Review of MS-C1620 Statistical Inference @Aalto University

Author: Xuan Binh, Taha Heidari

Email: [email protected]

🔝 Table of Contents

The aim of this repo is to build a foundation in Statistical Inference with R.

• Large of large Numbers

If you increase the number of observations (sample size), the sample moments will converge to the population moments (Law of Large numbers)

In the below example, we assume that we have a population of mean = 0 and standard deviation = 1 from a Normal Distribution. If you increase the number of samples you see that the histogram plot will get similar to the Normal Distribution with those population moments (mean,std).

hist(X,breaks=5)

Breaks: bins/class intervals (Custom Number of bins)

hist(X,main="struge")

Struge method for calculating the number of bins

#============================
#    Normal Distributions
#============================
# Create a vector of random numbers
# from a Normal Distribution
# with mean = 0, sd=1
X = rnorm(100,mean=0,sd=1)
# Calculate mean(X); sd(X); var(X); min(X); max(X)
mean(X); sd(X); var(X)
# Histogram plot with number of bins
# Breaks: bins/class intervals 
hist(X, breaks=5) 
hist(X,main="struge")

data("Salaries") 397 #sample size from a population
x <- Salaries
n <- length(x)
boxplot(x)
hist(x)
mean(x)
skewness(x)

#------
x2 <- rnorm(20)
boxplot(x2)
skewness(x2) #+ or -

#------
x2 <- rnorm(200)
boxplot(x2)
skewness(x2) #+ or -

sample statistic population parameter confidence interval

• population mean confidence interval with student-t
• bootstrap method for other population parameters like population mean, variance, skewness, standard deviation, ...

bootstrap

• create pseudo sample (random sampling with replacement: resampling) with the same size
• computationally cost (new) 100 or 1000 times

• ideally we wan to resample from the real population but bootstrapping seems like getting sth from nothing as it is resampling the only sample we have from the population

B <- 1000
res <- rep(0, B)

for (b in 1:B){
	res[b] <- mean(sample(x, n, replace = TRUE)) #skewness or other parameters
}

hist(res)

# 95%
ci_bs <- quatile(res, probs = c(0.025,0.975))
abline(v = ci_bs, lwd = 2, col = 2)
ci_bs


ci_ex <- c(mean(x)-1.96*sd(x)/sqrt(n), mean(x)+1.96*sd(x)/sqrt(n))
abline(v = ci_ex, lwd = 2, col = 3)
ci_ex

🔝 Table of Contents

using hist() to plot histogram

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