The document contains R code to generate three figures and a table. The code for Figures 1 and 2 generate time series plots (type='s') of vector X using a stochastic process where the value at each time step depends on the previous value plus random noise. The code for Table 3 estimates parameters of the same stochastic process using a linear regression approach, and compares the estimated values to the true parameters used to generate the data.

1. 1. R code for Figure 1
set.seed(2880)
tt=2:100
X=vector(length=100)
e1=rpois(1, lambda=1)
alpha=0.5
X[1]=e1
for(t in 2:100){
e=rpois(1, lambda=1)
nt=X[t-1]
S=vector(length=nt )
for(i in 1:nt){
S[i]=rbinom(1,1,0.5)
}
alpha.oX=sum(S)
X[t]= alpha.oX+e
}
X
plot(tt, X[2:100], type='s')
2. R code for Figure 2
set.seed(102)
tt=2:100
X=vector(length=100)
e1=rnorm(n=100, mean = 2, sd = 2)
alpha=0.5
X[1]=e1
for(t in 2:100){
e=rpois(1, lambda=1)
nt=X[t-1]
S=vector(length=nt )
for(i in 1:nt){
S[i]=rbinom(1,1,0.5)
}
alpha.oX=sum(S)
X[t]= alpha.oX+e
}
X
plot(tt, X[2:100], type='s')

2. 3. R code for Table 3
set.seed(2880)
tt=2:51
n=length(tt)
X=vector(length=n+2)
e1=rpois(1, lambda=1)
alpha=0.4
lambda=1
X[1]=e1
for(t in 2:(n+2)){
e=rpois(1, lambda=1)
nt=X[t-1]
S=vector(length=nt )
for(i in 1:nt){
S[i]=rbinom(1,1,0.5)
}
alpha.oX=sum(S)
X[t]= alpha.oX+e
}
X
xbar=mean(X)
Y=X-xbar
length(Y)
Z=Y[2:(n+2)]
xt=Z[1:n]
xt1=Z[2:(n+1)]
alpha_hat1=(t(xt)%*%xt1)/(t(Y)%*%Y)
alpha_hat1-alpha
epsilon_hat=xt1-alpha_hat1*xt
lambda_hat1=mean(epsilon_hat)
#abs(lambda_hat1)-lambda
((lambda/(1-alpha)-Z[(n+1)])/n)-(alpha_hat1-alpha)
alpha_hat2=(sum(xt1*xt)-(sum(xt1)*sum(xt)/n))/
(sum(xt^2)-(sum(xt)^2/n))
alpha_hat2-alpha
lambda_hat2=1/n*(sum(xt1)-alpha_hat2*sum(xt))
((lambda/(1-alpha)-Z[(n+1)])/n)-(alpha_hat2-alpha)