Statistical computing: JBM050 Quartile 4
2019-2020
Course organization
Books:
- Monte Carlo simulation and resampling methods for social science (lecture 1-4)
- An introduction to statistical learning (rest)
Lecture 1: Introduction
Learning from your data
Mathematical models for experiments are Analytics solutions and Simulations. For the simulations we
make data with our computer and study those values.
The data generating model in random experiments
Bernoulli distribution
Use the Bernoulli trial when you want the know the probability of, for example, head and coins. To
have a good approximation, you need a very large sample.
The outcomes of a coin toss are two values, so it is a binary variable (values 1 and 0). Each outcome
has a certain π probability; it is a binary random variable. Only 1 toss.
P ( X=1 )=π
P ( X=0 )=1−π
Random variables that are binary are a Bernoulli distribution. → X Bern(π )
Binomial distribution
It is an extension on the Bernoulli distribution, only now you toss several times. If we toss it five
times, our number of trials, n, is 5. The outcome can take values 0, 1, … ,n . The number of successes
is represented by a k.
n!
P ( X=k )= π k ( 1−π )n−k
k ! ( n−k ) !
n!
= n and it counts the number of ways k successes can occur in
The binomial coefficient
k ! ( n−k ) ! k()
a sequence of n trials. It can be described as → X Bin(n , π )
Hypergeometric distribution
The lady tasting tea: There are 8 cups of tea, in the first 4, milk was poured first, and in the last 4, tea
was poured first. She said that she could discriminate which drink was poured first.
She has 6 of the 8 cups right. But what is the probability of guessing 6 of 8 correct if you are just
guessing.
P ¿ ¿ guessing ¿=nr of sequences 6∨more ¿ ¿ =nr ¿ guessing ¿ ¿+ nr ¿ guessing ¿ ¿
total nr of possible sequences n
It is not binomial distributes, because the teacups are dependent of each other.
,Other random experiments
Monty hall paradox:
There are three doors, behind which 2 are goats, and one a car. One door is opened and has a goat
behind it. Now, the player chooses a door, and is asked to switch. Which is better? Switching. The
probability of winning after staying is 1/3 and for switching it is 2/3.
Random experiments with R
Generating data from discrete random variables (rolling dice or tossing coin)
- The sample R function
- Bernoulli, binomial, hypergeometric, multinomial, Poisson distribution
The R function sample:
[ marbles = c(‘red’, ‘blue’, ‘yellow’) ]
[ sample(x = marbles, size=2, replace=False) ]
Bernoulli and Binomial distribution in R
X Bin(n , π) is [ rbinom(n=5, size=1, prob=0.5) ]
To find the number of successes, run this line 5 times and count the number of times it has 4.
^
P ( X=4 )=^π
Pseudo code
Steps to create your own pseudo code:
1. Give the function a name and specify the arguments
2. Define the input and output
3. Show underlying procedure
a. Programming statements
b. Indentation to add clear structure
4. Make sure it returns the correct output
,Lecture 2: Basics: Sampling
Something is normally distributed with a mean and variance. Also
typed as: N ( μ , σ 2 ), they are the model parameters. It looks like this:
In statistics, a (point) estimator is an approximation of a population parameter that uses observed data.
It is also called a statistic. The rule = estimator and its value are the estimate.
Determine the population model and its values: (possibly a test question)
For example, coin tossing experiment:
- Population model: X Bern( π)
- Population parameter of interest: π
k nr .heads up
- Estimator/statistic: ^π = =
n total nr of tosses
- Estimate: ^π =0.53
The sampling distribution
It is the probability distribution of the sample statistic for a given sample size n. It describes how the
sample statistics varies over different samples of size n. the statistic is considered to be a random
variable.
Call random samples: [ rnorm(n=5, mean=x, sd=y) ]
2
σ
We know that the sample average is: ^x N (μ , )
n
The sample distribution of the average of a sample in size n is a normal distribution itself. The
standard deviation of a sample statistic is also called the standard error of that statistic.
Mean, bias, MSE, RE
Data is used to estimate the population parameter θ and yield the estimator θ^ .
The bias is θ−E( θ)^ .
2
The variance is E( ( E ( θ^ ) −θ^ ) ).
Its standard deviation is called the standard error, which is √ variance .
The MSE is the bia s 2+ variance .
, Lecture 3: Monte Carlo Simulation
Sampling distribution of the difference in means: Assume that two samples are drawn independently.
Analytical results are that under the assumptions, the sample distribution of the difference in sample
means x́ 2−x́ 1is a t-distribution with n1 +n 2−2 degrees of freedom, centered at μ2−μ 1 and a standard
2 2
1 1 ( n1 −1 ) s1 + ( n2−1 ) s 2 .
error equal to SE ( x́2 −x́1 ) =S p
√ + with S p=
n1 n 2 √n1+ n2−2
A t-test was invented to test on the difference between two independent samples:
x́ 2− x́ 1
t= t
SE (x́2 −x́1 ) n 1+n 2−2
Central limit theorem:
For large n, the sample mean of X coming from some probability distribution with
σ2
mean μ and variance σ 2, X N ( μ , σ 2 ) is approximately normal with mean μ and variance ,
n
σ2
x́ N ( μ , ).
n
Properties of statistical methods must be established so that the methods may be used with confidence.
Exact derivations are rarely possible. Analytical results may require assumptions. If these assumptions
are violated, the results may not be possible.
We can find out which assumptions are violated with MC Simulation. MC Simulation can be used for
1) QC for estimators and 2) QC for hypothesis testing. All in absence of analytical results.
MC Simulation for properties of estimators:
An estimator has a true sampling distribution under a particular set of conditions. We cannot derive
this true sampling distribution, for this we can use MC.
How to approximate:
1. Generate S independent data sets of given sample size n.
2. Compute the numerical value of the estimator for each data set.
3. If S is large enough, the summary statistics should be good approximations to the true
sampling properties of the estimator under the conditions of interest.
Compute for each of the statistics:
S
´ 1 (k)
- Average θ^ (k)= ∑ θ^ s
s s=1
- Bias ´
θ^ (k)−θ
S
1 ´^ (k ) 2
- Variance ∑ ( θ^ ¿ ¿(k)
s −θ ) ¿ ¿
S−1 s=1
S
1 ´ 2
- MSE ∑ ( θ^ ¿ ¿ (k )−θ ¿ ) ¿ ¿
S S =1
MSE ( θ^ (1))
Relative efficiency: ℜ=
MSE ( θ^ (2))
2019-2020
Course organization
Books:
- Monte Carlo simulation and resampling methods for social science (lecture 1-4)
- An introduction to statistical learning (rest)
Lecture 1: Introduction
Learning from your data
Mathematical models for experiments are Analytics solutions and Simulations. For the simulations we
make data with our computer and study those values.
The data generating model in random experiments
Bernoulli distribution
Use the Bernoulli trial when you want the know the probability of, for example, head and coins. To
have a good approximation, you need a very large sample.
The outcomes of a coin toss are two values, so it is a binary variable (values 1 and 0). Each outcome
has a certain π probability; it is a binary random variable. Only 1 toss.
P ( X=1 )=π
P ( X=0 )=1−π
Random variables that are binary are a Bernoulli distribution. → X Bern(π )
Binomial distribution
It is an extension on the Bernoulli distribution, only now you toss several times. If we toss it five
times, our number of trials, n, is 5. The outcome can take values 0, 1, … ,n . The number of successes
is represented by a k.
n!
P ( X=k )= π k ( 1−π )n−k
k ! ( n−k ) !
n!
= n and it counts the number of ways k successes can occur in
The binomial coefficient
k ! ( n−k ) ! k()
a sequence of n trials. It can be described as → X Bin(n , π )
Hypergeometric distribution
The lady tasting tea: There are 8 cups of tea, in the first 4, milk was poured first, and in the last 4, tea
was poured first. She said that she could discriminate which drink was poured first.
She has 6 of the 8 cups right. But what is the probability of guessing 6 of 8 correct if you are just
guessing.
P ¿ ¿ guessing ¿=nr of sequences 6∨more ¿ ¿ =nr ¿ guessing ¿ ¿+ nr ¿ guessing ¿ ¿
total nr of possible sequences n
It is not binomial distributes, because the teacups are dependent of each other.
,Other random experiments
Monty hall paradox:
There are three doors, behind which 2 are goats, and one a car. One door is opened and has a goat
behind it. Now, the player chooses a door, and is asked to switch. Which is better? Switching. The
probability of winning after staying is 1/3 and for switching it is 2/3.
Random experiments with R
Generating data from discrete random variables (rolling dice or tossing coin)
- The sample R function
- Bernoulli, binomial, hypergeometric, multinomial, Poisson distribution
The R function sample:
[ marbles = c(‘red’, ‘blue’, ‘yellow’) ]
[ sample(x = marbles, size=2, replace=False) ]
Bernoulli and Binomial distribution in R
X Bin(n , π) is [ rbinom(n=5, size=1, prob=0.5) ]
To find the number of successes, run this line 5 times and count the number of times it has 4.
^
P ( X=4 )=^π
Pseudo code
Steps to create your own pseudo code:
1. Give the function a name and specify the arguments
2. Define the input and output
3. Show underlying procedure
a. Programming statements
b. Indentation to add clear structure
4. Make sure it returns the correct output
,Lecture 2: Basics: Sampling
Something is normally distributed with a mean and variance. Also
typed as: N ( μ , σ 2 ), they are the model parameters. It looks like this:
In statistics, a (point) estimator is an approximation of a population parameter that uses observed data.
It is also called a statistic. The rule = estimator and its value are the estimate.
Determine the population model and its values: (possibly a test question)
For example, coin tossing experiment:
- Population model: X Bern( π)
- Population parameter of interest: π
k nr .heads up
- Estimator/statistic: ^π = =
n total nr of tosses
- Estimate: ^π =0.53
The sampling distribution
It is the probability distribution of the sample statistic for a given sample size n. It describes how the
sample statistics varies over different samples of size n. the statistic is considered to be a random
variable.
Call random samples: [ rnorm(n=5, mean=x, sd=y) ]
2
σ
We know that the sample average is: ^x N (μ , )
n
The sample distribution of the average of a sample in size n is a normal distribution itself. The
standard deviation of a sample statistic is also called the standard error of that statistic.
Mean, bias, MSE, RE
Data is used to estimate the population parameter θ and yield the estimator θ^ .
The bias is θ−E( θ)^ .
2
The variance is E( ( E ( θ^ ) −θ^ ) ).
Its standard deviation is called the standard error, which is √ variance .
The MSE is the bia s 2+ variance .
, Lecture 3: Monte Carlo Simulation
Sampling distribution of the difference in means: Assume that two samples are drawn independently.
Analytical results are that under the assumptions, the sample distribution of the difference in sample
means x́ 2−x́ 1is a t-distribution with n1 +n 2−2 degrees of freedom, centered at μ2−μ 1 and a standard
2 2
1 1 ( n1 −1 ) s1 + ( n2−1 ) s 2 .
error equal to SE ( x́2 −x́1 ) =S p
√ + with S p=
n1 n 2 √n1+ n2−2
A t-test was invented to test on the difference between two independent samples:
x́ 2− x́ 1
t= t
SE (x́2 −x́1 ) n 1+n 2−2
Central limit theorem:
For large n, the sample mean of X coming from some probability distribution with
σ2
mean μ and variance σ 2, X N ( μ , σ 2 ) is approximately normal with mean μ and variance ,
n
σ2
x́ N ( μ , ).
n
Properties of statistical methods must be established so that the methods may be used with confidence.
Exact derivations are rarely possible. Analytical results may require assumptions. If these assumptions
are violated, the results may not be possible.
We can find out which assumptions are violated with MC Simulation. MC Simulation can be used for
1) QC for estimators and 2) QC for hypothesis testing. All in absence of analytical results.
MC Simulation for properties of estimators:
An estimator has a true sampling distribution under a particular set of conditions. We cannot derive
this true sampling distribution, for this we can use MC.
How to approximate:
1. Generate S independent data sets of given sample size n.
2. Compute the numerical value of the estimator for each data set.
3. If S is large enough, the summary statistics should be good approximations to the true
sampling properties of the estimator under the conditions of interest.
Compute for each of the statistics:
S
´ 1 (k)
- Average θ^ (k)= ∑ θ^ s
s s=1
- Bias ´
θ^ (k)−θ
S
1 ´^ (k ) 2
- Variance ∑ ( θ^ ¿ ¿(k)
s −θ ) ¿ ¿
S−1 s=1
S
1 ´ 2
- MSE ∑ ( θ^ ¿ ¿ (k )−θ ¿ ) ¿ ¿
S S =1
MSE ( θ^ (1))
Relative efficiency: ℜ=
MSE ( θ^ (2))
