Consider an observed random sample of size n, w1, . . . , wn, from a normal distribution N(µ, σ2 ). To the 75 observations in the dataset Data-A1a.csv apply the EM algorithm to fit via maximum likelihood the two-component normal mixture density with common variances, Carry out a chi-squared goodness-of-fit test to assess the adequacy of the fit of the twocomponent normal mixture model with common variances to the n = 75 data points. use mclust of R studio
Question
Consider an observed random sample of size n, w1, . . . , wn, from a normal distribution N(µ, σ2 ). To the 75 observations in the dataset Data-A1a.csv apply the EM algorithm to fit via maximum likelihood the two-component normal mixture density with common variances, Carry out a chi-squared goodness-of-fit test to assess the adequacy of the fit of the twocomponent normal mixture model with common variances to the n = 75 data points. use mclust of R studio
Solution
Here are the steps to perform the EM algorithm and chi-squared goodness-of-fit test in R using the mclust package:
- Install and load the mclust package in RStudio:
install.packages("mclust")
library(mclust)
- Load your data into R. Assuming your data is in the same directory as your R script:
data <- read.csv("Data-A1a.csv")
- Apply the EM algorithm to fit the two-component normal mixture density with common variances. The Mclust function in the mclust package will do this:
model <- Mclust(data, G=2)
summary(model)
The G=2 argument specifies that we want to fit a two-component mixture model. The summary(model) command will give you the parameters of the fitted model.
- To carry out a chi-squared goodness-of-fit test, we first need to calculate the expected frequencies under the fitted model. We can do this using the
densityMclustfunction:
expected <- densityMclust(model, data)$z
- Next, we calculate the observed frequencies:
observed <- table(cut(data, breaks=seq(min(data), max(data), length.out=11)))
- Now we can perform the chi-squared test:
chisq.test(observed, p=expected)
This will return the chi-squared statistic and the p-value. If the p-value is less than your chosen significance level (e.g., 0.05), then you would reject the null hypothesis that the data follows the fitted two-component normal mixture model.
Please note that the above steps assume that your data is univariate. If your data is multivariate, you would need to adjust the steps accordingly. Also, the goodness-of-fit test is a simple one and may not be appropriate for all situations. It's always a good idea to visually inspect your data and the fitted model as well.
Similar Questions
Consider an observed random sample of size n, w1, . . . , wn, from a normal distribution N(µ, σ2 ). To the 75 observations in the dataset Data-A1a.csv apply the EM algorithm to fit via maximum likelihood the two-component normal mixture density with common variances, f(w; Ψ) = X 2 i=1 πi φ(w; µi , σ2 ), where φ(w; µ, σ2 ) = (2πσ2 ) −1/2 exp{−1 2 (w − µ) 2 /σ2 } and Ψ = (π1, µ1, µ2, σ2 ) T . To this end, (i) [1/2 mark] Specify the EM framework
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