Question

On R, I need to use two different methods to estimate the following function:

$f(x) = \alpha _1I(U\leq x)+\alpha _2I(U> x)$

where the two alphas are unknown constants, U is a random number in between the minimum and maximum values of x, and I is an indicator function (which equals 1 when the statement is true and 0 when false). X is a variable from a dataset, part of which I have included below:

X	Y
-0.66461	6.643187
0.397653	0.867769
-0.58379	4.874199
0.054239	-0.18325
-0.13224	0.736208
0.992707	6.305005
-0.02192	0.90596
0.98336	7.073595
0.152927	0.39065
0.961129	6.735509

How would I go about doing this in R? There are other parts to the question, which I can post if required.

Answer 1

The data input would be as below.

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> library(readr)
> dat <- read_delim("dat", "\t", escape_double = FALSE, trim_ws = TRUE)

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The two methods would only vary on the basis of different regression model. This is basically a type of dummy variable analysis. The two independent variables are basically dummy variables, such that .

IU <r) = 1 - IU >r)

1. The regression model would be as . Note that the intercept coefficient is not included, so that the regression model avoids the dummy variable trap. The following script would work for this model. Also note that different U's would yield different output.

Y = 1/( U X ) + 2/(U > X)

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# Select the random U
U <- sample(dat$X,1)

# Create the dummy variables
I1 <- as.numeric(U <= dat$X)
I2 <- as.numeric(U > dat$X)

# The regression without intercept . . .
summary(lm(dat$Y ~ 0 + I1 + I2))

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2. Putting
IU <r) = 1 - IU >r)
in the above regression, we have
Y = 0 (1 - IU >r)) + 2/(U > X)
or
Y = 01-01IU > r) + QIU > X)
or
Y = 01 +(02 - 01/(U >r)
or
Y = a1 +31(U >r)
. The script is as below.

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# Select the random U
U <- sample(dat$X,1)

# Create the dummy variable
I1 <- as.numeric(U > dat$X)

# The regression without intercept . . .
summary(lm(dat$Y ~ I1))

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