Statistics MCQs
Topic Notes: Statistics
MCQs and preparation resources for competitive exams, covering important concepts, past papers, and detailed explanations.
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
1
If the conditional distribution of a random sample given a statistic S is independent of the parameter theta, what is S called?
Answer:
Minimal sufficient statistic
The definition provided describes a sufficient statistic. While the option 'Minimal sufficient statistic' is selected, technically any sufficient statistic satisfies the condition of independence from the parameter in its conditional distribution. A minimal sufficient statistic is a specific type of sufficient statistic that is a function of all other sufficient statistics. There is a potential ambiguity here as the definition applies to all sufficient statistics, not just minimal ones.
2
In the context of sufficiency, which component must be absent from the conditional density h(x|T) for a statistic T to be considered sufficient?
Answer:
Parameter
According to the Fisher-Neyman Factorization Theorem, a statistic T is sufficient for a parameter if the conditional distribution of the data given T does not depend on the parameter. If the conditional density h(x|T) involves the parameter, then T does not capture all the information about the parameter contained in the sample.
3
In a uniform distribution, what property does the largest observation (Yn) possess?
Answer:
Complete
In the context of statistical inference for a uniform distribution, the maximum order statistic (Yn) is a sufficient and complete statistic for the parameter. Completeness is a property that ensures that the distribution of the statistic provides enough information to uniquely identify the parameter, which is a fundamental concept in estimation theory.
4
What condition must be met for a statistic s(x) to be considered a sufficient estimator for a parameter theta?
Answer:
Independent of parameter
A statistic s(x) is sufficient for a parameter theta if the conditional distribution of the sample data, given the value of the statistic s(x), does not depend on the parameter theta. This means that the statistic captures all the information available in the sample about the parameter, making any additional information in the sample redundant for the purpose of estimating theta.
5
What is the relationship between a Bayes' estimator and sufficient statistics?
Answer:
Minimal sufficient statistic
According to the Rao-Blackwell theorem and properties of Bayesian estimation, the Bayes' estimator is a function of the sufficient statistic. More specifically, it is often expressed in terms of the minimal sufficient statistic, which provides the most compact summary of the data containing all information about the parameter.
6
If the sum of all sample observations is a sufficient statistic for the population mean, what can be concluded about the sample mean?
Answer:
Sufficient
According to the Fisher-Neyman Factorization Theorem, if a statistic T is sufficient for a parameter, then any one-to-one function of T is also a sufficient statistic. Since the sample mean is a linear transformation (a one-to-one function) of the sum of observations, it inherits the property of sufficiency for the population mean.
7
If a statistic is sufficient for a parameter, what other property does it necessarily possess?
Answer:
Sufficient
By definition, if a statistic is sufficient for a parameter, it captures all the information in the sample about that parameter. The question asks for a property that is guaranteed; since sufficiency is the premise, the statistic is by definition sufficient. It does not necessarily have to be complete or unbiased.
8
What is the mathematical representation of the Neyman-Fisher Factorization Theorem?
Answer:
L(x; ?) = g (S ; ?) h(x)
The Neyman-Fisher Factorization Theorem states that a statistic T(X) is sufficient for a parameter theta if and only if the likelihood function L(x; theta) can be factored into a product of a function g(T(x), theta) and a function h(x) that does not depend on theta.
9
The Neyman-Fisher Factorization Theorem is primarily associated with which concept in statistical inference?
Answer:
Theorem of sufficient estimators
The Neyman-Fisher Factorization Theorem provides a necessary and sufficient condition for a statistic to be a sufficient statistic for a parameter. It states that a statistic is sufficient if and only if the likelihood function can be factored into a product of two functions, one depending on the data and the parameter, and the other depending only on the data.
10
According to the Fisher-Neyman Factorization Theorem, if the joint probability density function can be written as f(x; theta) = g(theta-hat; theta) * h(x), what is theta-hat?
Answer:
Sufficient
The Fisher-Neyman Factorization Theorem provides a necessary and sufficient condition for a statistic to be sufficient for a parameter. If the joint density of the sample can be factored into a product of a function that depends on the data only through the statistic theta-hat and a function that does not depend on the parameter theta, then theta-hat is a sufficient statistic for theta.