naginterfaces.library.stat.inv_​cdf_​students_​t_​vector

naginterfaces.library.stat.inv_cdf_students_t_vector(tail, p, df)

inv_cdf_students_t_vector returns a number of deviates associated with given probabilities of Student’s $t$-distribution with real degrees of freedom.

For full information please refer to the NAG Library document for g01tb

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01tbf.html

Parameters

tailstr, length 1, array-like, shape $\left(\text{ltail}\right)$

Indicates which tail the supplied probabilities represent. For $j=\left(mod(\text{i}-1,\text{ltail})\right)+1-1$, for $\text{i}=1,2,\dots ,max(\text{ltail},\text{lp},\text{ldf})$:

$tail\left[j\right]=\text{'L'}$

The lower tail probability, i.e., $p_i=(T_i\le t_{p_i}:{\nu }_i)$.

$tail\left[j\right]=\text{'U'}$

The upper tail probability, i.e., $p_i=(T_i\ge t_{p_i}:{\nu }_i)$.

$tail\left[j\right]=\text{'C'}$

The two tail (confidence interval) probability, i.e., $p_i=(T_i\le \left|t_{p_i}\right|:{\nu }_i)-(T_i\le -\left|t_{p_i}\right|:{\nu }_i)$.

$tail\left[j\right]=\text{'S'}$

The two tail (significance level) probability, i.e., $p_i=(T_i\ge \left|t_{p_i}\right|:{\nu }_i)+(T_i\le -\left|t_{p_i}\right|:{\nu }_i)$.

pfloat, array-like, shape $\left(\text{lp}\right)$

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

dffloat, array-like, shape $\left(\text{ldf}\right)$

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

Returns

tfloat, ndarray, shape $(max(\text{ltail},\text{lp}))$

$t_{p_i}$, the deviates for the Student’s $t$-distribution.

ivalidint, ndarray, shape $(max(\text{ltail},\text{lp}))$

$ivalid[i-1]$ indicates any errors with the input arguments, with

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

On entry, invalid value supplied in $tail$ when calculating $t_{p_i}$.

$ivalid[i-1]=2$

On entry, $p_i\le 0.0$, or, $p_i\ge 1.0$.

$ivalid[i-1]=3$

On entry, ${\nu }_i<1.0$.

$ivalid[i-1]=4$

The solution has failed to converge. The result returned should represent an approximation to the solution.

Raises

NagValueError

(errno $2$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ltail}>0$.

(errno $3$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{lp}>0$.

(errno $4$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ldf}>0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $tail$, $p$ or $df$ was invalid, or the solution failed to converge.

Check $ivalid$ for more information.

Notes

The deviate, $t_{p_i}$ associated with the lower tail probability, $p_i$, of the Student’s $t$-distribution with ${\nu }_i$ degrees of freedom is defined as the solution to

$$(T_i<t_{p_i}:{\nu }_i)=p_i=\frac{\Gamma \left(({\nu }_i+1)/2\right)}{\sqrt{{\nu }_i\pi }\Gamma \left({\nu }_i/2\right)}{\int }_{-\infty }^{t_{p_i}}{(1+\frac{T_i^2}{{\nu }_i})}^{-({\nu }_i+1)/2}d{T}_i\text{,}{\nu }_i\ge 1\text{;}-\infty <t_{p_i}<\infty \text{.}$$

For ${\nu }_i=1$ or $2$ the integral equation is easily solved for $t_{p_i}$.

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

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

The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See the G01 Introduction for further information.

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