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Is Cauchy distribution somehow an “unpredictable†distribution?
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Is Cauchy distribution somehow an "unpredictable" distribution?
I tried doing
cs <- function(n)
return(rcauchy(n,0,1))
in R for a multitude of n values and noticed that they generate quite unpredictable values occasionally.
Compare that to e.g.
as <- function(n)
return(rnorm(n,0,1))
which always seems to give a "compact" cloud of points.
By this pic it should look like the normal distribution? Yet it maybe only does for a subset of values. Or maybe the trick is that the Cauchy one converges much more slowly (to left and right) and thus is allows for more severe outliers, although at low probabilities?
Here as are normal rvs and cs are Cauchy rvs.
But by the extremity of the outliers, is it possible that the Cauchy standard deviations never converge? But go to infinity?
cauchy
asked 6 hours ago
mavavilj
1,019721
add a comment |Â
up vote
1
down vote
favorite
Is Cauchy distribution somehow an "unpredictable" distribution?
I tried doing
cs <- function(n)
return(rcauchy(n,0,1))
in R for a multitude of n values and noticed that they generate quite unpredictable values occasionally.
Compare that to e.g.
as <- function(n)
return(rnorm(n,0,1))
which always seems to give a "compact" cloud of points.
By this pic it should look like the normal distribution? Yet it maybe only does for a subset of values. Or maybe the trick is that the Cauchy one converges much more slowly (to left and right) and thus is allows for more severe outliers, although at low probabilities?
Here as are normal rvs and cs are Cauchy rvs.
But by the extremity of the outliers, is it possible that the Cauchy standard deviations never converge? But go to infinity?
cauchy
asked 6 hours ago
mavavilj
1,019721
1
1. Your question is vague/unclear, so it's hard to answer; e.g. what does "unpredictable" mean in your question? what do you mean by "the Cauchy standard deviations" and convergence near the end? You don't seem to be calculating standard deviations anywhere. standard deviations of what, exactly? 2. Many posts on site discuss properties of the Cauchy which may help you focus your question. It may also be worth checking Wikipedia. 3. I'd suggest avoiding the term "bell shaped"; both densities seem roughly shaped like a bell; just call them by their names.
– Glen_b♦
4 hours ago
Certainly the Cauchy is very heavy tailed.
– Glen_b♦
4 hours ago
I've posted a few facts; hopefully these will help you figure out what you want to know about so that you can refine your question.
– Glen_b♦
3 hours ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Is Cauchy distribution somehow an "unpredictable" distribution?
I tried doing
cs <- function(n)
return(rcauchy(n,0,1))
in R for a multitude of n values and noticed that they generate quite unpredictable values occasionally.
Compare that to e.g.
as <- function(n)
return(rnorm(n,0,1))
which always seems to give a "compact" cloud of points.
By this pic it should look like the normal distribution? Yet it maybe only does for a subset of values. Or maybe the trick is that the Cauchy one converges much more slowly (to left and right) and thus is allows for more severe outliers, although at low probabilities?
Here as are normal rvs and cs are Cauchy rvs.
But by the extremity of the outliers, is it possible that the Cauchy standard deviations never converge? But go to infinity?
cauchy
asked 6 hours ago
mavavilj
1,019721
Is Cauchy distribution somehow an "unpredictable" distribution?
I tried doing
cs <- function(n)
return(rcauchy(n,0,1))
in R for a multitude of n values and noticed that they generate quite unpredictable values occasionally.
Compare that to e.g.
as <- function(n)
return(rnorm(n,0,1))
which always seems to give a "compact" cloud of points.
By this pic it should look like the normal distribution? Yet it maybe only does for a subset of values. Or maybe the trick is that the Cauchy one converges much more slowly (to left and right) and thus is allows for more severe outliers, although at low probabilities?
Here as are normal rvs and cs are Cauchy rvs.
But by the extremity of the outliers, is it possible that the Cauchy standard deviations never converge? But go to infinity?
cauchy
cauchy
asked 6 hours ago
mavavilj
1,019721
asked 6 hours ago
mavavilj
1,019721
edited 5 hours ago
asked 6 hours ago
mavavilj
1,019721
asked 6 hours ago
mavavilj
1,019721
asked 6 hours ago
mavavilj
1,019721
1,019721
1
1. Your question is vague/unclear, so it's hard to answer; e.g. what does "unpredictable" mean in your question? what do you mean by "the Cauchy standard deviations" and convergence near the end? You don't seem to be calculating standard deviations anywhere. standard deviations of what, exactly? 2. Many posts on site discuss properties of the Cauchy which may help you focus your question. It may also be worth checking Wikipedia. 3. I'd suggest avoiding the term "bell shaped"; both densities seem roughly shaped like a bell; just call them by their names.
– Glen_b♦
4 hours ago
Certainly the Cauchy is very heavy tailed.
– Glen_b♦
4 hours ago
I've posted a few facts; hopefully these will help you figure out what you want to know about so that you can refine your question.
– Glen_b♦
3 hours ago
add a comment |Â
1
1. Your question is vague/unclear, so it's hard to answer; e.g. what does "unpredictable" mean in your question? what do you mean by "the Cauchy standard deviations" and convergence near the end? You don't seem to be calculating standard deviations anywhere. standard deviations of what, exactly? 2. Many posts on site discuss properties of the Cauchy which may help you focus your question. It may also be worth checking Wikipedia. 3. I'd suggest avoiding the term "bell shaped"; both densities seem roughly shaped like a bell; just call them by their names.
– Glen_b♦
4 hours ago
Certainly the Cauchy is very heavy tailed.
– Glen_b♦
4 hours ago
I've posted a few facts; hopefully these will help you figure out what you want to know about so that you can refine your question.
– Glen_b♦
3 hours ago
1
1
1. Your question is vague/unclear, so it's hard to answer; e.g. what does "unpredictable" mean in your question? what do you mean by "the Cauchy standard deviations" and convergence near the end? You don't seem to be calculating standard deviations anywhere. standard deviations of what, exactly? 2. Many posts on site discuss properties of the Cauchy which may help you focus your question. It may also be worth checking Wikipedia. 3. I'd suggest avoiding the term "bell shaped"; both densities seem roughly shaped like a bell; just call them by their names.
– Glen_b♦
4 hours ago
1. Your question is vague/unclear, so it's hard to answer; e.g. what does "unpredictable" mean in your question? what do you mean by "the Cauchy standard deviations" and convergence near the end? You don't seem to be calculating standard deviations anywhere. standard deviations of what, exactly? 2. Many posts on site discuss properties of the Cauchy which may help you focus your question. It may also be worth checking Wikipedia. 3. I'd suggest avoiding the term "bell shaped"; both densities seem roughly shaped like a bell; just call them by their names.
– Glen_b♦
4 hours ago
Certainly the Cauchy is very heavy tailed.
– Glen_b♦
4 hours ago
Certainly the Cauchy is very heavy tailed.
– Glen_b♦
4 hours ago
I've posted a few facts; hopefully these will help you figure out what you want to know about so that you can refine your question.
– Glen_b♦
3 hours ago
I've posted a few facts; hopefully these will help you figure out what you want to know about so that you can refine your question.
– Glen_b♦
3 hours ago
add a comment |Â
1 Answer
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oldest
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While a number of posts on site address various properties of the Cauchy, I didn't manage to locate one that really laid them out together. Hopefully this might be a good place to collect some. I may expand this.
Heavy tails
While the Cauchy is symmetric and roughly bell shaped, somewhat like the normal distribution, it has much heavier tails (and less of a "shoulder"). For example, there's a distinct probability that a Cauchy random variable will lay more than 1000 interquartile ranges from the median -- roughly of the same order as a normal random variable being about 2.5 interquartile ranges from its median.
Variance
The variance of the Cauchy is infinite.
Sample standard deviations exist, of course, but the larger the sample the larger they tend to be (e.g. the median sample standard deviation at n=10 is in the vicinity of 3.67 times the scale parameter (half the IQR), but at n=100 it's about 11.9).
Mean
The Cauchy distribution doesn't even have a finite mean; the integral for the mean doesn't converge. As a result, even the laws of large numbers don't apply -- as n grows, sample means don't converge to some fixed quantity (indeed there's nothing for them to converge to).
In fact, the distribution of the sample mean from a Cauchy distribution is the same as the distribution of a single observation(!). The tail is so heavy that adding more values into the sum makes a really extreme value likely enough as to just compensate for dividing by a larger denominator when taking the mean.
Predictability
You can certainly produce perfectly sensible prediction intervals for observations from a Cauchy distribution; there are simple, fairly efficient estimators that perform well for estimating location and scale and approximate prediction intervals can be constructed -- so in that sense, at least, Cauchy variates are 'predictable'. However, the tail extends very far, so that if you want a high-probability interval, it may be quite wide.
If you're trying to predict the center of the distribution (e.g. in a regression type model), that may in some sense be relatively easy to predict; the Cauchy is quite peaked (there's a lot of the distribution "close" to the center for a typical measure of scale), so the center can be relatively well-estimated if you have an appropriate estimator.
answered 3 hours ago
Glen_b♦
205k22389717
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
While a number of posts on site address various properties of the Cauchy, I didn't manage to locate one that really laid them out together. Hopefully this might be a good place to collect some. I may expand this.
Heavy tails
While the Cauchy is symmetric and roughly bell shaped, somewhat like the normal distribution, it has much heavier tails (and less of a "shoulder"). For example, there's a distinct probability that a Cauchy random variable will lay more than 1000 interquartile ranges from the median -- roughly of the same order as a normal random variable being about 2.5 interquartile ranges from its median.
Variance
The variance of the Cauchy is infinite.
Sample standard deviations exist, of course, but the larger the sample the larger they tend to be (e.g. the median sample standard deviation at n=10 is in the vicinity of 3.67 times the scale parameter (half the IQR), but at n=100 it's about 11.9).
Mean
The Cauchy distribution doesn't even have a finite mean; the integral for the mean doesn't converge. As a result, even the laws of large numbers don't apply -- as n grows, sample means don't converge to some fixed quantity (indeed there's nothing for them to converge to).
In fact, the distribution of the sample mean from a Cauchy distribution is the same as the distribution of a single observation(!). The tail is so heavy that adding more values into the sum makes a really extreme value likely enough as to just compensate for dividing by a larger denominator when taking the mean.
Predictability
You can certainly produce perfectly sensible prediction intervals for observations from a Cauchy distribution; there are simple, fairly efficient estimators that perform well for estimating location and scale and approximate prediction intervals can be constructed -- so in that sense, at least, Cauchy variates are 'predictable'. However, the tail extends very far, so that if you want a high-probability interval, it may be quite wide.
If you're trying to predict the center of the distribution (e.g. in a regression type model), that may in some sense be relatively easy to predict; the Cauchy is quite peaked (there's a lot of the distribution "close" to the center for a typical measure of scale), so the center can be relatively well-estimated if you have an appropriate estimator.
answered 3 hours ago
Glen_b♦
205k22389717
add a comment |Â
up vote
5
down vote
While a number of posts on site address various properties of the Cauchy, I didn't manage to locate one that really laid them out together. Hopefully this might be a good place to collect some. I may expand this.
Heavy tails
While the Cauchy is symmetric and roughly bell shaped, somewhat like the normal distribution, it has much heavier tails (and less of a "shoulder"). For example, there's a distinct probability that a Cauchy random variable will lay more than 1000 interquartile ranges from the median -- roughly of the same order as a normal random variable being about 2.5 interquartile ranges from its median.
Variance
The variance of the Cauchy is infinite.
Sample standard deviations exist, of course, but the larger the sample the larger they tend to be (e.g. the median sample standard deviation at n=10 is in the vicinity of 3.67 times the scale parameter (half the IQR), but at n=100 it's about 11.9).
Mean
The Cauchy distribution doesn't even have a finite mean; the integral for the mean doesn't converge. As a result, even the laws of large numbers don't apply -- as n grows, sample means don't converge to some fixed quantity (indeed there's nothing for them to converge to).
In fact, the distribution of the sample mean from a Cauchy distribution is the same as the distribution of a single observation(!). The tail is so heavy that adding more values into the sum makes a really extreme value likely enough as to just compensate for dividing by a larger denominator when taking the mean.
Predictability
You can certainly produce perfectly sensible prediction intervals for observations from a Cauchy distribution; there are simple, fairly efficient estimators that perform well for estimating location and scale and approximate prediction intervals can be constructed -- so in that sense, at least, Cauchy variates are 'predictable'. However, the tail extends very far, so that if you want a high-probability interval, it may be quite wide.
If you're trying to predict the center of the distribution (e.g. in a regression type model), that may in some sense be relatively easy to predict; the Cauchy is quite peaked (there's a lot of the distribution "close" to the center for a typical measure of scale), so the center can be relatively well-estimated if you have an appropriate estimator.
answered 3 hours ago
Glen_b♦
205k22389717
add a comment |Â
up vote
5
down vote
up vote
5
down vote
While a number of posts on site address various properties of the Cauchy, I didn't manage to locate one that really laid them out together. Hopefully this might be a good place to collect some. I may expand this.
Heavy tails
While the Cauchy is symmetric and roughly bell shaped, somewhat like the normal distribution, it has much heavier tails (and less of a "shoulder"). For example, there's a distinct probability that a Cauchy random variable will lay more than 1000 interquartile ranges from the median -- roughly of the same order as a normal random variable being about 2.5 interquartile ranges from its median.
Variance
The variance of the Cauchy is infinite.
Sample standard deviations exist, of course, but the larger the sample the larger they tend to be (e.g. the median sample standard deviation at n=10 is in the vicinity of 3.67 times the scale parameter (half the IQR), but at n=100 it's about 11.9).
Mean
The Cauchy distribution doesn't even have a finite mean; the integral for the mean doesn't converge. As a result, even the laws of large numbers don't apply -- as n grows, sample means don't converge to some fixed quantity (indeed there's nothing for them to converge to).
In fact, the distribution of the sample mean from a Cauchy distribution is the same as the distribution of a single observation(!). The tail is so heavy that adding more values into the sum makes a really extreme value likely enough as to just compensate for dividing by a larger denominator when taking the mean.
Predictability
You can certainly produce perfectly sensible prediction intervals for observations from a Cauchy distribution; there are simple, fairly efficient estimators that perform well for estimating location and scale and approximate prediction intervals can be constructed -- so in that sense, at least, Cauchy variates are 'predictable'. However, the tail extends very far, so that if you want a high-probability interval, it may be quite wide.
If you're trying to predict the center of the distribution (e.g. in a regression type model), that may in some sense be relatively easy to predict; the Cauchy is quite peaked (there's a lot of the distribution "close" to the center for a typical measure of scale), so the center can be relatively well-estimated if you have an appropriate estimator.
answered 3 hours ago
Glen_b♦
205k22389717
While a number of posts on site address various properties of the Cauchy, I didn't manage to locate one that really laid them out together. Hopefully this might be a good place to collect some. I may expand this.
Heavy tails
While the Cauchy is symmetric and roughly bell shaped, somewhat like the normal distribution, it has much heavier tails (and less of a "shoulder"). For example, there's a distinct probability that a Cauchy random variable will lay more than 1000 interquartile ranges from the median -- roughly of the same order as a normal random variable being about 2.5 interquartile ranges from its median.
Variance
The variance of the Cauchy is infinite.
Sample standard deviations exist, of course, but the larger the sample the larger they tend to be (e.g. the median sample standard deviation at n=10 is in the vicinity of 3.67 times the scale parameter (half the IQR), but at n=100 it's about 11.9).
Mean
The Cauchy distribution doesn't even have a finite mean; the integral for the mean doesn't converge. As a result, even the laws of large numbers don't apply -- as n grows, sample means don't converge to some fixed quantity (indeed there's nothing for them to converge to).
In fact, the distribution of the sample mean from a Cauchy distribution is the same as the distribution of a single observation(!). The tail is so heavy that adding more values into the sum makes a really extreme value likely enough as to just compensate for dividing by a larger denominator when taking the mean.
Predictability
You can certainly produce perfectly sensible prediction intervals for observations from a Cauchy distribution; there are simple, fairly efficient estimators that perform well for estimating location and scale and approximate prediction intervals can be constructed -- so in that sense, at least, Cauchy variates are 'predictable'. However, the tail extends very far, so that if you want a high-probability interval, it may be quite wide.
If you're trying to predict the center of the distribution (e.g. in a regression type model), that may in some sense be relatively easy to predict; the Cauchy is quite peaked (there's a lot of the distribution "close" to the center for a typical measure of scale), so the center can be relatively well-estimated if you have an appropriate estimator.
answered 3 hours ago
Glen_b♦
205k22389717
edited 27 mins ago
answered 3 hours ago
Glen_b♦
205k22389717
answered 3 hours ago
Glen_b♦
205k22389717
answered 3 hours ago
Glen_b♦
205k22389717
205k22389717
add a comment |Â
add a comment |Â
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1
1. Your question is vague/unclear, so it's hard to answer; e.g. what does "unpredictable" mean in your question? what do you mean by "the Cauchy standard deviations" and convergence near the end? You don't seem to be calculating standard deviations anywhere. standard deviations of what, exactly? 2. Many posts on site discuss properties of the Cauchy which may help you focus your question. It may also be worth checking Wikipedia. 3. I'd suggest avoiding the term "bell shaped"; both densities seem roughly shaped like a bell; just call them by their names.
– Glen_b♦
4 hours ago
Certainly the Cauchy is very heavy tailed.
– Glen_b♦
4 hours ago
I've posted a few facts; hopefully these will help you figure out what you want to know about so that you can refine your question.
– Glen_b♦
3 hours ago