Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The random variables $X_1,...X_n$ are independent draws from continuous unifirm distribution with support $[0,\theta]$. Derive a method of moments and maximum likelihood estimators of $\theta$. Your research assistant has constructed new random variables $Y_i$ such that:

$$Y_i = \begin{cases} 0, & \text{if} \, X_i \le k \\ 1, & \text{if} \, X_i \gt k\\ \end{cases}$$

where $k$ is a constant chosen by RA and known to you. What are the Maximum Likelihood and Method of Moments Estimators in this case? Assume $(k \lt \theta)$.

After trying, these are the answers which I am getting:

first part: By method of moments: $\hat{\theta} = 2\bar{X}$

By MLE: $\hat{\theta} = Max\{X_i\}$

Second Part: By Method of Moments: $\hat{\theta} = \frac{k}{1-\overline{Y}}$

By MLE: $\hat{\theta} =\frac{k}{1-\overline{Y}}$ (same as above)

Please tell me which answer is wrong so that I can redo that very part.

PS: This question was asked in an exam a couple of years back, so I dont know the correct answers.

share|cite|improve this question
I get the same results. Do you have some motive to suspect that some answer is wrong? – leonbloy Jun 25 '13 at 16:45
up vote 1 down vote accepted

All seem right to me.

The first part is well known.

For the second part, you can consider that $Y$ is a Bernoulli variable with probability of success $p= 1- k/\theta$. Hence, because $k$ is known, both $p$ and $\theta$ can be regarded as (complete) parameters of $Y$, related by the above function $p=g(\theta)$. And, recalling the important functional invariance property of the ML estimator, we can compute $\hat{p}_{ML}$ (which, we probably already know, for a Bernoulli is just the sample mean $\bar{Y}$), and so

$$ \widehat\theta_{ML}=\frac{k}{1-\hat{p}_{ML}}=\frac{k}{1-\bar{Y}}$$

share|cite|improve this answer
I like this (alternate) method more. Thanx – Mathy Jun 26 '13 at 22:37

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.