STAT 541: Centers and transformations

# Statistics 541: Centers and transformations

• Pass out handout

## 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

• standard deviation is most common (extreamly sensitive to outliers)
• MSE = s2
• 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.
• ayalpha + 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.