For binomial distribution (n, p), the pmf is,
For random samples X1, X2, ...., Xk, the likelihood function is,
The log-likelihood function is,
For maximum likelihood estimate,
where
(sample mean)
Thus the maximum likelihood estimate of p =
where
is the
sample mean of the random samples.
7. Find the maximum likelihood estimate of parameter p of the binomial distribution.
1. Let X b(n , 0 ), find the maximum likelihood estimate of the parameter 0 of the " corresponding binomial distribution. And prove the sample proportion is unbiased estimator of 0. 2. If are the values of a random sample from an exponential population, find the maximum likelihood estimator of its parameter 0.
1. Let X b(n , 0 ), find the maximum likelihood estimate of the parameter 0 of the " corresponding binomial distribution. And prove the sample...
6. Find the moment and maximum likelihood estimates of the parameter p of the negative -.. , Xn. Recall that the pmf is , and again when maximizing consider binomial distribution given an iid sample from it: X1, given by p(k) = ( )prqk-r for k = r, r + 1, the Bernoulli MLE
6. Find the moment and maximum likelihood estimates of the parameter p of the negative -.. , Xn. Recall that the pmf is , and again...
are independent variables from Negative Binomial distribution with parameters (known) and . Find the maximum likelihood estimator of .
Consider the Binomial distribution for x= 0,1,2,3,…..,n.Find the maximum likelihood estimator of p when a single observation is taken?
2. Let X1, X2, ...,Xbe i.i.d. Poisson with parameter .. (a) Find the maximum likelihood estimator of . Is the estimator minimum variance unbi- ased? (b) Derive the asymptotic (large-sample) distribution of the maximum likelihood estimator. (c) Suppose we are interested in the probability of a zero: Q = P(Xi = 0) = exp(-). Find the maximum likelihood estimator of O and its asymptotic distribution.
a. What is the maximum likelihood estimator for the parameter 2 of the poisson distribution for a sample of n poisson random variables?
2. i) Let B be a random variable with the Binomial (n, p) distribution, so that Write down the likelihood function L(p) for m independent observations xi,...,Inm 2 marks 6 marks ili) Compute the bias and the mean squared error of the corresponding maximum likeli- from B. Int ii) Show that the maximum likelihood estimate for pis-Σ.ri. mn [7 marks] hood estimator of p. iv) Let X be a continuous random variable with density function for x > 0, and...
5. Find the maximum likelihood est imat or of the unknown parameter 0 where X, X2,..., X from the distribution whose density function is is a sample e-(r-0) if r O )-{ fx(x) = otherwise
Let Y be a binomial random variable with parameter p. Find the sample size necessary to estimate p to within .05 with probability .95 if p is thought to be approximately .9.
To find the maximum likelihood estimate, suppose that, in
general, t animals are tagged. Then, of a second sample of
size m, r tagged animals are recaptured. We
estimate n by the maximizer of the likelihood:
Find the log-likelihood, and then indicate which terms of it
would not become zero if you took the derivative to find the MLE of
n.