Kerala PSC Statistical Assistant Exam-Part6
Kerala PSC Statistical Assistant Exam-Part6, let’s dive into the fascinating world of estimation in statistics!
Introduction to Estimation
Estimation is a fundamental aspect of statistics, where we aim to infer the value of an unknown parameter of a population based on sample data. Here are some key concepts:
Basic Properties of Estimators
- Unbiasedness: An estimator is unbiased if its expected value equals the true parameter value. In other words, it doesn’t systematically overestimate or underestimate the parameter.
- Consistency: An estimator is consistent if it converges to the true parameter value as the sample size increases.
- Efficiency: Among unbiased estimators, the one with the smallest variance is considered the most efficient.
- Sufficiency: An estimator is sufficient if it captures all the information about the parameter contained in the sample data.
Concepts of Sufficiency
- Sufficient Statistic: A statistic is sufficient for a parameter if the sample data can be reduced to this statistic without losing any information about the parameter. The concept is formalized by the Fisher-Neyman factorization theorem.
UMVUE (Uniformly Minimum Variance Unbiased Estimator)
- UMVUE: This is an unbiased estimator that has the lowest variance among all unbiased estimators for a parameter. It is the “best” unbiased estimator in terms of efficiency.
Completeness
- Complete Statistic: A statistic is complete if there are no other unbiased estimators that provide additional information about the parameter. Completeness is often used in conjunction with sufficiency to identify UMVUEs.
Methods of Estimation
- Method of Moments: This method involves equating sample moments (e.g., sample mean, sample variance) to population moments and solving for the parameter.
- Maximum Likelihood Estimation (MLE): This method finds the parameter value that maximizes the likelihood function, which measures how likely the observed sample is given the parameter.
- Bayesian Estimation: This method incorporates prior knowledge about the parameter in the form of a prior distribution and updates it with the sample data to obtain a posterior distribution.
Estimation is a powerful tool in statistics, enabling us to make informed decisions based on sample data.
Whether you’re working with simple or complex models, understanding these concepts will help you choose the best estimators for your analysis.
Kerala PSC Statistical Assistant Exam-Part6
Multiple Choice Questions
1.Which of the following is an unbiased estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
2.What does it mean for an estimator to be consistent?
- A) It has the smallest variance among all unbiased estimators.
- B) Its expected value equals the true parameter value.
- C) It converges to the true parameter value as the sample size increases.
- D) It captures all the information about the parameter contained in the sample data.
- Answer: C) It converges to the true parameter value as the sample size increases.
3.Which method of estimation involves maximizing the likelihood function?
- A) Method of Moments
- B) Maximum Likelihood Estimation (MLE)
- C) Bayesian Estimation
- D) Least Squares Estimation
- Answer: B) Maximum Likelihood Estimation (MLE)
4.A statistic is sufficient for a parameter if:
- A) It has the smallest variance among all unbiased estimators.
- B) It captures all the information about the parameter contained in the sample data.
- C) It converges to the true parameter value as the sample size increases.
- D) Its expected value equals the true parameter value.
- Answer: B) It captures all the information about the parameter contained in the sample data.
5.Which of the following is true about a complete statistic?
- A) It is always unbiased.
- B) It provides additional information about the parameter.
- C) There are no other unbiased estimators that provide additional information about the parameter.
- D) It is always efficient.
- Answer: C) There are no other unbiased estimators that provide additional information about the parameter.
6.What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
7.Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
8. In Bayesian estimation, what is the prior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: B) The distribution of the parameter before observing the sample data.
9.Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
Multiple Choice Questions
- Which of the following is an example of an efficient estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
- Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In Bayesian estimation, what is the posterior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: C) The distribution of the parameter after observing the sample data.
- Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
- What does UMVUE stand for?
- A) Uniformly Minimum Variance Unbiased Estimator
- B) Uniformly Maximum Variance Unbiased Estimator
- C) Uniformly Minimum Variance Unbiased Estimation
- D) Uniformly Maximum Variance Unbiased Estimation
- Answer: A) Uniformly Minimum Variance Unbiased Estimator
- Which method of estimation is based on the principle of equating sample moments to population moments?
- A) Method of Moments
- B) Maximum Likelihood Estimation (MLE)
- C) Bayesian Estimation
- D) Least Squares Estimation
- Answer: A) Method of Moments
- What is the primary goal of Bayesian estimation?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To update prior knowledge about the parameter with sample data.
- Answer: D) To update prior knowledge about the parameter with sample data.
- Which of the following is an example of a consistent estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In the context of estimation, what does efficiency refer to?
- A) The estimator’s ability to capture all the information about the parameter.
- B) The estimator’s ability to converge to the true parameter value as the sample size increases.
- C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- D) The estimator’s ability to be unbiased.
- Answer: C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- Which of the following is a property of a sufficient statistic?
- A) It is always unbiased.
- B) It captures all the information about the parameter contained in the sample data.
- C) It converges to the true parameter value as the sample size increases.
- D) It has the smallest variance among all unbiased estimators.
- Answer: B) It captures all the information about the parameter contained in the sample data.
- What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
- Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In Bayesian estimation, what is the prior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: B) The distribution of the parameter before observing the sample data.
- Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
- What does UMVUE stand for?
- A) Uniformly Minimum Variance Unbiased Estimator
- B) Uniformly Maximum Variance Unbiased Estimator
- C) Uniformly Minimum Variance Unbiased Estimation
- D) Uniformly Maximum Variance Unbiased Estimation
- Answer: A) Uniformly Minimum Variance Unbiased Estimator
- Which method of estimation is based on the principle of equating sample moments to population moments?
- A) Method of Moments
- B) Maximum Likelihood Estimation (MLE)
- C) Bayesian Estimation
- D) Least Squares Estimation
- Answer: A) Method of Moments
- What is the primary goal of Bayesian estimation?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To update prior knowledge about the parameter with sample data.
- Answer: D) To update prior knowledge about the parameter with sample data.
- Which of the following is an example of a consistent estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In the context of estimation, what does efficiency refer to?
- A) The estimator’s ability to capture all the information about the parameter.
- B) The estimator’s ability to converge to the true parameter value as the sample size increases.
- C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- D) The estimator’s ability to be unbiased.
- Answer: C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- Which of the following is a property of a sufficient statistic?
- A) It is always unbiased.
- B) It captures all the information about the parameter contained in the sample data.
- C) It converges to the true parameter value as the sample size increases.
- D) It has the smallest variance among all unbiased estimators.
- Answer: B) It captures all the information about the parameter contained in the sample data.
- What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
- Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In Bayesian estimation, what is the prior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: B) The distribution of the parameter before observing the sample data.
- Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
- What does UMVUE stand for?
- A) Uniformly Minimum Variance Unbiased Estimator
- B) Uniformly Maximum Variance Unbiased Estimator
- C) Uniformly Minimum Variance Unbiased Estimation
- D) Uniformly Maximum Variance Unbiased Estimation
- Answer: A) Uniformly Minimum Variance Unbiased Estimator
Multiple Choice Questions
- Which of the following is an example of an efficient estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
- Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In Bayesian estimation, what is the posterior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: C) The distribution of the parameter after observing the sample data.
- Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
- What does UMVUE stand for?
- A) Uniformly Minimum Variance Unbiased Estimator
- B) Uniformly Maximum Variance Unbiased Estimator
- C) Uniformly Minimum Variance Unbiased Estimation
- D) Uniformly Maximum Variance Unbiased Estimation
- Answer: A) Uniformly Minimum Variance Unbiased Estimator
- Which method of estimation is based on the principle of equating sample moments to population moments?
- A) Method of Moments
- B) Maximum Likelihood Estimation (MLE)
- C) Bayesian Estimation
- D) Least Squares Estimation
- Answer: A) Method of Moments
- What is the primary goal of Bayesian estimation?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To update prior knowledge about the parameter with sample data.
- Answer: D) To update prior knowledge about the parameter with sample data.
- Which of the following is an example of a consistent estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In the context of estimation, what does efficiency refer to?
- A) The estimator’s ability to capture all the information about the parameter.
- B) The estimator’s ability to converge to the true parameter value as the sample size increases.
- C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- D) The estimator’s ability to be unbiased.
- Answer: C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- Which of the following is a property of a sufficient statistic?
- A) It is always unbiased.
- B) It captures all the information about the parameter contained in the sample data.
- C) It converges to the true parameter value as the sample size increases.
- D) It has the smallest variance among all unbiased estimators.
- Answer: B) It captures all the information about the parameter contained in the sample data.
- What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
- Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In Bayesian estimation, what is the prior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: B) The distribution of the parameter before observing the sample data.
- Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
- What does UMVUE stand for?
- A) Uniformly Minimum Variance Unbiased Estimator
- B) Uniformly Maximum Variance Unbiased Estimator
- C) Uniformly Minimum Variance Unbiased Estimation
- D) Uniformly Maximum Variance Unbiased Estimation
- Answer: A) Uniformly Minimum Variance Unbiased Estimator
- Which method of estimation is based on the principle of equating sample moments to population moments?
- A) Method of Moments
- B) Maximum Likelihood Estimation (MLE)
- C) Bayesian Estimation
- D) Least Squares Estimation
- Answer: A) Method of Moments
- What is the primary goal of Bayesian estimation?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To update prior knowledge about the parameter with sample data.
- Answer: D) To update prior knowledge about the parameter with sample data.
- Which of the following is an example of a consistent estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In the context of estimation, what does efficiency refer to?
- A) The estimator’s ability to capture all the information about the parameter.
- B) The estimator’s ability to converge to the true parameter value as the sample size increases.
- C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- D) The estimator’s ability to be unbiased.
- Answer: C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- Which of the following is a property of a sufficient statistic?
- A) It is always unbiased.
- B) It captures all the information about the parameter contained in the sample data.
- C) It converges to the true parameter value as the sample size increases.
- D) It has the smallest variance among all unbiased estimators.
- Answer: B) It captures all the information about the parameter contained in the sample data.
- What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
- Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In Bayesian estimation, what is the prior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: B) The distribution of the parameter before observing the sample data.
- Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
- What does UMVUE stand for?
- A) Uniformly Minimum Variance Unbiased Estimator
- B) Uniformly Maximum Variance Unbiased Estimator
- C) Uniformly Minimum Variance Unbiased Estimation
- D) Uniformly Maximum Variance Unbiased Estimation
- Answer: A) Uniformly Minimum Variance Unbiased Estimator
Multiple Choice Questions
- Which of the following is an example of an efficient estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
- Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In Bayesian estimation, what is the posterior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: C) The distribution of the parameter after observing the sample data.
- Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
- What does UMVUE stand for?
- A) Uniformly Minimum Variance Unbiased Estimator
- B) Uniformly Maximum Variance Unbiased Estimator
- C) Uniformly Minimum Variance Unbiased Estimation
- D) Uniformly Maximum Variance Unbiased Estimation
- Answer: A) Uniformly Minimum Variance Unbiased Estimator
- Which method of estimation is based on the principle of equating sample moments to population moments?
- A) Method of Moments
- B) Maximum Likelihood Estimation (MLE)
- C) Bayesian Estimation
- D) Least Squares Estimation
- Answer: A) Method of Moments
- What is the primary goal of Bayesian estimation?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To update prior knowledge about the parameter with sample data.
- Answer: D) To update prior knowledge about the parameter with sample data.
- Which of the following is an example of a consistent estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In the context of estimation, what does efficiency refer to?
- A) The estimator’s ability to capture all the information about the parameter.
- B) The estimator’s ability to converge to the true parameter value as the sample size increases.
- C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- D) The estimator’s ability to be unbiased.
- Answer: C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- Which of the following is a property of a sufficient statistic?
- A) It is always unbiased.
- B) It captures all the information about the parameter contained in the sample data.
- C) It converges to the true parameter value as the sample size increases.
- D) It has the smallest variance among all unbiased estimators.
- Answer: B) It captures all the information about the parameter contained in the sample data.
- What is the primary goal of the method of moments?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To incorporate prior knowledge about the parameter.
- Answer: B) To equate sample moments to population moments.
- Which of the following is an example of a sufficient statistic?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In Bayesian estimation, what is the prior distribution?
- A) The distribution of the sample data.
- B) The distribution of the parameter before observing the sample data.
- C) The distribution of the parameter after observing the sample data.
- D) The distribution of the likelihood function.
- Answer: B) The distribution of the parameter before observing the sample data.
- Which of the following is a property of the maximum likelihood estimator (MLE)?
- A) It is always unbiased.
- B) It is always consistent.
- C) It is always efficient.
- D) It is always sufficient.
- Answer: B) It is always consistent.
- What does UMVUE stand for?
- A) Uniformly Minimum Variance Unbiased Estimator
- B) Uniformly Maximum Variance Unbiased Estimator
- C) Uniformly Minimum Variance Unbiased Estimation
- D) Uniformly Maximum Variance Unbiased Estimation
- Answer: A) Uniformly Minimum Variance Unbiased Estimator
- Which method of estimation is based on the principle of equating sample moments to population moments?
- A) Method of Moments
- B) Maximum Likelihood Estimation (MLE)
- C) Bayesian Estimation
- D) Least Squares Estimation
- Answer: A) Method of Moments
- What is the primary goal of Bayesian estimation?
- A) To maximize the likelihood function.
- B) To equate sample moments to population moments.
- C) To minimize the sum of squared errors.
- D) To update prior knowledge about the parameter with sample data.
- Answer: D) To update prior knowledge about the parameter with sample data.
- Which of the following is an example of a consistent estimator?
- A) Sample mean
- B) Sample variance
- C) Sample median
- D) Sample mode
- Answer: A) Sample mean
- In the context of estimation, what does efficiency refer to?
- A) The estimator’s ability to capture all the information about the parameter.
- B) The estimator’s ability to converge to the true parameter value as the sample size increases.
- C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- D) The estimator’s ability to be unbiased.
- Answer: C) The estimator’s ability to have the smallest variance among all unbiased estimators.
- Which of the following is a property of a sufficient statistic?
- A) It is always unbiased.
- B) It captures all the information about the parameter contained in the sample data.
- C) It converges to the true parameter value as the sample size increases.
- D) It has the smallest variance among all unbiased estimators.
- Answer: B) It captures all the information about the parameter contained in the sample data.
Hope this gives you a solid introduction to estimation! If you have any specific questions or need further details, feel free to ask.
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