# HotellingTSquareDistribution

represents Hotelling's distribution with dimensionality parameter p and m degrees of freedom.

# Details • The probability density for value in Hotelling's T-square distribution is proportional to for .
• HotellingTSquareDistribution allows p and m to be any positive real numbers such that .
• HotellingTSquareDistribution allows p and m to be dimensionless quantities.
• HotellingTSquareDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• represents a continuous statistical distribution defined over the interval and parameterized by two positive real numbers p and m>p-1. Here, p is called a "dimensionality parameter" and m a degrees of freedom parameter. The parameter m determines the height and steepness of the probability density function (PDF) of a Hotelling distribution. The general behavior of the PDF is determined by p and may be either monotonically decreasing with a potential singularity approaching the lower boundary of its domain (when ) or unimodal (for ). In addition, the tails of the PDF are "fat," in the sense that the PDF decreases algebraically rather than exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
• The Hotelling distribution dates to the early 1930s work of American mathematician Harold Hotelling that generalized the Student -distribution (StudentTDistribution) to cases involving hypothesis testing of p random variates. The Hotelling distribution forms the basis of the Hotelling test, which is a multivariate hypothesis test for the null hypothesis of equality among two unknown vectors of normally distributed variates having unknown covariance matrices. Since its creation, the Hotelling distribution has been used to model phenomena in agriculture, process control, principal component analysis, and quality control.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Hotelling distribution. Distributed[x,HotellingTSquareDistribution[p,m]], written more concisely as xHotellingTSquareDistribution[p,m], can be used to assert that a random variable x is distributed according to a Hotelling distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[HotellingTSquareDistribution[p,m],x] and CDF[HotellingTSquareDistribution[p,m],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a Hotelling distribution, EstimatedDistribution to estimate a Hotelling parametric distribution from given data, and FindDistributionParameters to fit data to a Hotelling distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Hotelling distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Hotelling distribution.
• TransformedDistribution can be used to represent a transformed Hotelling distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Hotelling distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Hotelling distributions.
• Hotelling's distribution is related to a number of other distributions. As previously noted, HotellingTSquareDistribution is linked to statistical testing involving NormalDistribution and MultinormalDistribution. It was devised as a generalization of StudentTDistribution and in particular, is proportional to . Under appropriate assumptions, is a special case of both PearsonDistribution (in the sense that its PDF is identical to that of PearsonDistribution[6,1,-((m(p-2))/(3+m-p)),2/(3+m-p),(2 m)/(3+m-p), 0]) and FRatioDistribution (in the sense that the CDF of is precisely that of FRatioDistribution[p,1 - p + m], x (m - p + 1)/(m p)]). HotellingTSquareDistribution is also related to ChiDistribution, ChiSquareDistribution, BetaDistribution, FisherZDistribution, and LaplaceDistribution.

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

## Scope(7)

Generate a sample of pseudorandom numbers from a Hotelling's T-square distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness:

The limiting value:

Kurtosis:

The limiting value:

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Hazard function:

Quantile function:

## Applications(1)

Hotelling's statistic is used to test whether multivariate data has a given mean:

For multivariate normal data of length with mean , the test statistic follows a HotellingTSquareDistribution[p,n-1] where is the dimension of the data:

Obtain the test statistic and -value for some data:

Alternatively, use TTest:

## Properties & Relations(3)

Relationships to other distributions: Hotelling's T-square distribution is a special case of FRatioDistribution:

Hotelling's T-square distribution is a special case of type 6 PearsonDistribution:

## Neat Examples(1)

PDFs for different p values with CDF contours: