MATH235 MATH235 Week 9 - Assessed problems (coursework) MATH235 Week 10 - Workshop problems

MATH235 Week 9 - Moodle Quiz-assessed problems

A random sample is drawn from a parametric distribution with parameter $\theta>0$ and resulted in a likelihood function $L(\theta)$ , which is suitably smooth. The plots of the likelihood function, the log likelihood function, $\ell(\theta)$ , and its first two derivatives, are displayed below. The plots all have $\theta$ on the horizontal axis, but the labels on the diagrams are missing and the plots are in a jumbled order.

Unnumbered Figure: Link

Q9.1

Panel (a) corresponds to

(a)

The likelihood function $L(\theta)$
(b)

The log-likelihood function $l(\theta)$
(c)

The first derivative of the log-likelihood, $l^{\prime}(\theta)$
(d)

The second derivative of the log-likelihood, $l^{\prime\prime}(\theta)$

[marks: 2]

Q9.2

Panel (b) corresponds to

(a)

The likelihood function $L(\theta)$
(b)

The log-likelihood function $l(\theta)$
(c)

The first derivative of the log-likelihood, $l^{\prime}(\theta)$
(d)

The second derivative of the log-likelihood, $l^{\prime\prime}(\theta)$

[marks: 2]

Q9.3

Panel (c) corresponds to

(a)

The likelihood function $L(\theta)$
(b)

The log-likelihood function $l(\theta)$
(c)

The first derivative of the log-likelihood, $l^{\prime}(\theta)$
(d)

The second derivative of the log-likelihood, $l^{\prime\prime}(\theta)$

[marks: 2]

Q9.4

Panel (d) corresponds to

(a)

The likelihood function $L(\theta)$
(b)

The log-likelihood function $l(\theta)$
(c)

The first derivative of the log-likelihood, $l^{\prime}(\theta)$
(d)

The second derivative of the log-likelihood, $l^{\prime\prime}(\theta)$

[marks: 2]

Q9.5

MLE
The maximum likelihood estimate of $\theta$ is

(a)

$-5.65$
(b)

$0.41$
(c)

$-25$
(d)

$3.6\times 10^{-3}$

[marks: 2]