Saturday, February 4, 2012

Minimizing the Length of a Confidence Interval

Right now I'm teaching an introductory course on statistical inference for Economics students. We've been dealing with confidence intervals, starting off (as usual) with one for the mean of a Normal population.

For a given confidence level, the shorter the interval is, the more "informative" it is. The question that then arises is how to make the interval as short as possible, everything else being equal? Good question!

First, let's review some basic results. Say I want to construct a 95% confidence interval for the mean of a Normal population. If the population variance, σ2 is known, then this interval will be constructed by using the properties of a standard Normal random variable. If the variance is unknown, then the interval will be constructed using the properties of a Student-t random variable.

Traditionally, in each case we center the (random) interval at the sample mean, and get the lower and upper limits by adding and subtracting an equal amount. When the population variance is known, this amount depends on the standard deviation of the population, the sample size, and the quantile for the Standard Normal distribution that assigns 2.5% to the right tail of the density. That quantile is roughly 1.96.

When the population variance is unknown, this amount depends on the estimated standard deviation of the population, the sample size, and the quantile for Student's t distribution that assigns 2.5% to the right tail of the density. That quantile depends on the degrees of freedom, (n - 1). For instance if n = 10, then this quantile is 2.262; while if n = 15, this quantile is 2.145

Now, why do we build the confidence interval so that it is symmetric about the sample mean, in each of these cases? That is, why do construct "equal tails" confidence intervals?

After all, for the example where the population variance is known, and I'm using quantiles from the Standard Normal distribution, I don't have to use -1.96σn-1/2 and +1.96σn-1/2 to get the lower and upper limits of the 95% confidence. You can easily check that I could subtract -1.6693σn-1/2 from the sample mean, and add  2.81σn-1/2, and I'd still get a confidence level (coverage probability) of 95%.

Indeed, given the continuity of the Normal density, there's actually an infinity of different asymmetric intervals that I could construct, each of which would be 95% confidence intervals! For any fixed degrees of freedom, the same point applies when we're using quantiles from Student's t distribution in case where the population variance is unknown.

So, the question remains - why do we traditionally construct "equal tails" intervals here?

Well, it's not just because of tradition! For the problem we're talking about, it can be shown that the "equal tails" confidence interval is also the shortest (and hence, most informative) interval. That's for a chosen confidence level, a fixed value of n, and a particular (actual or estimated) population standard deviation.

To illustrate (but certainly not prove) this point, take the example above based on the quantiles of the Normal distribution. The symmetric confidence interval has length 3.92σn-1/2, while the asymmetric interval I suggested has length 4.4793σn-1/2.

One thing that you'll have noticed about the particular interval estimation problem that I've been discussing so far is that the Standard Normal and Student's t distributions have density functions that are both uni-modal  and symmetric about their mode. Moreover, this mode is zero.

That's actually the key feature of the problem. If the density of the (pivotal) statistic that's being used as the starting point for the interval is uni-modal at zero, and symmetric about this point, then the "equal tails" confidence interval will be the shortest possible interval. In our problem, the statistic in question is either the standardized sample mean, which is pivotal whether we use the true or estimated population standard deviation. ("Pivotal" means that the statistic's distribution doesn't depend on the unknown parameters.) 

You'll find this discussed, for example, by Casella and Berger (2002), and taken up in detail by Ferentinos and Karakostas (2006).

Another sufficient condition for the "equal tails" and shortest confidence intervals to coincide is given by Kirmani (1990). If the distribution of the (pivotal) statistic is symmetric, and is concave on the right side of the point of symmetry, then the "equal tails" confidence interval will again be shortest.

Interesting! But the conditions given above are only sufficient conditions. Are they also necessary?

In other words, if the shortest confidence interval for a particular problem is also the "equal tails" confidence interval, does this mean that the distribution of the pivotal statistic has to be symmetric and uni-modal?

Surprisingly, this remains an open question! Ferentinos and Karakostas (2006) note that no formal proof is available, though there are informal reasons to believe that the conditions may also be necessary.

Some other interesting problems can arise, even when the distribution of the pivotal statistic is symmetric (but not uni-modal). For instance, the"shortest" confidence interval may not even exist! An example is given by Kirmani (1990).

Of course, once we move to situations where the distribution of the pivotal statistic is asymmetric, we open up a different can of worms.

The obvious example is when we want to construct a confidence interval for the variance of a Normal population. In this case, the pivotal statistic that we use to build the interval is (n - 1)s2/σ2. This statistic has a χ2 distribution, with (n - 1) degrees of freedom, and this distribution is asymmetric. It's skewed to the right.

In cases such as this, it's still traditional to construct an "equal tails" confidence interval. Now, of course, if we want a confidence level of (say) 95%, we determine the lower limit of the interval by using the quantile of the χ2 distribution that gives a 2.5% left-tail area for the density; and we determine the upper limit by using the quantile that gives a 2.5% right-tail area for the density.

For example, for 10 degrees of freedom, these quantiles are 3.247 and 20.483 respectively. As a result, the confidence interval that we construct will be asymmetric about s2 (the usual unbiased estimator of the population variance).

How do we construct a "shortest" confidence interval in situations like this?

Well, as long as the distribution of the pivotal statistic is uni-modal (as the χ2 distribution is), there is a result from Casella and Berger (2002) and Ferentinos and Karakostas (2007) that will help us. We choose the upper and lower quantiles from the distribution so as to ensure that the height to the density function is the same in each case, while still ensuring that our choice gives us the desired confidence level. As long as the chosen quantiles "straddle" the median of the distribution, we'll have the shortest confidence interval.

In practice, this is going to take quite a lot of effort! The mathematical problem that we face is one of solving two complicated equations for two unknowns - the latter being the two quantiles. One of the equations says that the value of the density function has to be the same when evaluated at the two unknowns. The other equation says that the sum of two areas under the density (two integrals) has to equal one minus the desired confidence level. When you consider the expression for the density function of the χ2 distribution with v degrees of freedom:

                      p(x) = [2(v/2) Γ(v/2)]-1 x(v/2)-1 exp(-x/2)  ;  x > 0  ,


you can see that we have an interesting problem.

That's why the "equal tails" confidence interval remains popular in cases such as this.



Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.


References

Casella, G. and R. Berger, 2001 Statistical Inference, 2nd. ed., Duxbury.

Ferentinos, K. K. and K. X. Karakostas, 2006. More on shortest and equal tails confidence intervals. Communications in Statistics - Theory and Methods, 35, 821-829.

Kirmani, S., 1990. On minimum length confidence intervals. International Journal of Mathematical Education in Science and Technology, 21, 791-793.


© 2012, David E. Giles

4 comments:

  1. I believe that you did fantastic resolution the moment when you choose this theme of the article here. Do you generally create your blog entries by yourself or maybe you have a writing partner or even an assistant?

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    1. I don't have a writing partner or an assistant - _ write the posts myself in my "spare" time.

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  2. What do you mean shortest has the most informative?

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    1. If I said my 95% C.I. was from $99 to $101, and yours 95% C.I. was from $10 to $190, which one is more useful (informative)?

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