STAT 541: Centers and transformations

# Statistics 541: Centers and transformations

## Admistrivia

## Centers

What is a center?
- distribution of Y = f(y;theta,lambda) = f(Y-theta;0,lamdba)
- symetry
- center is "obvious" center of density

Measures of center?
- Average
- best for normal
- also for many other exponential families (does this make it robust?)
- Not resistant

- trimmed mean (used in Olympic diving)
- 25% trimmed mean is midmean (50% data tossed!)
- median
- trimean = 25% + 50% + 75%

Example (see page 10.)
data: 1,2,3,4,5,6,7,8,9, oops, 20

Efficiency (see page 11)

## Scales

Scale is measure of spread
- standard deviation is most common (extreamly sensitive to outliers)
- MSE = s
^{2}
- E|X - EX| = (uncommon)
- E|X - median| = nicer theory
- MAD = median of distance to median (similar to IQR)
- IQR

## Transformations

Centers only make sense if symetric. If not symetric, maybe a small
change in the data will make it symetric.
- ay
^{alpha} + b (alpha not equal to zero)
- c log(y) + d (taking limit as alpha goes to zero)
- see page 15 for graphs
- Goal: after transformation data is symetric
- Goal: keep it natural! Logs, 1/Y, Y^2 are about all I use.

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Last modified: Thu Jan 18 08:43:09 2001

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