related to one or more random variables. The model is statistical as the variables are not deterministically but...
.
Among parameterized families
Parametric family
In mathematics and its applications, a parametric family or a parameterized family is a family of objects whose definitions depend on a set of parameters....
of distributions are the normal distributions, the Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
s, the binomial distributions, and the exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
s. The family of normal distributions has two parameters, the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
and the variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
: if these are specified, the distribution is known exactly. The family of chi-squared distributions, on the other hand, has only one parameter, the number of degrees of freedom.
In statistical inference
Statistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...
, parameters are sometimes taken to be unobservable, and in this case the statistician's task is to infer what he can about the parameter based on observations of random variables distributed according to the probability distribution in question, or, more concretely stated, based on a random sample taken from the population of interest. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out (for example, the number of degrees of freedom
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
in a Pearson's chi-squared test
Pearson's chi-squared test
Pearson's chi-squared test is the best-known of several chi-squared tests – statistical procedures whose results are evaluated by reference to the chi-squared distribution. Its properties were first investigated by Karl Pearson in 1900...
).
Even if a family of distributions is not specified, quantities such as the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
and variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
can still be regarded as parameters of the distribution of the population from which a sample is drawn. Statistical procedures can still attempt to make inferences about such population parameters. Parameters of this type are given names appropriate to their roles, including:
- location parameterLocation parameterIn statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...
- dispersion parameter or scale parameterScale parameterIn probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...
- shape parameterShape parameterIn probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.- Definition :...
Where a probability distribution has a domain over a set of objects that are themselves probability distributions, the term concentration parameter
Concentration parameter
In probability theory and statistics, a concentration parameter is a special kind of numerical parameter of a parametric family of probability distributions...
is used for quantities that index how variable the outcomes would be.
Quantities such as regression coefficients, are statistical parameters in the above sense, since they index the family of conditional probability distributions that describe how the dependent variables
Dependent and independent variables
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
are related to the independent variables.
Analogy
A parameter is to a populationStatistical population
A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...
as a statistic
Statistic
A statistic is a single measure of some attribute of a sample . It is calculated by applying a function to the values of the items comprising the sample which are known together as a set of data.More formally, statistical theory defines a statistic as a function of a sample where the function...
is to a sample.
See also
- Precision (statistics)Precision (statistics)In statistics, the term precision can mean a quantity defined in a specific way. This is in addition to its more general meaning in the contexts of accuracy and precision and of precision and recall....
, another parameter not specific to any one distribution - ParametrizationParametrizationParametrization is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object....
(i.e., coordinate systemCoordinate systemIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
) - Parsimony (with regards to the trade-off of many or few parameters in data fitting)