How to find which percentile is a value in a skewed normal distribution












0












$begingroup$


I have a skewed normal distribution for which I know the average, standard deviation, skewness & kurtosis (which is different from zero).



Given a number $X,$ how can estimate which percentile corresponds to that value? (I'm ok with getting an approximate value of this percentile.)



I used z-score tables in the past (before having skewed distributions), but they seem to apply only to non-skewed distributions.



Thanks for your help.






statistics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Assuming you mean a skewed normal distribution (as defined at en.wikipedia.org/wiki/Skew_normal_distribution) you could use the method of moments to estimate the parameters of the skewed normal distribution and then estimate any desired function of that distribution. The problem is that you won't know how good your estimate of any percentile will be unless maybe you also have the sample size.
    $endgroup$
    – JimB
    Jan 7 at 15:03












  • $begingroup$
    I have the sample size
    $endgroup$
    – Nisalon
    Jan 7 at 16:16










  • $begingroup$
    If you have the sample size, you could sample from the estimated distribution to get some idea as to how good the parameter estimates might be.
    $endgroup$
    – JimB
    Jan 7 at 16:59
















0












$begingroup$


I have a skewed normal distribution for which I know the average, standard deviation, skewness & kurtosis (which is different from zero).



Given a number $X,$ how can estimate which percentile corresponds to that value? (I'm ok with getting an approximate value of this percentile.)



I used z-score tables in the past (before having skewed distributions), but they seem to apply only to non-skewed distributions.



Thanks for your help.






statistics







share|cite|improve this question











$endgroup$












  • $begingroup$
    Assuming you mean a skewed normal distribution (as defined at en.wikipedia.org/wiki/Skew_normal_distribution) you could use the method of moments to estimate the parameters of the skewed normal distribution and then estimate any desired function of that distribution. The problem is that you won't know how good your estimate of any percentile will be unless maybe you also have the sample size.
    $endgroup$
    – JimB
    Jan 7 at 15:03












  • $begingroup$
    I have the sample size
    $endgroup$
    – Nisalon
    Jan 7 at 16:16










  • $begingroup$
    If you have the sample size, you could sample from the estimated distribution to get some idea as to how good the parameter estimates might be.
    $endgroup$
    – JimB
    Jan 7 at 16:59














0












0








0





$begingroup$


I have a skewed normal distribution for which I know the average, standard deviation, skewness & kurtosis (which is different from zero).



Given a number $X,$ how can estimate which percentile corresponds to that value? (I'm ok with getting an approximate value of this percentile.)



I used z-score tables in the past (before having skewed distributions), but they seem to apply only to non-skewed distributions.



Thanks for your help.






statistics







share|cite|improve this question











$endgroup$




I have a skewed normal distribution for which I know the average, standard deviation, skewness & kurtosis (which is different from zero).



Given a number $X,$ how can estimate which percentile corresponds to that value? (I'm ok with getting an approximate value of this percentile.)



I used z-score tables in the past (before having skewed distributions), but they seem to apply only to non-skewed distributions.



Thanks for your help.






statistics




statistics






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 7 at 14:57









amWhy

192k28225439




192k28225439










asked Jan 7 at 13:37









NisalonNisalon

162




162












  • $begingroup$
    Assuming you mean a skewed normal distribution (as defined at en.wikipedia.org/wiki/Skew_normal_distribution) you could use the method of moments to estimate the parameters of the skewed normal distribution and then estimate any desired function of that distribution. The problem is that you won't know how good your estimate of any percentile will be unless maybe you also have the sample size.
    $endgroup$
    – JimB
    Jan 7 at 15:03












  • $begingroup$
    I have the sample size
    $endgroup$
    – Nisalon
    Jan 7 at 16:16










  • $begingroup$
    If you have the sample size, you could sample from the estimated distribution to get some idea as to how good the parameter estimates might be.
    $endgroup$
    – JimB
    Jan 7 at 16:59


















  • $begingroup$
    Assuming you mean a skewed normal distribution (as defined at en.wikipedia.org/wiki/Skew_normal_distribution) you could use the method of moments to estimate the parameters of the skewed normal distribution and then estimate any desired function of that distribution. The problem is that you won't know how good your estimate of any percentile will be unless maybe you also have the sample size.
    $endgroup$
    – JimB
    Jan 7 at 15:03












  • $begingroup$
    I have the sample size
    $endgroup$
    – Nisalon
    Jan 7 at 16:16










  • $begingroup$
    If you have the sample size, you could sample from the estimated distribution to get some idea as to how good the parameter estimates might be.
    $endgroup$
    – JimB
    Jan 7 at 16:59
















$begingroup$
Assuming you mean a skewed normal distribution (as defined at en.wikipedia.org/wiki/Skew_normal_distribution) you could use the method of moments to estimate the parameters of the skewed normal distribution and then estimate any desired function of that distribution. The problem is that you won't know how good your estimate of any percentile will be unless maybe you also have the sample size.
$endgroup$
– JimB
Jan 7 at 15:03






$begingroup$
Assuming you mean a skewed normal distribution (as defined at en.wikipedia.org/wiki/Skew_normal_distribution) you could use the method of moments to estimate the parameters of the skewed normal distribution and then estimate any desired function of that distribution. The problem is that you won't know how good your estimate of any percentile will be unless maybe you also have the sample size.
$endgroup$
– JimB
Jan 7 at 15:03














$begingroup$
I have the sample size
$endgroup$
– Nisalon
Jan 7 at 16:16




$begingroup$
I have the sample size
$endgroup$
– Nisalon
Jan 7 at 16:16












$begingroup$
If you have the sample size, you could sample from the estimated distribution to get some idea as to how good the parameter estimates might be.
$endgroup$
– JimB
Jan 7 at 16:59




$begingroup$
If you have the sample size, you could sample from the estimated distribution to get some idea as to how good the parameter estimates might be.
$endgroup$
– JimB
Jan 7 at 16:59










1 Answer
1






active

oldest

votes


















0












$begingroup$

The Wiki page Skew normal distribution provides the information to estimate the parameters using the sample mean ($bar{x}$), standard deviation ($s$), and skewness ($hat{gamma}$). The 3 parameters to be estimated are $mu$, $sigma$, and $alpha$.



If $|hat{gamma}|<1$, then $hat{alpha}$ is found in two steps:



$$delta =sqrt{frac{pi left| hat{gamma} right| ^{2/3}}{2 left(left| hat{gamma} right| ^{2/3}+left(frac{4-pi }{2}right)^{2/3}right)}}$$



$$hat{alpha} = text{sgn}(hat{gamma})sqrt{frac{delta }{1-delta ^2}}$$



Otherwise $hat{alpha}$ is the solution to



$$hat{gamma} =frac{sqrt{2} (4-pi ) hat{alpha} ^3}{left((pi -2) hat{alpha} ^2+pi right)^{3/2}}$$



which needs to be performed numerically. Then $hat{mu}$ and $hat{sigma}$ are



$$hat{sigma} =frac{s}{sqrt{1-frac{2 hat{alpha} ^2}{pi left(hat{alpha} ^2+1right)}}}$$



$$hat{mu} =bar{x}-frac{sqrt{frac{2}{pi }} hat{alpha} hat{sigma} }{sqrt{hat{alpha} ^2+1}}$$



Now armed with the parameter estimates, then one can estimate the cumulative distribution function:



$$Pr(X le x)=Phileft(frac{x, -hat{mu} }{hat{sigma} }right)-2 Tleft(frac{x, -hat{mu} }{hat{sigma} },hat{alpha} right)$$



where $T$ is the Owen's T function: $T(x,a)=frac{int_0^a frac{exp left(-left(left(t^2+1right) x^2right)right)}{2 left(t^2+1right)} , dt}{2 pi }$ .



Here is an implementation using Mathematica:



Mathematica code



I know code should be given as text but in this case because it is unlikely that you have Mathematica (and it would look much messier as text), it should be instructive as to the process.



To estimate the percentage of the distribution no larger than a specified value you'll need to use the cumulative distribution function (CDF) described on the Wiki page. Using Mathematica for values of $X$ being 7.5 and 5.2:



Example of percentiles



If you have access to the statistical package R, then the sn package will perform these calculations.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Sorry, I'm not sure you actually answer my question : how can I get the percentile corresponding to one given number ?
    $endgroup$
    – Nisalon
    Jan 7 at 17:15










  • $begingroup$
    I've added the Mathematica code for that and mentioned the sn package in R.
    $endgroup$
    – JimB
    Jan 7 at 17:57










  • $begingroup$
    I don't want code (as this code will be integrated in PHP), I want to understand the process to be able to implement it
    $endgroup$
    – Nisalon
    Jan 7 at 20:40










  • $begingroup$
    Added a description of the process.
    $endgroup$
    – JimB
    Jan 7 at 21:39











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1 Answer
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1 Answer
1






active

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active

oldest

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active

oldest

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0












$begingroup$

The Wiki page Skew normal distribution provides the information to estimate the parameters using the sample mean ($bar{x}$), standard deviation ($s$), and skewness ($hat{gamma}$). The 3 parameters to be estimated are $mu$, $sigma$, and $alpha$.



If $|hat{gamma}|<1$, then $hat{alpha}$ is found in two steps:



$$delta =sqrt{frac{pi left| hat{gamma} right| ^{2/3}}{2 left(left| hat{gamma} right| ^{2/3}+left(frac{4-pi }{2}right)^{2/3}right)}}$$



$$hat{alpha} = text{sgn}(hat{gamma})sqrt{frac{delta }{1-delta ^2}}$$



Otherwise $hat{alpha}$ is the solution to



$$hat{gamma} =frac{sqrt{2} (4-pi ) hat{alpha} ^3}{left((pi -2) hat{alpha} ^2+pi right)^{3/2}}$$



which needs to be performed numerically. Then $hat{mu}$ and $hat{sigma}$ are



$$hat{sigma} =frac{s}{sqrt{1-frac{2 hat{alpha} ^2}{pi left(hat{alpha} ^2+1right)}}}$$



$$hat{mu} =bar{x}-frac{sqrt{frac{2}{pi }} hat{alpha} hat{sigma} }{sqrt{hat{alpha} ^2+1}}$$



Now armed with the parameter estimates, then one can estimate the cumulative distribution function:



$$Pr(X le x)=Phileft(frac{x, -hat{mu} }{hat{sigma} }right)-2 Tleft(frac{x, -hat{mu} }{hat{sigma} },hat{alpha} right)$$



where $T$ is the Owen's T function: $T(x,a)=frac{int_0^a frac{exp left(-left(left(t^2+1right) x^2right)right)}{2 left(t^2+1right)} , dt}{2 pi }$ .



Here is an implementation using Mathematica:



Mathematica code



I know code should be given as text but in this case because it is unlikely that you have Mathematica (and it would look much messier as text), it should be instructive as to the process.



To estimate the percentage of the distribution no larger than a specified value you'll need to use the cumulative distribution function (CDF) described on the Wiki page. Using Mathematica for values of $X$ being 7.5 and 5.2:



Example of percentiles



If you have access to the statistical package R, then the sn package will perform these calculations.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Sorry, I'm not sure you actually answer my question : how can I get the percentile corresponding to one given number ?
    $endgroup$
    – Nisalon
    Jan 7 at 17:15










  • $begingroup$
    I've added the Mathematica code for that and mentioned the sn package in R.
    $endgroup$
    – JimB
    Jan 7 at 17:57










  • $begingroup$
    I don't want code (as this code will be integrated in PHP), I want to understand the process to be able to implement it
    $endgroup$
    – Nisalon
    Jan 7 at 20:40










  • $begingroup$
    Added a description of the process.
    $endgroup$
    – JimB
    Jan 7 at 21:39
















0












$begingroup$

The Wiki page Skew normal distribution provides the information to estimate the parameters using the sample mean ($bar{x}$), standard deviation ($s$), and skewness ($hat{gamma}$). The 3 parameters to be estimated are $mu$, $sigma$, and $alpha$.



If $|hat{gamma}|<1$, then $hat{alpha}$ is found in two steps:



$$delta =sqrt{frac{pi left| hat{gamma} right| ^{2/3}}{2 left(left| hat{gamma} right| ^{2/3}+left(frac{4-pi }{2}right)^{2/3}right)}}$$



$$hat{alpha} = text{sgn}(hat{gamma})sqrt{frac{delta }{1-delta ^2}}$$



Otherwise $hat{alpha}$ is the solution to



$$hat{gamma} =frac{sqrt{2} (4-pi ) hat{alpha} ^3}{left((pi -2) hat{alpha} ^2+pi right)^{3/2}}$$



which needs to be performed numerically. Then $hat{mu}$ and $hat{sigma}$ are



$$hat{sigma} =frac{s}{sqrt{1-frac{2 hat{alpha} ^2}{pi left(hat{alpha} ^2+1right)}}}$$



$$hat{mu} =bar{x}-frac{sqrt{frac{2}{pi }} hat{alpha} hat{sigma} }{sqrt{hat{alpha} ^2+1}}$$



Now armed with the parameter estimates, then one can estimate the cumulative distribution function:



$$Pr(X le x)=Phileft(frac{x, -hat{mu} }{hat{sigma} }right)-2 Tleft(frac{x, -hat{mu} }{hat{sigma} },hat{alpha} right)$$



where $T$ is the Owen's T function: $T(x,a)=frac{int_0^a frac{exp left(-left(left(t^2+1right) x^2right)right)}{2 left(t^2+1right)} , dt}{2 pi }$ .



Here is an implementation using Mathematica:



Mathematica code



I know code should be given as text but in this case because it is unlikely that you have Mathematica (and it would look much messier as text), it should be instructive as to the process.



To estimate the percentage of the distribution no larger than a specified value you'll need to use the cumulative distribution function (CDF) described on the Wiki page. Using Mathematica for values of $X$ being 7.5 and 5.2:



Example of percentiles



If you have access to the statistical package R, then the sn package will perform these calculations.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Sorry, I'm not sure you actually answer my question : how can I get the percentile corresponding to one given number ?
    $endgroup$
    – Nisalon
    Jan 7 at 17:15










  • $begingroup$
    I've added the Mathematica code for that and mentioned the sn package in R.
    $endgroup$
    – JimB
    Jan 7 at 17:57










  • $begingroup$
    I don't want code (as this code will be integrated in PHP), I want to understand the process to be able to implement it
    $endgroup$
    – Nisalon
    Jan 7 at 20:40










  • $begingroup$
    Added a description of the process.
    $endgroup$
    – JimB
    Jan 7 at 21:39














0












0








0





$begingroup$

The Wiki page Skew normal distribution provides the information to estimate the parameters using the sample mean ($bar{x}$), standard deviation ($s$), and skewness ($hat{gamma}$). The 3 parameters to be estimated are $mu$, $sigma$, and $alpha$.



If $|hat{gamma}|<1$, then $hat{alpha}$ is found in two steps:



$$delta =sqrt{frac{pi left| hat{gamma} right| ^{2/3}}{2 left(left| hat{gamma} right| ^{2/3}+left(frac{4-pi }{2}right)^{2/3}right)}}$$



$$hat{alpha} = text{sgn}(hat{gamma})sqrt{frac{delta }{1-delta ^2}}$$



Otherwise $hat{alpha}$ is the solution to



$$hat{gamma} =frac{sqrt{2} (4-pi ) hat{alpha} ^3}{left((pi -2) hat{alpha} ^2+pi right)^{3/2}}$$



which needs to be performed numerically. Then $hat{mu}$ and $hat{sigma}$ are



$$hat{sigma} =frac{s}{sqrt{1-frac{2 hat{alpha} ^2}{pi left(hat{alpha} ^2+1right)}}}$$



$$hat{mu} =bar{x}-frac{sqrt{frac{2}{pi }} hat{alpha} hat{sigma} }{sqrt{hat{alpha} ^2+1}}$$



Now armed with the parameter estimates, then one can estimate the cumulative distribution function:



$$Pr(X le x)=Phileft(frac{x, -hat{mu} }{hat{sigma} }right)-2 Tleft(frac{x, -hat{mu} }{hat{sigma} },hat{alpha} right)$$



where $T$ is the Owen's T function: $T(x,a)=frac{int_0^a frac{exp left(-left(left(t^2+1right) x^2right)right)}{2 left(t^2+1right)} , dt}{2 pi }$ .



Here is an implementation using Mathematica:



Mathematica code



I know code should be given as text but in this case because it is unlikely that you have Mathematica (and it would look much messier as text), it should be instructive as to the process.



To estimate the percentage of the distribution no larger than a specified value you'll need to use the cumulative distribution function (CDF) described on the Wiki page. Using Mathematica for values of $X$ being 7.5 and 5.2:



Example of percentiles



If you have access to the statistical package R, then the sn package will perform these calculations.






share|cite|improve this answer











$endgroup$



The Wiki page Skew normal distribution provides the information to estimate the parameters using the sample mean ($bar{x}$), standard deviation ($s$), and skewness ($hat{gamma}$). The 3 parameters to be estimated are $mu$, $sigma$, and $alpha$.



If $|hat{gamma}|<1$, then $hat{alpha}$ is found in two steps:



$$delta =sqrt{frac{pi left| hat{gamma} right| ^{2/3}}{2 left(left| hat{gamma} right| ^{2/3}+left(frac{4-pi }{2}right)^{2/3}right)}}$$



$$hat{alpha} = text{sgn}(hat{gamma})sqrt{frac{delta }{1-delta ^2}}$$



Otherwise $hat{alpha}$ is the solution to



$$hat{gamma} =frac{sqrt{2} (4-pi ) hat{alpha} ^3}{left((pi -2) hat{alpha} ^2+pi right)^{3/2}}$$



which needs to be performed numerically. Then $hat{mu}$ and $hat{sigma}$ are



$$hat{sigma} =frac{s}{sqrt{1-frac{2 hat{alpha} ^2}{pi left(hat{alpha} ^2+1right)}}}$$



$$hat{mu} =bar{x}-frac{sqrt{frac{2}{pi }} hat{alpha} hat{sigma} }{sqrt{hat{alpha} ^2+1}}$$



Now armed with the parameter estimates, then one can estimate the cumulative distribution function:



$$Pr(X le x)=Phileft(frac{x, -hat{mu} }{hat{sigma} }right)-2 Tleft(frac{x, -hat{mu} }{hat{sigma} },hat{alpha} right)$$



where $T$ is the Owen's T function: $T(x,a)=frac{int_0^a frac{exp left(-left(left(t^2+1right) x^2right)right)}{2 left(t^2+1right)} , dt}{2 pi }$ .



Here is an implementation using Mathematica:



Mathematica code



I know code should be given as text but in this case because it is unlikely that you have Mathematica (and it would look much messier as text), it should be instructive as to the process.



To estimate the percentage of the distribution no larger than a specified value you'll need to use the cumulative distribution function (CDF) described on the Wiki page. Using Mathematica for values of $X$ being 7.5 and 5.2:



Example of percentiles



If you have access to the statistical package R, then the sn package will perform these calculations.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 7 at 21:39

























answered Jan 7 at 16:57









JimBJimB

47537




47537












  • $begingroup$
    Sorry, I'm not sure you actually answer my question : how can I get the percentile corresponding to one given number ?
    $endgroup$
    – Nisalon
    Jan 7 at 17:15










  • $begingroup$
    I've added the Mathematica code for that and mentioned the sn package in R.
    $endgroup$
    – JimB
    Jan 7 at 17:57










  • $begingroup$
    I don't want code (as this code will be integrated in PHP), I want to understand the process to be able to implement it
    $endgroup$
    – Nisalon
    Jan 7 at 20:40










  • $begingroup$
    Added a description of the process.
    $endgroup$
    – JimB
    Jan 7 at 21:39


















  • $begingroup$
    Sorry, I'm not sure you actually answer my question : how can I get the percentile corresponding to one given number ?
    $endgroup$
    – Nisalon
    Jan 7 at 17:15










  • $begingroup$
    I've added the Mathematica code for that and mentioned the sn package in R.
    $endgroup$
    – JimB
    Jan 7 at 17:57










  • $begingroup$
    I don't want code (as this code will be integrated in PHP), I want to understand the process to be able to implement it
    $endgroup$
    – Nisalon
    Jan 7 at 20:40










  • $begingroup$
    Added a description of the process.
    $endgroup$
    – JimB
    Jan 7 at 21:39
















$begingroup$
Sorry, I'm not sure you actually answer my question : how can I get the percentile corresponding to one given number ?
$endgroup$
– Nisalon
Jan 7 at 17:15




$begingroup$
Sorry, I'm not sure you actually answer my question : how can I get the percentile corresponding to one given number ?
$endgroup$
– Nisalon
Jan 7 at 17:15












$begingroup$
I've added the Mathematica code for that and mentioned the sn package in R.
$endgroup$
– JimB
Jan 7 at 17:57




$begingroup$
I've added the Mathematica code for that and mentioned the sn package in R.
$endgroup$
– JimB
Jan 7 at 17:57












$begingroup$
I don't want code (as this code will be integrated in PHP), I want to understand the process to be able to implement it
$endgroup$
– Nisalon
Jan 7 at 20:40




$begingroup$
I don't want code (as this code will be integrated in PHP), I want to understand the process to be able to implement it
$endgroup$
– Nisalon
Jan 7 at 20:40












$begingroup$
Added a description of the process.
$endgroup$
– JimB
Jan 7 at 21:39




$begingroup$
Added a description of the process.
$endgroup$
– JimB
Jan 7 at 21:39


















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The Binding of Isaac: Rebirth/Afterbirth

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Please consider supporting The Cutting Room Floor on Patreon. Thanks for all your support! The Binding of Isaac: Rebirth/Afterbirth From The Cutting Room Floor < The Binding of Isaac: Rebirth Jump to: navigation, search This is a sub-page of The Binding of Isaac: Rebirth . Afterbirth is the first expansion pack for Rebirth and features a variety of improvements. Contents 1 Unused Graphics 1.1 Old Man Keeper 1.2 Skinless Hush 1.3 Miscellaneous Unused Hush Sprites 1.4 Generosity achievement 1.5 Afterbirth+ Room Editor 1.6 Bearded Ultra Greed 1.7 Early Ultra Greed 1.8 Unused Brownie Expression 1.9 Frail's Spinning Head 1.10 Mom's Dead Hand Alt 2 Unused Audio 2.1 Unused Dead Womb Layer 2.2 Unused Hush Battle Outro 2.3 The Stain Grunt Unused Graphics Old Man Keeper Image file: gfxuimain me...
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What does “Dominus providebit” mean?

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5 Does "Dominus providebit" mean "The Lord will provide" or "The Lord provides"? I once had a bit of an argument with a guy who studied Latin over that. It's an inscription that appears on the rim of some old Swiss coins that I have. meaning latin-to-english-translation grammar-identification share | improve this question edited Jan 18 at 15:27 V2Blast 105 4 asked Jan 17 at 22:49 Donald K. Donald K. 26 1 ...
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