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Statistics: Running Simulations for Five Methods

Statistics
Statistics: Running Simulations for Five Methods
Given a linear model Y=β0∗+X∗⊤β∗+ε, ε∼N (0,σ∗2).
The summation ϵ is an independent and identically distributed or an iid with normal random
variables having a mean of 0 and a variance of σ2. Arguably, this model comprises of three fixed
parameters that can be estimated: β0, β1, and σ2 unknown constants. In the above linear model,
there is a slight modification of the notation here. We have applied the notation to be Y since the
model will be fitted using a set of n data points ranging from i=1, 2… onwards.
1. Running simulations for five methods:
• LASSO (Tibshirani; 1996)
Summary
• true model is y=Xβ+ϵy=Xβ+ϵ
• where ϵ∼Nn (0, I) ϵ∼Nn (0, I)
Example 1
Small signal. Lots of noise.
• β=(115…,04085)Tβ=(1,…,1⏟15,0,…,0⏟4085)T
• p=5000>n=1000p=5000>n=1000
• Uncorrelated predictors:
o Xi∼iidN(0,I)
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