In probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the noncentral t-distribution (also known as the singly noncentral t-distribution) generalizes Student's t-distribution using a noncentrality parameter

Noncentrality parameter

Noncentrality parameters are parameters of families of probability distributions which are related to other "central" families of distributions. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family...

. Like the central t-distribution, the noncentral t-distribution is primarily used 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...

, although it may also be used in robust modeling

Robust statistics

Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

 for data

Data

The term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...

. In particular, the noncentral t-distribution arises in power analysis

Statistical power

The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...

.

Characterization

If is a normally distributed random variable with unit variance and zero mean, and is a Chi-squared distributed random variable with 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...

 that is statistically independent

Statistical independence

In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

 of , then


is a noncentral t-distributed random variable with degrees of freedom and noncentrality parameter

Noncentrality parameter

Noncentrality parameters are parameters of families of probability distributions which are related to other "central" families of distributions. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family...

 . Note that the noncentrality parameter may be negative.

Cumulative distribution function

The cumulative distribution function

Cumulative distribution function

In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...

 of noncentral t-distribution with degrees of freedom and noncentrality parameter can be expressed as


where is the regularized incomplete beta function,
and is the cumulative distribution function of the standard normal distribution.

Alternatively, the noncentral t-distribution CDF can be expressed as:
where is the gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

 and is the regularized incomplete beta function.

Although there are other forms of the cumulative distribution function, the first form presented above is very easy to evaluate through recursive computing

Recursion

Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...

. In statistical software R

R (programming language)

R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....

, the cumulative distribution function is implemented as pt.

Probability density function

The probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 for the noncentral t-distribution with degrees of freedom and noncentrality parameter can be expressed in several forms.

The confluent hypergeometric function

Confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...

 form of the density function is

where is a confluent hypergeometric function

Confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity...

.

An alternative integral form is

A third form of the density is obtained using its cumulative distribution functions, as follows.

This is the approach implemented by the dt function in R

R (programming language)

R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....

.

Moments of the Noncentral t-distribution

In general, the th raw moment of the non-central t-distribution is



In particular, the mean and variance of the noncentral t-distribution are

and

Asymmetry

The noncentral t-distribution is asymmetric unless μ is zero, i.e., a central t-distribution. The right tail will be heavier than the left when μ > 0, and vice versa. However, the usual skewness is not generally a good measure of asymmetry for this distribution, because if the degrees of freedom is not larger than 3, the third moment does not exist at all. Even if the degrees of freedom is greater than 3, the sample estimate of the skewness is still very unstable unless the sample size is very large.

Mode

The noncentral t-distribution is always unimodal and bell shaped, but the mode is not analytically available, although it always lies in the interval

when and

when

Moreover, the mode always has the same sign as the noncentrality parameter and the negative of the mode is exactly the mode for a noncentral t-distribution with the same number of degrees of freedom but noncentrality parameter

The mode is strictly increasing with when and strictly decreasing with when In the limit, when approaches zero, the mode is approximated by

and when approaches infinity, the mode is approximated by

Use in power analysis

Suppose we have an independent and identically distributed sample , each of which is normally distributed with mean and variance , and we are interested in testing the null hypothesis

Null hypothesis

The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...

  vs. the alternative hypothesis . We can perform a one sample t-test

Student's t-test

A t-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known...

 using the test statistic

Test statistic

In statistical hypothesis testing, a hypothesis test is typically specified in terms of a test statistic, which is a function of the sample; it is considered as a numerical summary of a set of data that...

where is the sample mean and is the unbiased sample variance. Since the right hand side of the second equality exactly matches the characterization of a noncentral t-distribution as described above, has a noncentral t-distribution with degrees of freedom and noncentrality parameter .

If the test procedure rejects the null hypothesis whenever , where is the upper quantile of the (central) Student's t-distribution for a pre-specified , then the power of this test is given by


Similar applications of the noncentral t-distribution can be found in the power analysis

Statistical power

The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...

 of the general normal-theory linear models

General linear model

The general linear model is a statistical linear model.It may be written aswhere Y is a matrix with series of multivariate measurements, X is a matrix that might be a design matrix, B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors or...

, which includes the above one sample t-test

Student's t-test

A t-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is supported. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known...

 as a special case.

Related distributions

• Central t distribution: The central t-distribution can be converted into a location
  Location parameter
  In 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...
  
  /scale
  Scale parameter
  In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...
  
   family. This family of distributions is used in data modeling to capture various tail behaviors. The location/scale generalization of the central t-distribution is a different distribution from the noncentral t-distribution discussed in this article. In particular, this approximation does not respect the asymmetry of the noncentral t-distribution. However, the central t-distribution can be used as an approximation to the non-central t-distribution.

• If is noncentral t-distributed with degrees of freedom and noncentrality parameter and , then has a noncentral -distribution
  Noncentral F-distribution
  In probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a generalization of the F-distribution...
  
   with 1 numerator degree of freedom, denominator degrees of freedom, and noncentrality parameter .

• If is noncentral t-distributed with degrees of freedom and noncentrality parameter and , then has a normal distribution with mean and unit variance.

• When the denominator noncentrality parameter of a doubly noncentral t-distribution is zero, then it becomes a noncentral t-distribution.

Special cases

• When , the noncentral t-distribution becomes the central (Student's) t-distribution with the same degrees of freedom.

External links

• Eric W. Weisstein. "Noncentral Student's t-Distribution." From MathWorld—A Wolfram Web Resource

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.