4. Point
estimation
One of the main goals of statistics
is to estimate unknown
parameters
To approximate these parameters,
we choose an estimator, which is
simply any function of randomly
sampled observations
5. Point estimation
To illustrate this idea, we will estimate the value of by uniformly
dropping samples on a square containing an inscribed circle. Notice
that the value of can be expressed as a ratio of areas.
6. Point estimation
We can estimate this ratio with our samples. Let m be the number of
samples within our circle and n the total number of samples dropped.
We define our estimator as:
8. Point
estimation in
R
We may also sample from a reference population and see if
our sample mean is a good approximation to the reference
population
In this case, we say that the sample mean is a point estimator
of the true population parameter
Let us sample 10 samples from a
normal distribution with mean
of 0 and standard deviation of 1
sample <- rnorm(10, 0, 1)
mean(sample)
Is this sample mean close to the true population mean?
Now, try with 100, 1000 and
10,000
Are we getting closer?
10. Confidence
interval
Another way of estimating parameter from a
population is to define a range of possible
values instead of just using 1 point estimate
This range is an interval, and is associated with
a confidence level
The confidence level is the probability that this
range will contain the true population
parameter
This is known formally as the confidence
interval (CI)
12. Confidence
interval in R
We will make some assumptions for
what we might find in an experiment
and find the resulting confidence
interval using a normal distribution.
In this example we will use a 95%
confidence level and wish to find the
confidence interval.
15. Bootstrap
The computational technique known as
the Bootstrap provides a convenient way
to estimate properties of an estimator via
resampling.
In this example, we resample with
replacement from the empirical
distribution function in order to estimate
the standard error of the sample mean.
17. Bootstrap in R
Using rnorm and possibly a loop or two. Write a function called
my_firstbootstrap(n,m,x,y), where n is the sampling size and m is the
number of resampling. x and y are the means and sd of a normal
distribution
Devise a way to evaluate the bootstrap outcomes as you increment
sampling size, and number of resampling
Hint: you can evaluate the sampling mean to the true population
mean
18. Bootstrap in R
my_firstbootstrap(10,100,10,1)
my_firstbootstrap(100,100,10,1)
my_firstbootstrap(1000,100,10,1)
my_firstbootstrap(10,500,10,1)
my_firstbootstrap(10,1000,10,1)
my_firstbootstrap(10,2000,10,1)
my_firstbootstrap(10,3000,10,1)
