# Basu's Theorem Quiz

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| By Zlatan Aleksic
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Zlatan Aleksic
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Jump into our Basu's Theorem Quiz and put your skills to the test. This quiz is all about one of the key ideas in statistics, Basu's Theorem, which tells us about certain types of independence in statistics. We've put together a series of questions to see how well you understand this theorem and its applications. Whether you've just started learning about it or you've been studying statistics for a while, this quiz is a great way to check what you know and learn a bit more along the way.

Our Basu's Theorem Quiz covers everything from the basic concepts to Read moremore detailed questions about how and when to use the theorem in real statistical problems. Each question will challenge you to think and apply what you know in a clear way. Give our quiz a try and see how much you really know about Basu's Theorem!

## Basu's Theorem Questions and Answers

• 1.

### What does Basu's Theorem primarily relate to?

• A.

Independence

• B.

Bias

• C.

Efficiency

• D.

Variance

A. Independence
Explanation
Basu's Theorem primarily relates to the independence of statistics. It specifically states that any ancillary statistic, which by definition does not depend on the parameter, is independent of every complete sufficient statistic. This independence is critical as it allows statisticians to use ancillary statistics for purposes other than parameter estimation, such as validating models or checking experimental designs, without influencing the estimates of parameters derived from sufficient statistics.

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• 2.

### Which type of statistic is central to Basu's Theorem?

• A.

Ancillary

• B.

Sufficient

• C.

Biased

• D.

Robust

A. Ancillary
Explanation
The type of statistic central to Basu's Theorem is the ancillary statistic. Ancillary statistics are those that are not influenced by the parameters of a model. Basu’s Theorem highlights the unique property of these statistics being independent of complete sufficient statistics, thus they can provide useful information about the study design or data collection process without affecting parameter estimation.

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• 3.

### What must a statistic be to satisfy Basu's Theorem?

• A.

Complete

• B.

Minimal

• C.

Unbiased

• D.

Efficient

A. Complete
Explanation
For a statistic to satisfy Basu's Theorem, it must be complete. A complete statistic is one for which no non-trivial function of it has an expectation that is zero for all parameter values unless the function itself is zero almost surely. This completeness is essential for ensuring that the statistic captures all the information about the parameter, which is a prerequisite for establishing the independence stipulated by Basu’s Theorem.

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• 4.

### Which characteristic defines an ancillary statistic?

• A.

Parameter-independent

• B.

Variable-dependent

• C.

Consistent

• D.

Sufficient

A. Parameter-independent
Explanation
An ancillary statistic is defined by its characteristic of being parameter-independent. This means its distribution does not vary with the parameter of the underlying model. Such a statistic is valuable because it remains constant across different parameter values, making it useful for assessing the variability and structure of the data without confounding the effects due to parameter changes.

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• 5.

### What ensures a statistic's independence according to Basu?

• A.

Completeness

• B.

Bias

• C.

Efficiency

• D.

None

A. Completeness
Explanation
According to Basu's Theorem, the independence of a statistic, specifically an ancillary statistic from a complete sufficient statistic, is ensured by the completeness of the sufficient statistic. Completeness here is critical because it confirms that the sufficient statistic has absorbed all information about the parameter, thus allowing the ancillary statistic to be genuinely independent in its distribution, free from the influence of parameter variations.

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• 6.

### Which term best describes a statistic unaffected by parameters?

• A.

Ancillary

• B.

Sufficient

• C.

Efficient

• D.

Consistent

A. Ancillary
Explanation
Ancillary statistics are best described as being unaffected by parameters. This independence from parameters means that the distribution of an ancillary statistic remains constant regardless of changes in the parameter values of the population from which data are drawn. This property is essential for using ancillary statistics to provide insights into aspects of the data collection process or experimental design without biasing statistical analyses focused on parameter estimation.

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• 7.

### What does Basu's Theorem guarantee about complete and ancillary statistics?

• A.

Independence

• B.

Dependence

• C.

Correlation

• D.

None

A. Independence
Explanation
Basu's Theorem guarantees the independence of complete sufficient and ancillary statistics. This theorem establishes that these two types of statistics do not affect each other. This independence is crucial because it means that the data can be partitioned into parts that provide all necessary information about the parameters and parts that can be used for other purposes like checking the randomness of the sample.

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• 8.

### Which property is not influenced by Basu’s Theorem?

• A.

Bias

• B.

Independence

• C.

Efficiency

• D.

Completeness

A. Bias
Explanation
Bias is not influenced by Basu’s Theorem, which deals with the relationship between independence, sufficiency, and ancillarity. Bias in an estimator relates to whether or not the expected value of the estimator equals the parameter being estimated, irrespective of the estimator’s sufficiency or its relationship with ancillary statistics. Therefore, Basu’s Theorem does not address issues of bias directly but rather focuses on the statistical independence properties.

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• 9.

### What feature distinguishes ancillary from sufficient statistics?

• A.

Independence

• B.

Efficiency

• C.

Sufficiency

• D.

Completeness

A. Independence
Explanation
The feature that distinguishes ancillary from sufficient statistics is their independence from the parameters of the model. While sufficient statistics are valued for containing all information necessary for parameter estimation, ancillary statistics are notable for their lack of dependency on these parameters, providing useful side information about the data collection process or the experimental design without overlapping with the information contained in sufficient statistics.

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• 10.

### What is not a direct result of Basu's Theorem?

• A.

Correlation

• B.

Independence

• C.

Bias reduction

• D.

All of the above

C. Bias reduction
Explanation
Bias reduction is not a direct result of Basu’s Theorem, which focuses on establishing the independence of ancillary statistics from complete sufficient statistics. Basu’s Theorem clarifies the roles and properties of different types of statistics but does not involve mechanisms that directly reduce bias in estimators. Instead, its contribution lies in enhancing understanding of statistical independence and its implications for using different statistical tools effectively within inference processes.

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• Current Version
• Aug 07, 2024
Quiz Edited by
ProProfs Editorial Team
• Mar 14, 2018
Quiz Created by
Zlatan Aleksic

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