MATH235 Week 8  - Moodle Quiz-assessed problems

likelihood
Let $X_1,\mathrm{\dots },X_n$ be mutually independent random variables, each with pdf

$$f(x|\theta )=2\theta x\exp (-\theta x^2),\text{for}x>0.$$

What is the log-likelihood of $\theta$?

1. (a)

   $2\theta \prod x_i\exp (-\theta x_i^2)$

2. (b)

   $n\log (2\theta )+\sum \log (x_i)-\theta \sum x_i^2$

3. (c)

   $2^n{\theta }^n\prod x_i\exp (-\theta \sum x_i^2)$

4. (d)

   $n\log (2\theta )+\prod \log (x_i)-\theta \sum x_i^2$

5. (e)

   $n\log (2\theta )+\sum \log (x_i)-\theta \prod x_i^2$

[marks: 2]

MLE
Continuing with the example in Q1, the MLE of $\theta$ is

1. (a)

   $\frac{\sum x_i^2+\sum \log (x_i)}{n}$

2. (b)

   $\frac{n}{\sum x_i^2+\sum \log (x_i)}$

3. (c)

   $\frac{\sum x_i^2}{n}$

4. (d)

   $\frac{n}{\theta -2\sum x_i}$

5. (e)

   $\frac{n}{\sum x_i^2}$

[marks: 2]

Observed Information
Continuing the same example, the observed information at the MLE is

1. (a)

   0

2. (b)

   $n/{\widehat{\theta }}^2$

3. (c)

   $n/{\theta }^2$

4. (d)

   $-n/{\theta }^2$

5. (e)

   $-n/{\widehat{\theta }}^2$

[marks: 2]

Pins A sequence of ten independent pin drops, each with probability $\theta$ of landing point uppermost, results in $UDUUUUDUUD$. The observed information at the MLE based on these data, $\widehat{\theta }$, is

1. (a)

   33.333

2. (b)

   31.033

3. (c)

   13.333

4. (d)

   41.667

5. (e)

   47.619

[marks: 2]

Paired Pins Now suppose the results are paired: $UD$ $UU$ $UU$ $DU$ $UD$, and instead of recording the full result only the occurrence or non-occurrence of two point uppermosts is noted. So you see $NYYNN$. The observed information at the MLE based on these data, $\widehat{\theta }$, is

1. (a)

   41.667

2. (b)

   47.619

3. (c)

   33.333

4. (d)

   13.333

5. (e)

   31.033

[marks: 2]