U Statistics

Summary

This lecture explores the concept of estimability in statistics, focusing on the degree of a parameter's estimability within a family of distributions. It introduces the idea that a parameter is estimable to the degree R if a sample size of R is sufficient to estimate it. The lecture discusses the construction of symmetric kernels and their role in creating unbiased estimators, exemplified through U-statistics. Practical applications include estimating population mean and variance, with the sample mean and sample variance serving as U-statistics. The lecture concludes with a look ahead to survival analysis in future sessions.

Takeaways

• 🔢 Estimability of a parameter is the capability to estimate that parameter from a given sample size.
• 🎲 The degree of estimability (R) indicates the minimum sample size required to estimate a parameter.
• 📊 A parameter is considered estimable if there exists a function (estimator) that, when applied to a sample, yields an unbiased estimate of the parameter.
• 📚 The concept of a 'kernel' function is introduced as a way to create unbiased estimators from sample data.
• 🔄 The importance of symmetric kernels is emphasized to ensure that the order of observations does not affect the estimation.
• 🧮 U-statistics are introduced as a method to estimate parameters using symmetric kernels applied to all possible combinations of sample data.
• 📉 U-statistics can be used to estimate various parameters, such as the mean or variance, depending on the chosen kernel function.
• 📝 The script provides examples of how to construct symmetric kernels and U-statistics for estimating the mean and variance of a distribution.
• 🔍 The expectation of U-statistics is shown to be equal to the parameter being estimated, confirming their unbiasedness.
• 📚 The lecture concludes with a discussion on the application of U-statistics in survival analysis in future lectures.

Q & A

What is the concept of estimability degree of a parameter?

The estimability degree of a parameter, denoted as 'R', refers to the minimum sample size required to estimate that parameter within a given family of distributions. If a parameter 'gamma' is estimable of degree 'R', it means that a sample size of at least 'R' is necessary to obtain an estimate for 'gamma'.

How is a parameter considered estimable?

A parameter is considered estimable if there exists a function, often referred to as a kernel, that when applied to a sample of the appropriate size, results in an expectation equal to the parameter's value. This function must be symmetric to ensure that the order of observations does not affect the estimate.

What is a symmetric kernel in the context of parameter estimation?

A symmetric kernel is a function that yields the same result regardless of the order of the arguments it receives. It is used in the context of parameter estimation to ensure that the estimator is unbiased and does not depend on the order in which the sample observations are taken.

Why is it necessary to use a symmetric kernel for estimating parameters?

Using a symmetric kernel is necessary to ensure that the estimator of a parameter is unbiased and that the order of observations in the sample does not affect the estimate. This is important for maintaining consistency and reliability in the estimation process.

What is the role of 'H' in the estimation process?

In the script, 'H' represents a symmetric kernel function used to estimate the parameter 'gamma'. It is applied to all possible combinations of the sample of size 'R', and the average of these applications is used to form a U-statistic, which serves as an unbiased estimator of 'gamma'.

How is the U-statistic defined for a parameter with estimability degree 'R'?

The U-statistic for a parameter with estimability degree 'R' is defined as the average of the symmetric kernel function 'H' applied to all possible combinations of 'R' observations from a larger sample of size 'n'. It is calculated as 1/(n choose R) times the sum of 'H' over all such combinations.

What is the significance of the sample size 'n' in relation to the estimability degree 'R'?

The sample size 'n' must be at least as large as the estimability degree 'R' to ensure that the parameter can be estimated. If 'n' is less than 'R', the parameter cannot be estimated based on that sample. If 'n' is greater than or equal to 'R', the parameter can be estimated, and the precision of the estimation may increase with larger sample sizes.

Can you provide an example of a kernel function for estimating the mean of a distribution?

Yes, for estimating the mean ('gamma') of a distribution, a simple kernel function could be the identity function, where 'a_star(x1) = x1'. The symmetric kernel 'H' in this case would be 'H(x) = x', and the U-statistic would be the sample mean, which is an unbiased estimator of the population mean.

How is the variance of a distribution estimated using U-statistics?

To estimate the variance ('Sigma square') of a distribution, a kernel function such as 'a_star(x1, x2) = (x1 - x2)^2' can be used. The symmetric kernel 'H' would then average the values of 'a_star' over all permutations of the sample pairs, resulting in a U-statistic that is an unbiased estimator of the variance.

What is the practical implication of using U-statistics in survival analysis?

In survival analysis, U-statistics are used to estimate parameters such as survival probabilities or hazard rates. The practical implication is that U-statistics provide a robust method for making inferences about the survival distribution even when the data is subject to censoring or other forms of incompleteness.