Two continuous distributions with same 1% & 5% quantiles however distributions are notably different in shape, location etc?

Question

Previously I asked this question to R-help (finance group) however, could not get a satisfactory reply (probably this is not directly related to R.) Here is my question:

Is there any possible way to get 2 continuous distributions (defined on the real line) such that, both distributions have same 1% & 5% quantiles (or, any one of these two. However to me this is a weak condition!), however those distributions are notably different in shape, location etc.

Can somebody please point me any example of such 2 distribution?

Answer 1

Would you consider it a notable difference if one distribution has a negative mean, and the other has a positive mean? This difference may be critical in some situations.

Let $X$ be the constant $-1$. $E[X] = -1.$

Let $Y$ be $-1$ with probability $99.5\%$, and $399$ with probability $0.5\%$. You can think of $Y$ as the net value of a raffle ticket you bought for $\$1$ for a $\$400$ prize. $E[Y] = 1$. If you take gambles like this repeatedly, you will win in the long run with high probability.

These are not continuous, but you can make slight adjustments to make them continuous. Let $X'$ have density $1/2$ from $-2$ to $0$, and let $Y'$ have density $1/2$ from $-2$ to $-1/100$ and from $400-1/100$ to $400.$ These exhibit the same behavior.

More subtly, you can make the difference between a winning and a losing gamble occur between, say, the $55$th and $60$th percentiles. Suppose you risk $3$ to gain $4,$ e.g., in a poker game you call a $\$3$ overbet into a pot of $\$1$ on the last round. If you lose at most $4/7 \approx 57\%$ then you have a winning gamble. The $5\%$ quantiles can't see the difference between losing $56\%$ of the time (a good gamble) and losing $59\%$ (a bad gamble). This example can also be made continuous.

Answer 2

Take two distribution functions $F_1$, $F_2$ (increasing functions from $\mathbb{R} \to [0,1]$...) such that $F_1(0.01)=F_2(0.01)$ and $F_1(0.05)=F_2(0.05)$. There are infinitely many possibilities !

For instance you can fix some values for the $1\%$ and $5\%$ quantiles and use the optim() R function to determine the parameters of a normal distribution which fits these values and on the other hand determine the parameters of a log-normal distribution which fits these values.