Assignment Task

Linear Predictor for AR(1) Process: Finding Optimal Prediction Function for Missing Values

1. Suppose we are dealing with 𝑌𝑡Yt as an AR(1) process. Given the observations 𝑌1Y1 and 𝑌3Y3, we aim to estimate the missing value 𝑌2Y2. What is the best linear predictor for 𝑌2Y2 based on 𝑌1Y1 and 𝑌3Y3? Provide mathematical proofs, calculations, or steps to support your answer.

2. In your mathematical statistics class, your professor tasks you with finding a prediction function $h(x)$ that minimizes 𝐻=𝐸[(𝑦−ℎ(𝑥))2]H=E[(y−h(x))2], where $x$ and $y$ are jointly distributed random variables with density function $f(x,y)$.

   a) Determine the appropriate $h(x)$ to minimize $H$.

   b) Apply the above result to the model 𝑦=𝑥2+𝑧y=x2+z, where $x$ and $z$ are normal random variables with mean zero and variance 1. Show that $H=1$.

   c) Suppose your professor limits $h(x)$ choices to linear functions $h(x)=ax+b$. Find $a$ and $b$ that minimize $H$, showing that $a=1$ and 𝑏=𝐸(𝑥𝑦)/𝐸(𝑥2)=0b=E(xy)/E(x2)=0.

3. Consider a (weakly) stationary process $Y$ with zero mean and autocovariance function denoted by $acv(m)$.

   a) Given 𝑌𝑡Yt, determine the best predictor 𝑌𝑡^+1Yt^+1.

   b) Given 𝑌𝑡Yt and 𝑌𝑡−1Yt−1, find the best linear predictor 𝑌𝑡~+1Yt~+1 of 𝑌𝑡+1Yt+1.

   c) Define a transformation called Difference of Mean Squared Errors (DOMSE):

   𝐷𝑂𝑀𝑆𝐸=𝐸{(𝑌𝑡+1−𝑌𝑡^+1)2}−𝐸{(𝑌𝑡+1−𝑌𝑡~+1)2}DOMSE=E{(Yt+1−Yt^+1)2}−E{(Yt+1−Yt~+1)2}

   Derive an expression for DOMSE and evaluate it for 𝑌𝑡=cos⁡(𝑛𝑈)Yt=cos(nU) where $U$ is uniform on $[-\pi ,\pi ]$, and for an AR model where $acv(k)=\alpha |k|$ with $|\alpha |<1$.

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