Studentized range distribution

Studentized range distribution
Parameters	k > 1 — the number of groups ν > 0 — degrees of freedom
Support	q ∈ [0; +∞)
PDF	${\begin{matrix}f(q;k,\nu )={\frac {{\sqrt {2\pi }}k(k-1)\nu ^{\frac {\nu }{2}}}{\Gamma \left({\frac {\nu }{2}}\right)2^{{\frac {\nu }{2}}-1}}}\int _{0}^{\infty }x^{\nu }\phi ({\sqrt {\nu }}x)\times \\[0.5em]\left[\int _{-\infty }^{\infty }\phi (u)\phi (u-qx)(\Phi (u)-\Phi (u-qx))^{k-2}\,{\text{d}}u\right]{\text{d}}x\end{matrix}}$
CDF	${\begin{matrix}F(q;k,\nu )={\frac {k\nu ^{\frac {\nu }{2}}}{\Gamma \left({\frac {\nu }{2}}\right)2^{{\frac {\nu }{2}}-1}}}\int _{0}^{\infty }x^{\nu -1}e^{\frac {-\nu x^{2}}{2}}\times \\[0.5em]\left[\int _{-\infty }^{\infty }\phi (u)(\Phi (u)-\Phi (u-qx))^{k-1}\,{\text{d}}u\right]{\text{d}}x\end{matrix}}$

In probability and statistics, Studentized range distribution is a continuous probability distribution that arises when estimating the range of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ) and suppose that ${\bar {y}}$ _min is the smallest of these sample means and ${\bar {y}}$ _max is the largest of these sample means, and suppose S² is the pooled sample variance from these samples. Then the following random variable has a Studentized range distribution.

q={\frac {{\overline {y}}_{\max }-{\overline {y}}_{\min }}{S{\sqrt {2/n}}}}

Definition

Probability density function

Differentiating the cumulative distribution function with respect to q gives the probability density function.

f(q;k,\nu )={\frac {{\sqrt {2\pi }}k(k-1)\nu ^{\frac {\nu }{2}}}{\Gamma \left({\frac {\nu }{2}}\right)2^{{\frac {\nu }{2}}-1}}}\int _{0}^{\infty }x^{\nu }\phi ({\sqrt {\nu }}x)\left[\int _{-\infty }^{\infty }\phi (u)\phi (u-qx)(\Phi (u)-\Phi (u-qx))^{k-2}\,{\text{d}}u\right]{\text{d}}x

Cumulative distribution function

The cumulative distribution function is given by ^[1]

F(q;k,\nu )={\frac {k\nu ^{\frac {\nu }{2}}}{\Gamma \left({\frac {\nu }{2}}\right)2^{{\frac {\nu }{2}}-1}}}\int _{0}^{\infty }x^{\nu -1}e^{\frac {-\nu x^{2}}{2}}\left[\int _{-\infty }^{\infty }\phi (u)(\Phi (u)-\Phi (u-qx))^{k-1}\,{\text{d}}u\right]{\text{d}}x

Special cases

When the degrees of freedom approach infinity, the standard normal distribution can be used for the general equation above. If k is 2 or 3,^[2] the studentized range probability distribution function can be directly evaluated, where $\phi (z)$ is the standard normal probability density function.

f(q;k=2)={\sqrt {2}}\phi \left({\frac {q}{\sqrt {2}}}\right)

f(q;k=3)=6{\sqrt {2}}\phi \left({\frac {q}{\sqrt {2}}}\right)\left(\Phi \left({\frac {q}{\sqrt {6}}}\right)-{\frac {1}{2}}\right)

When the degrees of freedom approaches infinity the studentized range cumulative distribution can be calculated at all k using the standard normal distribution.

F(q;k)=k\int _{-\infty }^{\infty }\phi (z)[\Phi (z)-\Phi (z-q)]^{k-1}\,{\text{d}}z

How the studentized range distribution arises

For any probability distribution f, the range probability distribution is:^[2]

f(r;k)=k(k-1)\int _{-\infty }^{\infty }f\left(t+{\frac {1}{2}}r\right)f\left(t-{\frac {1}{2}}r\right)\left(\int _{t-{\frac {1}{2}}r}^{t+{\frac {1}{2}}r}f(x){\text{d}}x\right)^{k-2}\,{\text{d}}t

What this means, is that we are adding up the likelihood that, given k draws from a distribution, two of them differ by r, and the remaining k-2 draws all fall between the two extreme values. If we use u substitution where $u=t+{\frac {r}{2}}$ and define F as the cumulative distribution function of f, then the equation can be simplified.

f(r;k)=k(k-1)\int _{-\infty }^{\infty }f(u-r)f(u)(F(u)-F(u-r))^{k-2}\,{\text{d}}u

In order to create the studentized range distribution, we first use the standard normal distribution for f and F, and change the variable r to q.

f(q;k)=Sk(k-1)\int _{-\infty }^{\infty }\phi (u-Sq)\phi (u)\left[\Phi (u)-\Phi (u-Sq)\right]^{k-2}\,{\text{d}}u

The chi distribution is:

f(x;\nu )={\frac {2^{1-\nu /2}x^{\nu -1}e^{-x^{2}/2}}{\Gamma (\nu /2)}}

If we apply a change of variables we see it can also be expressed as:

f(S;\nu )={\frac {\nu ^{\frac {\nu }{2}}S^{\nu -1}e^{-{\frac {\nu S^{2}}{2}}}}{2^{\nu /2-1}\Gamma (\nu /2)}}

Multiplying the two and integrating over S gives:

f(q;k,\nu )={\frac {\nu ^{\frac {\nu }{2}}}{2^{\nu /2-1}\Gamma ({\frac {\nu }{2}})}}\int _{0}^{\infty }S^{\nu }e^{-{\frac {\nu S^{2}}{2}}}k(k-1)\int _{-\infty }^{\infty }\phi (u-Sq)\phi (u)\left[\Phi (u)-\Phi (u-Sq)\right]^{k-2}{\text{d}}u{\text{d}}S

Uses

Critical values of the studentized range distribution are used in Tukey's range test.

References

↑ Lund, R. E.; Lund, J. R. (1983). "Algorithm AS 190: Probabilities and Upper Quantiles for the Studentized Range". Journal of the Royal Statistical Society. 32 (2): 204–10. JSTOR 2347300.
1 2 A. T. McKay (1933). "A Note on the Distribution of Range in Samples of n". Biometrika. 25 (3): 415–20. JSTOR 2332292.

Pearson, E. S.; Hartley, H. O. (1942). "The Probability Integral of the Range in Samples of N Observations From a Normal Population.". Biometrika. 32 (3): 301–10. JSTOR 2332134.

Hartley, H. O. (1942). "The Range in Random Samples". Biometrika. 32 (3): 334–48. JSTOR 2332137.

Dunlap, W. P.; Powell, R. S.; Konnerth, T. K. (1977). "A FORTRAN IV function for calculating probabilities associated with the studentized range statistic". Behavior Research Methods & Instrumentation. 9 (4): 373–75.

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