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Let say, I have 2 continuous random variables X1 & X2. Both have same location parameters. Other parameters may be same or may not.

Now say, the q1-th quantile of X1 is less than the q1-th quantile of x2. But the q2-th quantile of x1 is more than the q2th quantile of x2.

My question is, is that possible? Is there any example of x1 & x2 which have that property?

I will be really grateful if someone can give me some pointer.

Edit: At this point, I realize the question I asked for was not correctly specified.

I'm particularly interested in the case where the two quantiles being considered are on the same side of the location parameter.

Glen_b
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Ron
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    Two normal distributions centered around 0, one with larger variance (compare 0.1 and 0.9 quantile)? – Michael M Oct 13 '13 at 20:02
  • I am not sure if this can happen. Quantiles for Normal distributions are basically the scaled up version of the corresponding SD. Therefore if for the 1st Normal, 0.1 quantile is higher then 0.9 quantile will also be higher. What I am looking for is opposite. I am looking for this scenario: 0.1 quantile is higher for 1st r.v. than 2nd r.v. but, 0.9 quantile is lower for 1st than 2nd. – Ron Oct 13 '13 at 20:26
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    @Michael Mayer is correct. It might help to draw a picture of the two CDFs overlaid on one another. – whuber Oct 13 '13 at 21:03

1 Answers1

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Since you appear to doubt the example offered, I have included a diagram. As Michael Mayer said, two normal distributions centered around 0, one with larger variance, is sufficient.

In the diagram, we compare the 0.1 and the 0.9 quantiles for $\sigma=1$ (blue) and $\sigma=0.8$ (dark orange)

normcdfs sigma=1, sigma=0.8

Michael Mayer's example fulfills the requirements of your question with $q_1=0.1$, $q_2=0.9$ and $X_1$ being the one with larger variance.

Edit:

For the case where $q_1$ and $q_2$ must both be on the same side of whatever the measure of location is, let's take two symmetric distributions, which share the same mean and median.

Let $X_1$ be $\sim \text{N}(0,1^2)$ and let $X_2$ be an equal mixture of a $\text{N}(-0.8,0.1^2)$ and a $\text{N}(0.8,0.1^2)$, and let $q_1 = 0.6$ and $q_2 = 0.9$:

normal 0,1 vs symmetric mixture of normals with small s.d.

This example fulfills the new requirements of your question with $q_1=0.6$, $q_2=0.9$ and $X_1$ being the one with only a single normal component (shown in blue above).

Further, you should note that 'location parameter' isn't sufficiently specified. I could parameterize normal distributions by their 5th percentile and their standard deviation, and call the parameter based on the 5th percentile the location parameter (it's just a shift of the mean by $1.645\sigma$. and can work as a perfectly valid location parameter). Then Michael's original example suffices even under the new conditions. If that contradicts your intention, your intention needs to be stated specifically enough to exclude it.

Glen_b
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  • Thanks everyone for your quick pointer. However at this point, I realize the question I asked for is not properly specified. I should mention a caveat that, the quantiles should be of the same side of Origin (assuming location parameters for both r.v.s are zero). For example, let say 5th and 1st quantile of 2 random variables. My question is that, given 2 real valued r.v.s, if 5th quantile of a r.v. is smaller than 5th quantile of 2nd r.v. then 1st quantiles also will be of similar order? Is it a necessary property? Or there exist random variables for which such rule may not follow? – Ron Oct 14 '13 at 17:29
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    Ron, I have added your change to your question and responded to it. Please note that the way you worded it in your comment doesn't make sense (what's the "fifth quantile"? What's the "first quantile"?), but with that part left out, we can get somewhere. – Glen_b Oct 14 '13 at 21:52