Sampling Distribution and Properties of Summary Statistics
ο»ΏLecture 3
Properties of Summary Statistics: Sampling Distribution
Main Theme
How can we use math to justify that our numerical summaries from the sample are good
summaries of the population?
Lecture Summary
? Today, we focus on two summary statistics of the sample and study
its theoretical properties
?
Sample
mean:
X
=
1
=1
?
Sample
variance:
S2
=
1 -1
=1
-
2
? They are aimed to get an idea about the population mean and the population variance (i.e. parameters)
? First, we'll study, on average, how well our statistics do in estimating the parameters
? Second, we'll study the distribution of the summary statistics, known as sampling distributions.
Setup
? Let 1, ... , be i.i.d. samples from the population
? : distribution of the population (e.g. normal) with features/parameters
? Often the distribution AND the parameters are unknown. ? That's why we sample from it to get an idea of them!
? i.i.d.: independent and identically distributed
? Every time we sample, we redraw members from the population and obtain (1, ... , ). This provides a representative sample of the population.
? : dimension of
? For simplicity, we'll consider univariate cases (i.e. = 1)
Loss Function
? How "good" are our numerical summaries (i.e. statistics) in capturing features of the population (i.e. parameters)?
? Loss Function: Measures how good the statistic is in estimating the parameter
? 0 loss: the statistic is the perfect estimator for the parameter ? loss: the statistic is a terrible estimator for the parameter
? Example: , = - 2 where is the statistic and is the parameter. Called square-error loss
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