naginterfaces.library.stat.inv_​cdf_​students_​t

naginterfaces.library.stat.inv_cdf_students_t(p, df, tail='L')

inv_cdf_students_t returns the deviate associated with the given tail probability of Student’s $t$-distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01fb

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01fbf.html

Parameters

pfloat

$p$, the probability from the required Student’s $t$-distribution as defined by $tail$.

dffloat

$\nu$, the degrees of freedom of the Student’s $t$-distribution.

tailstr, length 1, optional

Indicates which tail the supplied probability represents.

$tail=\text{'U'}$

The upper tail probability, i.e., $P(T\ge t_p:\nu )$.

$tail=\text{'L'}$

The lower tail probability, i.e., $P(T\le t_p:\nu )$.

$tail=\text{'S'}$

The two tail (significance level) probability, i.e., $P(T\ge \left|t_p\right|:\nu )+P(T\le -\left|t_p\right|:\nu )$.

$tail=\text{'C'}$

The two tail (confidence interval) probability, i.e., $P(T\le \left|t_p\right|:\nu )-P(T\le -\left|t_p\right|:\nu )$.

Returns

xfloat

The deviate associated with the given tail probability of Student’s $t$-distribution with real degrees of freedom.

Raises

NagValueError

(errno $1$)

On entry, $tail=⟨value⟩$.

Constraint: $tail=\text{'L'}$, $\text{'U'}$, $\text{'S'}$ or $\text{'C'}$.

(errno $2$)

On entry, $p=⟨value⟩$.

Constraint: $p<1.0$.

(errno $2$)

On entry, $p=⟨value⟩$.

Constraint: $p>0.0$.

(errno $3$)

On entry, $df=⟨value⟩$.

Constraint: $df\ge 1.0$.

Warns

NagAlgorithmicWarning

(errno $5$)

The solution has failed to converge. However, the result should be a reasonable approximation.

Notes

The deviate, $t_p$ associated with the lower tail probability, $p$, of the Student’s $t$-distribution with $\nu$ degrees of freedom is defined as the solution to

$$P(T<t_p:\nu )=p=\frac{\Gamma \left((\nu +1)/2\right)}{\sqrt{\nu \pi }\Gamma \left(\nu /2\right)}{\int }_{-\infty }^{t_p}{(1+\frac{T^2}{\nu })}^{-(\nu +1)/2}dT\text{,}\nu \ge 1\text{;}-\infty <t_p<\infty \text{.}$$

For $\nu =1$ or $2$ the integral equation is easily solved for $t_p$.

For other values of $\nu <3$ a transformation to the beta distribution is used and the result obtained from inv_cdf_beta().

For $\nu \ge 3$ an inverse asymptotic expansion of Cornish–Fisher type is used. The algorithm is described by Hill (1970).

References

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth

Hill, G W, 1970, Student’s $t$ -distribution, Comm. ACM (13(10)), 617–619