Quiz: QZ-20260908-28670

Quiz Details

QZ-20260908-28670

Topics:

Maximum likehood estimators Minimal sufficient statistics

Difficulty: Level 5 - Very Hard

Questions: 15

Language: English (English)

Generated: September 08, 2026 at 10:45 AM

Generated by: NANZIRI PATIENCE

Instructions: Select an answer for each question and click "Check Answer" to see if you're correct. Then view the explanation to learn more!

1 What is the primary property of a maximum likelihood estimator (MLE)?

A) It provides an unbiased estimate of the parameters.

B) It is always consistent for finite samples.

C) It minimizes the expected value of the squared error.

D) It maximizes the likelihood function based on observed data.

Correct Answer: D
Explanation: The primary property of an MLE is that it maximizes the likelihood function, which represents the probability of the observed data given the parameters.

2 In the context of sufficient statistics, what does the Rao-Blackwell theorem state?

A) A sufficient statistic can be derived from any estimator.

B) All sufficient statistics are minimal sufficient statistics.

C) Sufficient statistics are always consistent estimators.

D) Any unbiased estimator can be improved using a sufficient statistic.

Correct Answer: D
Explanation: The Rao-Blackwell theorem states that any unbiased estimator can be improved by conditioning it on a sufficient statistic, leading to a lower variance estimator.

3 Which of the following is NOT a property of MLE?

A) Sufficiency

B) Asymptotic Normality

C) Consistency

D) Invariance

Correct Answer: A
Explanation: Sufficiency is a property related to statistics that can provide complete information about a parameter, but it is not inherently a property of MLE.

4 For which of the following distributions is the sample mean a minimal sufficient statistic?

A) Binomial distribution

B) Poisson distribution

C) Normal distribution with known variance

D) Exponential distribution

Correct Answer: C
Explanation: The sample mean is a minimal sufficient statistic for the normal distribution with known variance, as it captures all necessary information about the mean parameter.

5 What is the likelihood function for independent and identically distributed observations?

A) Difference of individual likelihoods

B) Sum of individual likelihoods

C) Average of individual likelihoods

D) Product of individual likelihoods

Correct Answer: D
Explanation: The likelihood function for independent and identically distributed observations is the product of the individual likelihoods, reflecting the joint probability of all observations.

6 Which of the following statements about minimal sufficient statistics is true?

A) They are always unique.

B) They must be complete.

C) They do not contain all information about the parameter.

D) They can be derived from any sufficient statistic.

Correct Answer: D
Explanation: Minimal sufficient statistics can always be derived from any sufficient statistic, and they retain all the information about the parameter while being the simplest form.

7 What is the maximum likelihood estimator of the parameter p in a Bernoulli trial?

A) 1-x/n

B) n/x

C) x/n

D) 1/n

Correct Answer: C
Explanation: The MLE of parameter p in a Bernoulli trial is given by the ratio of successes (x) to the total trials (n), which is x/n.

8 Which of the following is an example of a complete statistic?

A) Sample variance from a uniform distribution

B) Maximum of a sample from a uniform distribution

C) Sample mean from a normal distribution

D) Sum of a sample from a Poisson distribution

Correct Answer: D
Explanation: The sum of a sample from a Poisson distribution is a complete statistic since it captures all the necessary information about the parameter of the distribution.

9 In the context of MLE, what does the term 'invariance' refer to?

A) The MLE is invariant to outliers in the data.

B) The MLE is invariant to transformations of the data.

C) The MLE is invariant under parametric transformations.

D) The MLE does not change with sample size.

Correct Answer: C
Explanation: The invariance property of MLE states that if you transform parameters using a one-to-one function, the maximum likelihood estimator of the transformed parameter is the transformation of the MLE of the original parameter.

10 What is a key characteristic of minimal sufficient statistics?

A) They are unique for every distribution.

B) They can be derived from non-sufficient statistics.

C) They summarize data without loss of information.

D) They require larger sample sizes to be effective.

Correct Answer: C
Explanation: Minimal sufficient statistics summarize the data in such a way that no information about the parameter is lost, retaining all necessary information.

11 Which of the following is a condition for a statistic to be sufficient for a parameter θ?

A) The statistic must have minimum variance.

B) The statistic must be a function of the likelihood ratio.

C) The conditional distribution of the data must not depend on θ.

D) The statistic must be unbiased.

Correct Answer: C
Explanation: For a statistic to be sufficient for a parameter θ, the conditional distribution of the data given the statistic must not depend on θ, meaning it captures all the information about θ.

12 What does the term 'asymptotic efficiency' mean in the context of MLE?

A) The MLE is consistent for any sample size.

B) The MLE achieves the lowest variance among all estimators for large samples.

C) The MLE approaches the true parameter value as sample size increases.

D) The MLE is unbiased for finite sample sizes.

Correct Answer: B
Explanation: Asymptotic efficiency refers to the property that the MLE achieves the lowest variance among all consistent estimators as the sample size approaches infinity.

13 Which method is commonly used to find the MLE?

A) Bayesian estimation

B) Least squares estimation

C) Method of moments

D) Gradient ascent

Correct Answer: D
Explanation: The MLE is often found using optimization techniques such as gradient ascent, which maximizes the likelihood function.

14 In the context of sufficient statistics, what does the term 'factorization theorem' imply?

A) Sufficient statistics must always be minimal.

B) Any sufficient statistic must be a factor of the likelihood function.

C) Every likelihood function factors uniquely.

D) A statistic is sufficient if the likelihood function can be factored into two parts.

Correct Answer: D
Explanation: The factorization theorem states that a statistic is sufficient for a parameter if the likelihood function can be expressed as a product of two functions, one of which depends only on the statistic.

15 What is the primary challenge in finding MLEs in practice?

A) Lack of sufficient data.

B) Difficulty in deriving the likelihood function.

C) The presence of biased estimators.

D) Computational complexity in evaluating the likelihood function.

Correct Answer: D
Explanation: Finding MLEs can be challenging due to computational complexity, especially when the likelihood function is complex or high-dimensional.

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