Let Θ̂M L be the maximum likelihood estimator for the mean of an exponential random variable. Su

step 1 Hello, this is problem 9 .81. We know that 1 is distributed as an exponential random variable wi

Let Θ̂M L be the maximum likelihood estimator for the mean...

Key Concepts

Maximum Likelihood Estimation

This is a statistical method for estimating the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. In the context of the exponential distribution, the maximum likelihood estimator for the mean is the sample average, which is used as a primary statistic in deriving further properties.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between independent events that occur at a constant rate. Key properties include its mean, which serves as a scale parameter, and its variance, which is the square of the mean. These relationships provide the basis for parameter estimation and inference in problems that involve exponential data.

Central Limit Theorem

The central limit theorem is a fundamental theorem in probability theory stating that, for a large number of independent observations drawn from any distribution with finite mean and variance, the sampling distribution of the sample mean will approximate a normal distribution. This theorem is crucial when justifying normal approximations for the sampling distributions of estimators, including those derived from the exponential distribution.

Delta Method

The delta method is a technique used in statistics to derive the approximate distribution of a function of an estimator. By using a first-order Taylor expansion, it allows determination of the asymptotic behavior (mean and variance) of non-linear transformations of asymptotically normal estimators, which is essential when considering functions like the square of an estimator as in the provided problem.

Transcript

00:01 Hello, this is problem 9 .81.

00:04 We know that 1 is distributed as an exponential random variable with parameter data.

00:16 Okay, so we're trying to find the mle of data.

00:28 We have that f of y is equal to 1 over theta, e to negative y data.

00:38 So this is the pdf of the exponential random variable.

00:44 First we find the likelihood.

00:46 So capital l of theta is equal to the product from i is equal to 1 to n of 1 over theta.

00:57 The only thing that changes here is we're going to put e to the negative of y -s of i divided by theta.

01:06 Because there's a lot of ys we're looking at all in total.

01:15 So from here we want to simplify things.

01:19 So we'll do 1 over data to the n because there's n thetas and then we do e to the negative 1 over data.

01:31 So we could just take up the 1 over data.

01:35 And then since we're going to be adding the ys of i n of them so we do the summation of i is equal to 1 to n of y of i so this is the first step we're doing this and then we need to take the natural log of the likelihood so if we do that we'll get negative n natural log and of theta because we could just bring this little end to the top and it becomes a negative and the end will come in the front but it's a negative so that's how we get the negative end there and then the natural log of data and then we know that the natural log of e it will cancel out so you just have this left the negative one over data in i sqa 1 to n of y so i so this is the second step now the third step now we need to take a derivative so we're going to take the partial derivative of what we have before the natural log of the likelihood of data so partial derivative of that we respect to data and it's going to be equal to well, the negative n is a constant.

03:16 The theta is a parameter.

03:19 So the real natural log of theta is just 1 over theta.

03:24 So negative n over theta.

03:28 But we're going to go a step further.

03:31 At this point, we could put a little hat.

03:36 And so that's what we're trying to look for.

03:38 We're trying to look for theta hat.

03:40 So we keep going.

03:42 So now we want to take the derivative of, this part here.

03:49 So we could just bring the theta to the top.

03:52 So v theta to negative 1.

03:54 So then we multiply the negative 1 times the negative so becomes a positive...