Question

Let \(Y\) be binomial with probability \(\pi=e^{\lambda} /\left(1+e^{\lambda}\right)\) and denominator \(m\). (a) Show that \(m-Y\) is binomial with \(\lambda^{\prime}=-\lambda\). Consider $$ \tilde{\lambda}=\log \left(\frac{Y+c_{1}}{m-Y+c_{2}}\right) $$ as an estimator of \(\lambda\). Show that in order to achieve consistency under the transformation \(Y \rightarrow m-Y\), we must have \(c_{1}=c_{2}\) (b) Write \(Y=m \pi+\sqrt{m \pi(1-\pi)} Z\), where \(Z=O_{p}(1)\) for large \(m\). Show that $$ \mathrm{E}\\{\log (Y+c)\\}=\log (m \pi)+\frac{c}{m \pi}-\frac{1-\pi}{2 m \pi}+O\left(m^{-3 / 2}\right) $$ Find the corresponding expansion for \(\mathrm{E}\\{\log (m-Y+c)\\}\), and with \(c_{1}=c_{2}=c\) find the value of \(c\) for which \(\tilde{\lambda}\) is unbiased for \(\lambda\) to order \(m^{-1}\). What is the connection to the empirical logistic transform? (Cox, 1970, Section 3.2)

Short answer

Set \( c_1 = c_2 = \frac{1}{2} \) for a consistent, unbiased estimator. The connection is through bias-corrected empirical logistic transform.

Step-by-step solution

Define the Binomial Complement

Given a binomial random variable \( Y \) with parameters \( m \) and success probability \( \pi \), we have \( Y \sim \text{Binomial}(m, \pi) \). The complement \( m-Y \) is also a binomial random variable, specifically \( m-Y \sim \text{Binomial}(m, 1-\pi) \).

Transform Parameters

Express the success probability \( \pi \) in terms of \( \lambda \): \( \pi = \frac{e^{\lambda}}{1 + e^{\lambda}} \). Then, for \( \-\lambda \), the success probability becomes \( 1 - \pi = \frac{1}{1 + e^{\lambda}} \), consistent with \( \pi \) under \( \lambda^{\prime} = -\lambda \).

Consistency of Estimator

Consider the estimator \( \tilde{\lambda} = \log\left( \frac{Y+c_1}{m-Y+c_2} \right) \). For it to be consistent when \( Y \rightarrow m-Y \), the transformation should not affect the estimator's distribution. This gives the condition \( c_1 = c_2 \).

Approximate Y using Central Limit Theorem

For large \( m \), approximate \( Y \) as \( Y = m\pi + \sqrt{m\pi(1-\pi)}Z \) where \( Z = O_p(1) \) (a convergence rate notation).

Expand Logarithmic Expectation

Given \( \mathrm{E}[\log(Y+c)] \), apply a Taylor expansion around \( m\pi \):\[ \mathrm{E}[\log(Y+c)] = \log(m\pi) + \frac{c}{m\pi} - \frac{1-\pi}{2m\pi} + O\left(m^{-3/2}\right) \]

Log Transformation for Complement

Find the expectation for \( \log(m-Y+c) \) similarly:\[ \mathrm{E}[\log(m-Y+c)] = \log(m(1-\pi)) + \frac{c}{m(1-\pi)} - \frac{\pi}{2m(1-\pi)} + O\left(m^{-3/2}\right) \]

Equate Expectations for Unbiased Estimator

To ensure \( \tilde{\lambda} \) is unbiased, set the expansions equal and solve for \( c \):From consistency, \( c \) should balance the bias introduced by switching \( Y \) to \( m-Y \):\( c = \frac{1}{2} \).

Connection to Empirical Logistic Transform

The form of \( \tilde{\lambda} \) relates to the empirical logistic transform, which involves taking the natural log of odds ratios adjusted by constants for bias correction. Setting \( c_1 = c_2 = \frac{1}{2} \) aligns the estimator with the typical bias-corrected empirical logistic estimator.

Key concepts

Binomial Distribution

In statistical models, the **Binomial Distribution** is a key concept. It describes the number of successes in a fixed number of independent and identically distributed Bernoulli trials, each with its own success probability. Think of it like flipping a biased coin multiple times and counting how many times it lands on heads.

For a more formal definition, if we say a random variable \( Y \) follows a Binomial Distribution with parameters \( m \) (the number of trials) and \( \pi \) (the probability of success on each trial), we denote it as \( Y \sim \text{Binomial}(m, \pi) \). The probability mass function for \( Y \) is given by:

• \( P(Y = k) = \binom{m}{k} \pi^k (1-\pi)^{m-k} \)

Here, \( k \) is the number of successes and \( \binom{m}{k} \) is the binomial coefficient. This function tells us the probability of achieving exactly \( k \) successes out of \( m \) trials.

In some exercises, we might deal with the **complement** of a binomial random variable, such as \( m-Y \). This simply switches the roles of success and failure, leading to another binomial variable with success probability \( 1-\pi \).

Estimator Consistency

**Estimator Consistency** is a crucial property in statistics. An estimator is consistent if, as the sample size increases, the estimator converges in probability to the true parameter value it estimates. This assures us that given enough data, the estimator will yield results very close to the actual parameter value.

Consider an estimator \( \tilde{\lambda} = \log \left( \frac{Y+c_1}{m-Y+c_2} \right) \) from our exercise. This is used to estimate \( \lambda \). For the estimator to be consistent, the transformation \( Y \rightarrow m-Y \), which inverts successes and failures, should not affect its expected value. That's why we require that \( c_1 = c_2 \).

These constants equalizing ensures that the estimator is unbiased, even when switching \( Y \) to its complement. This lack of bias is essential for consistency because it means that the average value of the estimator equals the true parameter over many samples.

Empirical Logistic Transform

The **Empirical Logistic Transform** is a fascinating tool in statistical analysis. It involves transforming the odds ratio for a more intuitive and useful interpretation.

In our context, we have an estimator \( \tilde{\lambda} \) designed to estimate a parameter \( \lambda \). This estimator is expressed as the natural log of a transformed odds ratio. Specifically:

• \( \tilde{\lambda} = \log \left( \frac{Y+c_{1}}{m-Y+c_{2}} \right) \)

This form resembles the empirical logistic transform. It is often used to transition from raw data such as counts or probabilities into a domain where linear relationships are more evident. This helps simplify statistical inferences.

Choosing \( c_1 = c_2 = \frac{1}{2} \) corrects for potential bias and aligns with conventional approaches in logistic regression. This makes the estimator \( \tilde{\lambda} \) unbiased up to the order of \( m^{-1} \), an improvement over simpler transforms. All in all, it enhances the reliability of parameter estimation when working with logistic models.