On confidence interval estimation of normal percentiles

Abstract

Chakraborti and Li (Am Stat 61:331–336, 2007) considered several methods for confidence interval estimation of a normal distribution percentile. In this paper, we comment on two of their methods based on maximum likelihood and Bayesian inference. We show that the use of some approximations and simulations in the two methods is not necessary and some improvements and modifications can be made. Also, we compare the modified methods with methods in Chakraborti and Li (2007).

Appendices

Appendices

The explicit expression for the $I_{n}$ of $F_{T_{1}}(t)$ when $n=11$ is too long to be listed in the main part of the paper. It is given below. The same goes to the $I_{n}$ for $f_{T_{1}}(t)$, $P\left( \kappa _{\text {p}}\mid y\right)$, and $p\left( \kappa _{\text {p}}\mid y\right)$. Note that the $I_{n}$ of $P\left( \kappa _{\text {p}}\mid y\right)$ is the same as that of $F_{T_{1}}(t)$. The $I_{n}$ of $p\left( \kappa _{\text {p}}\mid y\right)$ is the same as that of $f_{T_{1}}(t)$. The results were generated by Matlab.

For $F_{T_{1}}(t)$ and $P\left( \kappa _{\text {p}}\mid y\right)$,

$$ I_{11}= {} 384\ \mathop {\mathrm {erf}}\nolimits \left( c\right) + \frac{384\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}}{\sqrt{d^2 + \frac{1}{2}}} + \frac{96\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}}{{\left( d^2 + \frac{1}{2}\right) }^{\frac{3}{2}}} + \frac{36\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}}{{\left( d^2 + \frac{1}{2}\right) }^{\frac{5}{2}}} + \frac{15\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}}{{\left( d^2 + \frac{1}{2}\right) }^{\frac{7}{2}}} + \frac{105\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}}{16\ {\left( d^2 + \frac{1}{2}\right) }^{\frac{9}{2}}} + \frac{192\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{\sqrt{d^2 + \frac{1}{2}}} + \frac{48\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{\sqrt{d^2 + \frac{1}{2}}} + \frac{48\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{{\left( d^2 + \frac{1}{2}\right) }^{\frac{3}{2}}} + \frac{12\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{{\left( d^2 + \frac{1}{2}\right) }^{\frac{3}{2}}} + \frac{18\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{{\left( d^2 + \frac{1}{2}\right) }^{\frac{5}{2}}} $$

$$+ \frac{9\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{2\ {\left( d^2 + \frac{1}{2}\right) }^{\frac{5}{2}}} + \frac{15\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{2\ {\left( d^2 + \frac{1}{2}\right) }^{\frac{7}{2}}} + \frac{8\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{6\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^3} - \frac{3\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^4}\right) }{\sqrt{d^2 + \frac{1}{2}}} + \frac{2\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{6\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^3} - \frac{3\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^4}\right) }{{\left( d^2 + \frac{1}{2}\right) }^{\frac{3}{2}}} + \frac{d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{24\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^4} - \frac{12\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^5}\right) }{\sqrt{d^2 + \frac{1}{2}}} + \frac{48\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^2}{\sqrt{d^2 + \frac{1}{2}}} + \frac{8\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^3}{\sqrt{d^2 + \frac{1}{2}}} + \frac{d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^4}{\sqrt{d^2 + \frac{1}{2}}} + \frac{3\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ {\left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }^2}{\sqrt{d^2 + \frac{1}{2}}} + \frac{12\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^2}{{\left( d^2 + \frac{1}{2}\right) }^{\frac{3}{2}}} + \frac{2\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^3}{{\left( d^2 + \frac{1}{2}\right) }^{\frac{3}{2}}} + \frac{9\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^{\frac{5}{2}}} + \frac{6\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^2\ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{\sqrt{d^2 + \frac{1}{2}}}- \frac{192\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^2} - \frac{120\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^3} - \frac{66\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^4} - \frac{279\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{8\ \sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^5} + \frac{24\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) \ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{\sqrt{d^2 + \frac{1}{2}}} + \frac{6\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) \ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{{\left( d^2 + \frac{1}{2}\right) }^{\frac{3}{2}}} $$

$$+ \frac{4\ d\ {\text {erfc}} \left( \frac{c\ d}{\sqrt{d^2 + \frac{1}{2}}}\right) \ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) \ \left( \frac{6\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^3} - \frac{3\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^4}\right) }{\sqrt{d^2 + \frac{1}{2}}} + \frac{48\ c^3\ d^4\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^4} + \frac{52\ c^3\ d^4\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^5} + \frac{163\ c^3\ d^4\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{4\ \sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^6} - \frac{8\ c^5\ d^6\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^6} - \frac{25\ c^5\ d^6\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{2\ \sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^7} + \frac{c^7\ d^8\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^8} - \frac{96\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^2}- \frac{60\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^3} - \frac{24\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^2} - \frac{33\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^4} - \frac{15\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^3}- \frac{4\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{6\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^3} - \frac{3\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^4}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^2} - \frac{24\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^2}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^2}- \frac{4\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^3}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^2} - \frac{15\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^2}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^3} + \frac{24\ c^3\ d^4\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^4} + \frac{26\ c^3\ d^4\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^5} + \frac{6\ c^3\ d^4\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^4} - \frac{4\ c^5\ d^6\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^6} + \frac{6\ c^3\ d^4\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ {\left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) }^2}{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^4} - \frac{12\ c\ d^2\ \mathrm {e}^{-\frac{c^2}{2\ \left( d^2 + \frac{1}{2}\right) }}\ \mathrm {e}^{-\frac{c^2\ d^2}{d^2 + \frac{1}{2}}}\ \left( \frac{c^2}{d^2 + \frac{1}{2}} - \frac{c^2}{2\ {\left( d^2 + \frac{1}{2}\right) }^2}\right) \ \left( \frac{2\ c^2}{{\left( d^2 + \frac{1}{2}\right) }^2} - \frac{c^2}{{\left( d^2 + \frac{1}{2}\right) }^3}\right) }{\sqrt{\pi }\ {\left( d^2 + \frac{1}{2}\right) }^2}.$$

For $f_{T_{1}}(t)$ and $p\left( \kappa _{\text {p}}\mid y \right)$,

$$\begin{aligned} I_{11}= & {} \frac{3991211251234741\ }{4503599627370496\ \sqrt{d}} \left( \frac{945\ \mathrm {e}^{\frac{c^2}{4\ d}}\ {\text {erfc}} \left( \frac{c}{2\ \sqrt{d}}\right) }{32\ d^5} - \frac{2895\ c}{32\ \sqrt{\pi }\ d^{\frac{11}{2}}} - \frac{165\ c^3}{4\ \sqrt{\pi }\ d^{\frac{13}{2}}} \right. \\&\left. - \frac{147\ c^5}{32\ \sqrt{\pi }\ d^{\frac{15}{2}}} - \frac{11\ c^7}{64\ \sqrt{\pi }\ d^{\frac{17}{2}}} - \frac{c^9}{512\ \sqrt{\pi }\ d^{\frac{19}{2}}}\right) \\&+ \frac{3991211251234741\ }{4503599627370496\ \sqrt{d}}\left( \frac{4725\ c^2\ \mathrm {e}^{\frac{c^2}{4\ d}}\ {\text {erfc}} \left( \frac{c}{2\ \sqrt{d}}\right) }{64\ d^6} + \frac{1575\ c^4\ \mathrm {e}^{\frac{c^2}{4\ d}}\ {\text {erfc}} \left( \frac{c}{2\ \sqrt{d}}\right) }{64\ d^7} \right. \\&\left. + \frac{315\ c^6\ \mathrm {e}^{\frac{c^2}{4\ d}}\ {\text {erfc}} \left( \frac{c}{2\ \sqrt{d}}\right) }{128\ d^8} + \frac{45\ c^8\ \mathrm {e}^{\frac{c^2}{4\ d}}\ {\text {erfc}} \left( \frac{c}{2\ \sqrt{d}}\right) }{512\ d^9} \right) \\&+ \frac{3991211251234741\ }{4503599627370496\ \sqrt{d}}\left( \frac{c^{10}\ \mathrm {e}^{\frac{c^2}{4\ d}}\ {\text {erfc}} \left( \frac{c}{2\ \sqrt{d}}\right) }{1024\ d^{10}}\right) . \end{aligned}$$