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A study of daily default rates allows me to conclude that they are distributed as a normal one. Previously, I had to eliminate some of the effects of stationarity.

I have following two questions:

How can it be justified that if the daily default rate is normal, then the annual default rate will also follow a Normal distribution? or this assumption is not possible at all? Eliminating the stationarity effect implies the normal distribution assumption is not right?

Add:My daily default rate for a 3 year sample is normal, regardless of the summer months and weekends. My problem is that I do not know if it is possible to increase the temporality of the variable rate of default and assume that it is normal, because it is another variable.

kjetil b halvorsen
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Mark
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    1. A sum of marginally normal random variables is not necessarily normal; if that's what your question is, it's [a duplicate](https://stats.stackexchange.com/questions/162428/if-x-and-y-are-normally-distributed-random-variables-what-kind-of-distribut/162440#162440). A sum of *jointly* normal random variables will be normal, however. ... 2. What is the basis on which you assert that your *data* arise from a normal distribution? (i.e. how do you know you have normality?) ... 3. You should clarify what you did to your data and how that affects the variables you're asking about. ... ctd – Glen_b Apr 17 '17 at 23:43
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    ctd ... i.e. define the variables you're asking about(daily vs annual defaults) and the ones you have observations for and the transformed ones you created ... and explain how all those variables are related – Glen_b Apr 17 '17 at 23:43
  • My daily default rate for a 3 year sample is normal, regardless of the summer months and weekends. My problem is that I do not know if it is possible to increase the temporality of the variable rate of default and assume that it is normal, because it is another variable.Thank you !! – Mark Apr 24 '17 at 17:14

2 Answers2

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Sum of two normally distributed independent random variables is always normally distributed. I suggest you simulate some data and see for yourself. So if your daily values are normally distributed yearly values will be as well.

EDIT:

Let's use simulation to see this. We used two extremely separated normal distributions to prove the point:

a<-rnorm(n,0,1)

b<-rnorm(n,6,0.5)

hist(a+b)

When n tends to infinity, a+b tends to normality.

Community
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Vivaldi
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  • If the means for two normally distributed data sets are six standard deviations from each other, the resulting distribution will *not* be normal. The statement is only true if the means are centered near the population mean. – Tavrock Apr 18 '17 at 00:47
  • This answer as well as the comment by @Tavrock is incorrect. See [this question](https://stats.stackexchange.com/q/30159/6633) and its answers. – Dilip Sarwate Apr 18 '17 at 03:32
  • @DilipSarwate that question states adding two Gaussian distributions and returning with a *bivariate Gaussian* distribution. The question on this thread is asking, as far as I can discern, about adding multiple Gaussian distributions together and calling the answer a *univariate Gaussian* distribution. – Tavrock Apr 19 '17 at 00:56
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    @Tavrock you are wrong. Try simulating. – Vivaldi Apr 24 '17 at 09:52
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    @DilipSarwate he has daily deafult rates and want to conclude if yearly deafult rates are also normal. He is actually summing 365 normal distributions. – Vivaldi Apr 24 '17 at 09:52
  • @Tavrock as others already said, you are wrong, please see https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables – Tim Apr 24 '17 at 10:34
  • Vivaldi, you have edited your answer to include the word _independent_ in your description of the normal random variables being summed. My comment, and the link I pointed out applies to the _original_ version of your answer; the claim in the edited answer is correct, though unnecessarily restrictive. Also, the simulation is misleading: the sum of two independent normal random variables is _always_ normal, and one does not need to use "widely separated" distributions to "prove" this to those who do not believe anything until it is "verified" by simulation. – Dilip Sarwate Apr 24 '17 at 14:20
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It depends. For example Let's suppose you collect data on Monday and you run the descriptive stats for that day and found out the skewness and kurtosis. From that you arrived at a conclusion that data for Monday is normal.

Now on Tuesday you again collected data found out the skewness,kurtosis and concluded that data for Tuesday is normal.

Similarly you collected data for all seven days of a week and after collecting data for all seven days you again ran a test.To your surprise it was not normal

ALERT:The daily normal distribution is local whereas week distribution is a superposition of them.

In simpler words Mean,St.Dev on Monday can be 50 and 2,whereas on Tuesday it can be 100,2. Even if you superpose these two days data it is still not normal.

Aadish Chopra
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