A sufficient statistic is a statistic that summarizes all of the information in a sample about a chosen parameter. For example, the sample mean, x̄, estimates the Population mean, μ. x̄ is a sufficient statistic if it retains all of the information about the population mean that was contained in the original data points.
Let X1, X2, …, Xn be a random sample from a Probability distribution with unknown parameter θ. Then, the statistic:
Y=u(X1,X2,…,Xn)
is said to be sufficient for θ if the conditional distribution of X1, X2, …, Xn, given the statistic Y, does not depend on the parameter θ.
Guidelines to solve questions
In each of the questions below consists of a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements and give answer.
If the data in statement I alone are sufficient to answer the question, while the data in statement II alone are not sufficient to answer the question
If the data in statement II alone are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question
If the data either in statement I alone or in statement II alone are sufficient to answer the question
If the data given in both statements I and II together are not sufficient to answer the question and
If the data in both statements I and II together are necessary to answer the question.
Example of Data sufficiency
Question: In which year was Rahul born ?
Statements:
Rahul at present is 25 years younger to his mother.
Rahul’s brother, who was born in 1964, is 35 years younger to his mother.
From both I and II, we find that Rahul is (35 – 25) = 10 years older than his brother, who was born in 1964. So, Rahul was born in 1954.
,
In statistics, sufficiency is a property of a statistic that summarizes a dataset in such a way that all the information about the distribution of the data that is relevant for a particular statistical inference is contained in the statistic. Sufficient statistics are often used in hypothesis testing, estimation, and prediction.
A sufficient statistic is a function of the data that contains all the information about the distribution of the data that is relevant for a particular statistical inference. In other words, a sufficient statistic is a statistic that summarizes the data in such a way that no other statistic can provide any additional information about the distribution of the data.
A complete statistic is a statistic that is sufficient for the parameter of interest and for which no other statistic is sufficient. In other words, a complete statistic is a statistic that summarizes the data in such a way that all the information about the parameter of interest is contained in the statistic, and no other statistic can provide any additional information about the parameter of interest.
A minimal sufficient statistic is a sufficient statistic that is not a function of any other sufficient statistic. In other words, a minimal sufficient statistic is a statistic that summarizes the data in such a way that all the information about the distribution of the data that is relevant for a particular statistical inference is contained in the statistic, and no other statistic can provide any additional information about the distribution of the data that is not already contained in the minimal sufficient statistic.
An ancillary statistic is a statistic that is independent of the parameter of interest. In other words, an ancillary statistic is a statistic that does not provide any information about the parameter of interest.
An exponential family is a family of probability distributions that can be written in the form
for some functions $\eta$, $t_1, \ldots, t_k$, and $\beta_1, \ldots, \beta_k$.
Suffciency is important in statistics because it allows us to make inferences about the parameter of interest without having to consider all the information in the data. Sufficient statistics can be used in hypothesis testing, estimation, and prediction.
In hypothesis testing, sufficient statistics can be used to construct tests that are more powerful than tests that do not use sufficient statistics. In estimation, sufficient statistics can be used to construct estimators that are more efficient than estimators that do not use sufficient statistics. In prediction, sufficient statistics can be used to construct predictors that are more accurate than predictors that do not use sufficient statistics.
Suffciency is a powerful tool in statistics that can be used to make inferences about the parameter of interest without having to consider all the information in the data. Sufficient statistics can be used in hypothesis testing, estimation, and prediction.
In machine Learning, sufficiency is often used in the context of feature selection. Feature selection is the process of identifying a subset of features from a dataset that are most relevant for a particular task. Sufficient statistics can be used to identify a subset of features that are sufficient for a particular task. This can be done by using a greedy algorithm that starts with all the features and then iteratively removes features that are not necessary for the task.
Suffciency is also used in machine learning in the context of dimensionality reduction. Dimensionality reduction is the process of reducing the number of features in a dataset without losing too much information. Sufficient statistics can be used to identify a lower-dimensional representation of the data that is still sufficient for a particular task. This can be done by using a projection method such as principal component analysis or singular value decomposition.
What is data sufficiency?
Data sufficiency is the concept that a set of data is sufficient to answer a question. In statistics, this means that the data contains enough information to make a confident inference about the population.
What are the different types of data sufficiency?
There are two main types of data sufficiency: statistical sufficiency and decision theoretic sufficiency. Statistical sufficiency is concerned with whether the data contains enough information to estimate a population parameter. Decision theoretic sufficiency is concerned with whether the data contains enough information to make a decision about a population parameter.
What are the different methods for assessing data sufficiency?
There are a number of different methods for assessing data sufficiency. One common method is to use a likelihood ratio test. This test compares the likelihood of the data under two different models, one of which is the null model and the other of which is the alternative model. If the likelihood ratio test is significant, then the data is not sufficient to support the null model.
What are the different applications of data sufficiency?
Data sufficiency is used in a variety of different applications, including:
What are the different challenges associated with data sufficiency?
One challenge associated with data sufficiency is that it can be difficult to determine whether a set of data is sufficient to answer a question. This is because the amount of data needed to answer a question can vary depending on the question itself and the characteristics of the population.
Another challenge associated with data sufficiency is that it can be difficult to collect enough data to be sure that the data is sufficient. This is because data collection can be expensive and time-consuming.
What are the different future directions for research in data sufficiency?
One future direction for research in data sufficiency is to develop new methods for assessing data sufficiency. These methods could be more efficient and more accurate than current methods.
Another future direction for research in data sufficiency is to develop new applications for data sufficiency. These applications could include new methods for statistical inference, decision making, data mining, and machine learning.
Sure. Here are some multiple choice questions about statistics:
A survey of 1000 people found that 60% of them support a new law. What is the margin of error for this survey? (A) 3% (B) 4% (C) 5% (D) 6%
A coin is tossed 10 times. What is the probability of getting 6 heads? (A) 1/1024 (B) 1/1023 (C) 1/1022 (D) 1/1021
A study found that people who eat breakfast are more likely to be overweight than people who don’t eat breakfast. What is the cause of this correlation? (A) Eating breakfast causes people to be overweight. (B) Being overweight causes people to skip breakfast. (C) There is no cause-and-effect relationship between eating breakfast and being overweight. (D) There is a third variable that is causing both eating breakfast and being overweight.
A study found that people who take vitamin C are less likely to get the common cold. What is the conclusion of this study? (A) Vitamin C prevents the common cold. (B) People who take vitamin C are more likely to wash their hands, which prevents the common cold. (C) There is no cause-and-effect relationship between vitamin C and the common cold. (D) There is a third variable that is causing both vitamin C intake and the common cold.
A study found that people who exercise regularly are less likely to get cancer. What is the conclusion of this study? (A) Exercise prevents cancer. (B) People who exercise regularly are more likely to eat a healthy diet, which prevents cancer. (C) There is no cause-and-effect relationship between exercise and cancer. (D) There is a third variable that is causing both exercise and cancer.
A study found that people who are married are happier than people who are single. What is the conclusion of this study? (A) Marriage causes happiness. (B) Happy people are more likely to get married. (C) There is no cause-and-effect relationship between marriage and happiness. (D) There is a third variable that is causing both marriage and happiness.
A study found that people who watch more TV are more likely to be obese. What is the conclusion of this study? (A) Watching TV causes obesity. (B) Obese people are more likely to watch TV. (C) There is no cause-and-effect relationship between watching TV and obesity. (D) There is a third variable that is causing both watching TV and obesity.
A study found that people who drink more coffee are more likely to have heart disease. What is the conclusion of this study? (A) Drinking coffee causes heart disease. (B) People with heart disease are more likely to drink coffee. (C) There is no cause-and-effect relationship between coffee consumption and heart disease. (D) There is a third variable that is causing both coffee consumption and heart disease.
A study found that people who eat more fruits and vegetables are less likely to get cancer. What is the conclusion of this study? (A) Eating fruits and vegetables prevents cancer. (B) People who eat more fruits and vegetables are more likely to exercise, which prevents cancer. (C) There is no cause-and-effect relationship between eating fruits and vegetables and cancer. (D) There is a third variable that is causing both eating fruits and vegetables and cancer.
A study found that people who are born in the winter are more likely to be left-handed. What is the conclusion of this study? (A) Being born in the winter causes left-handedness. (B) Left-handed people are more likely to be born in the winter. (C) There is no cause-and-effect relationship between birth month and handedness. (D) There is a third variable that is causing both birth month and handedness.