Let Θ̂M L be the maximum likelihood estimator for the mean αof a Poisson random vart able. Suppo

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

Key Concepts

Delta Method

The Delta Method is a technique used to approximate the distribution of a function of an asymptotically normal estimator. By employing a first-order Taylor expansion, it helps in deriving the variance and standard error of non-linear transformations of parameters, making it particularly useful when studying estimators that are functions of estimates, such as the exponential transformation used in this problem.

Asymptotic Normality

Asymptotic normality refers to the property that, as the sample size increases, the distribution of certain estimators, like the MLE, tends to a normal distribution regardless of the underlying distribution of the data. This property allows for the use of normal approximations in inference, including the construction of confidence intervals and hypothesis testing with large samples.

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by finding the values that maximize the likelihood function given the observed data. It is widely used because of its desirable asymptotic properties, such as consistency and efficiency, especially as the sample size increases.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, with these events happening at a known constant mean rate and independently of the time since the last event. A notable property used in such estimation problems is that the probability of zero events is given by an exponential function involving the distribution's mean.

Transcript

00:01 This question is given with an example which follows by zone distribution and we are asked to determine maximum likelihood estimate for theta and show that it is an unbiased estimate.

00:14 So the probability mass function for x is the probability mass function for x is equal to x theta is equal to we have to copy the theta power theta power x divide of our x equal to 1 3 3 the natural the natural log the natural law applies the function and the likelihood function is likely function is likely function is d a 4 n of theta is equal to ln of n theta x1 x2 up to x2 up to xm which is equal to n and of b power and theta theta power x1 plus x2 plus x3 up to xm divide the 5 x1 victorial x2 factorial x3 up to xm up to x so so this is equal to minus n theta plus n x bar l and of theta minus n of x1 victoria x2 factorial x3 factorial up to x n -2 here x1 plus x2 plus x3 plus x -3 plus x -n is equal to n x differentiation gives us differentiating the above statement or the above equation will give us differentiation gives lm ha is equal to theta is a quarter minus m theta minus x bar divided by theta solving of theta a time of theta minus theta qta q qa minus x bar divided by theta kore is equal to 0 given theta is equal to since l i mat theta is equal to minus x sorry minus n x bar divided by theta is equal to sorry this then zero we see that theta cape magazine mize and of theta kep, therefore the maximum is theta care which is equal to x bar which is equal to sum of x i i from 1 to x i divided by so so the parameter theta is the expected value of the present and the medium but we see that expected value of theta cap you see that expected value of theta cap is equal to expected value of x -by is equal to expected value of x which is equal to theta so it shows that the mn e estimated for theta is unbiased so m -l -e estimator for theta is embraced.

11:02 Now coming to the next part of the question, b part of the question...

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