Let X1, X2, …, Xn be a random sample from a distribution...

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a distribution with one of two pdfs. If $\theta=1$, then $f(x ; \theta=1)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2},-\infty<x<\infty$. If $\theta=2$, then $f(x ; \theta=2)=1 /\left[\pi\left(1+x^{2}\right)\right],-\infty<x<\infty$. Find the mle of $\theta$.

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a distribution with one of two pdfs. If $\theta=1$, then $f(x ; \theta=1)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2},-\infty<x<\infty$. If $\theta=2$, then
$f(x ; \theta=2)=1 /\left[\pi\left(1+x^{2}\right)\right],-\infty<x<\infty$. Find the mle of $\theta$.

Key Concepts

Model Selection

This concept involves choosing between different candidate models or distributions to best represent the data. When the parameter determines which model is used, the selection is made by comparing the likelihoods under each model, and choosing the one that maximizes the likelihood function.

Discrete Parameter Space

In problems where the parameter can take only a limited number of values, the estimation procedure involves comparing the likelihoods for each possible value of the parameter. The estimated value is the one that yields the highest likelihood among the specified candidates.

Maximum Likelihood Estimation (MLE)

This is a method of estimating the parameters of a statistical model by finding the parameter value that maximizes the likelihood function, which represents the probability of observing the given data. The MLE is often used because it has desirable properties such as consistency and efficiency under regularity conditions.

Likelihood Function

The likelihood function is a function of the parameter(s) of the model, given the observed data. It is constructed as the product (or sum of the logs) of the probabilities or probability densities associated with each observation. Maximizing this function over the parameter space identifies the parameter value(s) under which the observed data is most probable.

Transcript

00:02 Hi, here it is given that x1, x2, dot dot x n, this random samples follows a distribution with one of its two pdfs, okay, this follows fx theta, and theta takes only two values, theta equals to 1, 2, and for theta equals to 1, see, and fx theta is given, and, theta, even in such a way that fx theta equals to 1 by root 2 pi into the power minus x square whole divided by 2 minus infinity less than x less than infinity and theta equals to 1 and another pdf is 1 by pi into 1 whole divided by 1 plus x square minus infinity less than x less than infinity and theta equals to two see when theta equals to one xi follows standard normal distribution and when theta equals to two xi follows standard caution distribution okay now we have to find the mali of theta so to find the amy of theta first we construct the likelihood function which is l theta x equal equals to product of i runs from 1 to n f x i.

01:51 So here we have two likelihood functions.

01:57 For theta equals to 1 we have 1 by root 2 pi whole to the power n e to the power minus summation x i square.

02:14 Minus infinity less than x i less than infinity and theta equals to one and irons from one to okay another likelihood function is one by pi to the power n one whole divided by product of one plus x iance from one to n and here also minus infinity less than x i less than infinity and here theta equals to two.

02:52 Two likelihood functions we have.

02:55 We'll better write it like this.

02:59 This is l theta equals to 1 colon x curl and this is l theta equals to 2 colon x curl.

03:09 And we have to find the maximum likelihood estimated of theta...

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