Unit 13: robust statistics and estimation

robust statistics and estimation

what are some limitations of standard estimators?

sensitivity to outliers, assumptions of normality, influence of variance

What is the primary goal, outlier impact, and example of standard estimators?

primary goal- efficiency under normality

outlier impact- high (sensitivity)

example- arithmetic mean

What is the primary goal, outlier impact, and example of robust estimators?

primary goal- stability across distributions

outlier impact- low (resistance)

example- median trimmed mean, M-estimators, winsorized variance

a statistic is considered robust if..

• small departures from assumptions (like normality) do not affect it much

• a few extreme observations do not drastically change its value

what are different ways we can know a statistic is robust? like what to look at

1. finite sample breakdown point

2. standard error

3. probability coverage + control of alpha

4. power

what is meant by finite breakdown point

a measure of “resistance to contamination” —- the smallest proportion of values that have a large impact on parameter estimate

What is the breakdown point of a sample mean and sample median?

sample mean— 1/n

sample median —- .5

what does the SE need to look like when a statistic is robust?

reduced SE due to minimizing variance from outliers

what does biased mean and inflated SE create?

biased mean (due to outliers) and inflated SE “create” significance that is likely just noise/error

what are examples of robust estimators?

trimmed mean

bootstrap resampling

what is the process of trimmed mean?

sot values and remove the % of data points from either tail

10-20% is generally considered optimal

what is bootstrap resampling? and what does it allow for us?

• generate distributions under the null

  • allows us to get accurate p-values without requiring the data to follow a particular shape (normality and maintain type I error rate

• distribution of mean differences to calculate CI’s

  • allows us to get accurate CIs that preserve the true shape of the different between group means (magnitude and reliability of the effect)

how can we do testing individual effects?

if you have repeated measures, you can generate individual-level disrtibrutiiosns and use these to inferentially test effects

what are some limitations of standard estimators?

traditional parametric tests rely on strict assumptions like normality, standard estimators like the arithmetic mean and highly sensitive to outlier, which can bias results and inflate standard errors.

what are the benefits of robust stats?

robust stats such as the trimmed mean or median, provide stability across different distributions and resist “contamination” from extreme observations

• these method help maintain accurate type I error rte

• preserves statistical power when data deviates from normality