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we have skew normal distribution with location=0, scale =1 and shape =0 then it is same as standard normal distribution with mean 0 and variance 1.but if we change the shape parameter say shape=5 then mean and variance also changes.how can we fix mean and variance with different values of shape parameter

Amber
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1 Answers1

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Just look after how the mean and variance of a skew normal distribution can be computed and you got the answer! Knowing that the mean looks like:

alt text    and    alt text

You can see, that with a xi=0 (location), omega=1 (scale) and alpha=0 (shape) you really get a standard normal distribution (with mean=0, standard deviation=1):

alt text

If you only change the alpha (shape) to 5, you can except the mean will differ a lot, and will be positive. If you want to hold the mean around zero with a higher alpha (shape), you will have to decrease other parameters, e.g.: the omega (scale). The most obvious solution could be to set it to zero instead of 1. See: alt text

Mean is set, we have to get a variance equal to zero with a omega set to zero and shape set to 5. The formula is known:

alt text

With our known parameters:

alt text

Which is insane :) That cannot be done this way. You may also go back and alter the value of xi instead of omega to get a mean equal to zero. But that way you might first compute the only possible value of omega with the formula of variance given.

alt text

Then the omega should be around 1.605681 (negative or positive).

Getting back to mean:

alt text

So, with the following parameters you should get a distribution you was intended to:

location = 1.256269 (negative or positive), scale = 1.605681 (negative or positive) and shape = 5.

Please, someone test it, as I might miscalculated somewhere with the given example.

daroczig
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    Ok, I could not help doing the tests also in R (with `sn` package), and looks like everything is just fine with the given parameters (location and scale should have opposite sign): mean=0, var=1 and looks skew (with a skewness around 0.8619845)! – daroczig Jan 11 '11 at 23:42
  • i did not understand the meaning of your last statement about test in R.can u elaborate it one more time?thanks – Amber Jan 12 '11 at 02:55
  • if we have 3 equation of mean,variance and skewness then how can we fix location, scale and shape parameter.can u explain about it? – Amber Jan 12 '11 at 03:33
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    @Amber: I did some tests to see if my computation was right, you have nothing to do with it. But about your other comment: you really should read about the subject, as I think there is no sense in speaking about a subject you are not familiar with. You might start with wikipedia: http://en.wikipedia.org/wiki/Skew_normal_distribution or with any basic statistics book. – daroczig Jan 12 '11 at 08:28
  • how did u calculate the skew parameter(with a skewness around 0.8619845)?because i am not getting the same as u got?i understand the article but i m not figuring out it? – Amber Jan 13 '11 at 05:58
  • @Amber: It was calculated on the test data, that is why I wrote "around". Computing by the formula (see on e.g. Wikipedia linked above), you should get a skewness ~0.850965. Another issu: as I think I gave a correct answer, are you willing to accept it, or do you see any problem with it? – daroczig Jan 13 '11 at 10:14
  • if i calculate the location ,scale and shape parameter in terms of mean ,variance and skewness then how can i get the skew normal distributed random numbers. – Amber Jan 17 '11 at 02:45
  • @Amber: Wikipedia has some idea about the theory of generating random numbers based on spec. distributions (http://en.wikipedia.org/wiki/Random_number_generation#Generation_from_a_probability_distribution), but I am sure Matlab would have a function making this exercise a lot simpler. Unfortunately I am not familiar with Matlab, though I could help you in R. – daroczig Jan 17 '11 at 13:02
  • actually in R if we use 'sn' package then we can generate the multivariate skew normal random variables but i am not much familiar with R.i have opened the installed this package but unfortunately i am not getting hoe to use 'rsn' and "mrsn" function, these are the functions for uni variate and multivariate skew normal distribution.when i type it on the command window it says me that i don't have these function so how to work with these? – Amber Jan 18 '11 at 10:37
  • `rsn(n=100, location=1.256269, scale=1.605681, shape=5)` will generate 100 random values from the desired distribution. Please open new questions for other topics in the future. – daroczig Jan 18 '11 at 15:34