a question about maximum likelihood estimation (MLE)

Suppose we know a certain transformation of x, say y = theta * x^3,
satisfy a normal distribution N(beta, 1), where theta and beta are two
parameters to be estimated.

We have two independent observations x1 and x2. That is, there are two
observations for y: y1 = theta * x1^3 and y2 = theta * x2^3. Now we
want to use the MLE method to estimate the two parameters.

Denote the pdf of y as p_y(y) and that of x as p_x(x). We have p_x(x)
= p_y(theta * x^3) * 3*theta*x^2.

One way to estimate theta and beta is to maximize p_x(x1) * p_x(x2):
max_{theta, beta} p_x(x1) * p_x(x2)
= max_{theta, beta} p_y(theta * x1^3) * p_y(theta * x2^3) *
3*theta*x1^2 * 3*theta*x2^2

The other way is to maximize p_y(y1) * p_y(y2):
max_{theta, beta} p_y(theta * x1^3)

Obviously, these two methods are different because of the addition
term, 3*theta*x1^2 * 3*theta*x2^2, in the first method. Intuitively, I
know this is because an additional term (the derivative of y with
respect to x: 3*theta*x^2) is added to generate p_x(x) from p_y(y).
This term, however, does not appear when we transform from x1, x2 to
y1, y2.

In fact, if both x and y are discrete random variables, we simple use
p_x(x) = p_y(theta * x^3), instead of p_x(x) = p_y(theta * x^3) *
3*theta*x^2 -- here, p_x(x) and p_y(y) are probability mass
functions. In this case, the two methods above lead to the same
result.

But, how can we explain the difference of two methods when x and y are
both continuous variables?

.

Relevant Pages

  • a question about maximum likelihood estimation (MLE)
    ... Suppose we know a certain transformation of x, say y = theta * x^3, ... satisfy a normal distribution N(beta, 1), where theta and beta are two ... In fact, if both x and y are discrete random variables, we simple use ...
    (sci.stat.math)
  • a question about maximum likelihood estimation (MLE)
    ... Suppose we know a certain transformation of x, say y = theta * x^3, ... satisfy a normal distribution N(beta, 1), where theta and beta are two ... In fact, if both x and y are discrete random variables, we simple use ...
    (sci.stat.math)
  • a question about maximum likelihood estimation (MLE)
    ... Suppose we know a certain transformation of x, say y = theta * x^3, ... satisfy a normal distribution N(beta, 1), where theta and beta are two ... In fact, if both x and y are discrete random variables, we simple use ...
    (sci.math)
  • a question about maximum likelihood estimation (MLE)
    ... Suppose we know a certain transformation of x, say y = theta * x^3, ... satisfy a normal distribution N(beta, 1), where theta and beta are two ... In fact, if both x and y are discrete random variables, we simple use ...
    (sci.econ)
  • a question about maximum likelihood estimation (MLE)
    ... Suppose we know a certain transformation of x, say y = theta * x^3, ... satisfy a normal distribution N(beta, 1), where theta and beta are two ... In fact, if both x and y are discrete random variables, we simple use ...
    (sci.stat.math)
Loading