Confidence Interval For The Mode With A Single Data Point
It is a common but false belief that from a single observation x we can not gain information about the variability in the population and that consequently that finite length confidence intervals for mean and/or variance are impossible even in principle.
It is possible for an unknown unimodal distribution to estimate a confidence interval for the mode with a sample size of 1. This was first shown by Abbot and Rosenblatt and extended by Blachman and Machol. This confidence interval can be sharpened if the distribution can be assumed to be symmetrical. It is further possible to sharpen this interval if the distribution is normally distributed.
Let the confidence interval be 1 - α. Then the confidence intervals for the general, symmetric and normally distributed variates respectively are
where X is the variate, θ is the mode and || is the absolute value.
These estimates are conservative. The confidence intervals for the mode at the 90% level given by these estimators are X ± 19 | X - θ |, X ± 9 | X - θ | and X ± 5.84 | X - θ | for the general, symmetric and normally distributed variates respectively. The 95% confidence interval for a normally distributed variate is given by X ± 10.7 | X - θ |. It may be worth noting that the mean and the mode coincide if the variates are normally distributed.
The 95% bound for a normally distributed variate has been improved and is now known to be X ± 9.68 | X - θ | The bound for a 99% confidence interval is X ± 48.39 | X - θ'|
- Note
Machol has shown that that given a known density symmetrical about 0 that given a single sample value (x) that the 90% confidence intervals of population mean are
where ν is the population median.
If the precise form of the distribution is not known but it is known to be symmetrical about zero then we have
where X is the variate, μ is the population mean and a and k are arbitrary real numbers.
It is also possible to estimate a confidence interval for the standard deviation from a single observation if the distribution is symmetrical about 0. For a normal distribution the with an unknown variance and a single data point (X) the 90%, 95% and 99% confidence intervals for the standard deviation are, and . These intervals may be shorted if the mean is known to be bounded by a multiple of the standard deviation.
If the distribution is known to be normal then it is possible to estimate a confidence interval for both the mean and variance from a simple value. The 90% confidence intervals are
The confidence intervals can be estimated for any chosen range.
This method is not limited to the normal distribution but can be used with any known distribution.
Read more about this topic: Mode (statistics)
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