Mode (statistics) - Confidence Interval For The Mode With A Single Data Point

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)

Famous quotes containing the words confidence, interval, mode, single, data and/or point:

    False shame accompanies a man that is poor, shame that either harms a man greatly or profits him; shame is with poverty, but confidence with wealth.
    Hesiod (c. 8th century B.C.)

    I was interested to see how a pioneer lived on this side of the country. His life is in some respects more adventurous than that of his brother in the West; for he contends with winter as well as the wilderness, and there is a greater interval of time at least between him and the army which is to follow. Here immigration is a tide which may ebb when it has swept away the pines; there it is not a tide, but an inundation, and roads and other improvements come steadily rushing after.
    Henry David Thoreau (1817–1862)

    I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?
    Henry David Thoreau (1817–1862)

    A writer is in danger of allowing his talent to dull who lets more than a year go past without finding himself in his rightful place of composition, the small single unluxurious “retreat” of the twentieth century, the hotel bedroom.
    Cyril Connolly (1903–1974)

    This city is neither a jungle nor the moon.... In long shot: a cosmic smudge, a conglomerate of bleeding energies. Close up, it is a fairly legible printed circuit, a transistorized labyrinth of beastly tracks, a data bank for asthmatic voice-prints.
    Susan Sontag (b. 1933)

    He left the name, at which the world grew pale,
    To point a moral, or adorn a tale.
    Samuel Johnson (1709–1784)