Question

Let \(Y_{1}, \ldots, Y_{n}\) be independent exponential variables with hazard \(\lambda\) subject to Type I censoring at time \(c\). Show that the observed information for \(\lambda\) is \(D / \lambda^{2}\), where \(D\) is the number of the \(Y_{j}\) that are uncensored, and deduce that the expected information is \(i(\lambda \mid c)=n\\{1-\exp (-\lambda c)\\} / \lambda^{2}\) conditional on \(c\) Now suppose that the censoring time \(c\) is a realization of a random variable \(C\), whose density is gamma with index \(v\) and parameter \(\lambda \alpha\) : $$ f(c)=\frac{(\lambda \alpha)^{v} c^{v-1}}{\Gamma(v)} \exp (-c \lambda \alpha), \quad c>0, \alpha, v>0 $$ Show that the expected information for \(\lambda\) after averaging over \(C\) is $$ i(\lambda)=n\left\\{1-(1+1 / \alpha)^{-v}\right\\} / \lambda^{2} $$ Consider what happens when (i) \(\alpha \rightarrow 0\), (ii) \(\alpha \rightarrow \infty\), (iii) \(\alpha=1, v=1\), (iv) \(v \rightarrow \infty\) but \(\mu=v / \alpha\) is held fixed. In each case explain qualitatively the behaviour of \(i(\lambda)\).

Short answer

The expected information for \( \lambda \) is \( \frac{n\{1-(1+1/\alpha)^{-v}\}}{\lambda^{2}} \), reflecting mix of uncensored events and censoring.

Step-by-step solution

Understand Type I Censoring

Type I censoring occurs when observations are cut off at a certain time point, resulting in some complete and some censored observations. Here, if the event occurs before time \( c \), we observe it, otherwise, it is censored.

Derive Observed Information

For an exponential distribution, the likelihood function for censored data is a combination of the likelihoods for the uncensored and censored data. The observed information for \( \lambda \) is calculated using the second derivative of the log-likelihood function with respect to \( \lambda \). For uncensored, \( Y_j < c \), the contribution per observation to the log-likelihood is \( -\lambda Y_j + \log \lambda \). For censored, \( Y_j = c \), it is \( -\lambda c \). Deriving these gives that the observed information is \[ I_O(\lambda) = \frac{D}{\lambda^2} \] where \( D \) is the number of uncensored observations \( D \).

Calculate Expected Information Conditional on \( c \)

Given that a proportion of the observations will be uncensored and a proportion observed, observed minus the censoring, the expected information is the expected value of the observed information over the randomness induced by censoring. This gives: \[ i(\lambda \mid c)=n\{1-exp(-\lambda c)\} / \lambda^{2} \] This result reflects the fact that we're weighting only the contribution of uncensored observations.

Introduce Gamma Distributed Censoring Time

The censoring time \( c \) is assumed to follow a gamma distribution, which is defined by the formula given for \( f(c) \). Gamma distribution allows \( c \) to be variable, characterized by the shape \( v \) and scale \( \lambda \alpha \) parameters.

Compute Expected Information Averaging over \( C \)

To find the expectation over \( C \), integrate the conditional expected information over \( c \), weighted by the gamma density:\[ i(\lambda) = \int_{0}^{\infty} i(\lambda \mid c) f(c) \, dc = n \left\{ 1 - \left(1 + \frac{1}{\alpha}\right)^{-v} \right\} / \lambda^{2} \] This calculation accounts for the average effect of variable censoring times on the information about \( \lambda \).

Analyze Special Cases

Analysis of the given cases involves considering the limits and properties of the expected information formula:1. \( \alpha \rightarrow 0 \): Censoring almost never happens, so information equals \( n/\lambda^2 \).2. \( \alpha \rightarrow \infty \): Censoring happens immediately, giving no information, \( i(\lambda) = 0 \).3. \( \alpha = 1, v = 1 \): Results in a moderate level of censoring; use the formula directly.4. \( v \rightarrow \infty \) while \( \mu = v/\alpha\): Information reflects a balance, with proportion \( \mu/(1+\mu) \) of censoring influencing \( i(\lambda) \).

Key concepts

Type I Censoring

Type I Censoring is a statistical technique used when dealing with incomplete data. Imagine conducting a study measuring time until an event occurs. **Type I Censoring** occurs when the observation period ends at a predefined time.
If the event hasn't occurred by this time, the data is considered censored.
This approach is useful for controlling study duration or when resources are limited. This censoring splits data points into:

• **Uncensored observations** - where the event occurred before the cut-off time.
• **Censored observations** - the event did not occur before the cut-off.

Understanding when and how Type I Censoring occurs helps in analyzing censored data correctly and enhances decision-making regarding parameter estimation.

Observed Information

In statistical analysis, **Observed Information** is crucial for estimating parameters like the rate in exponential distributions. Observed information indicates how much data support our parameter estimates using actual observations. When dealing with censored data, this involves distinguishing between censored and uncensored data to refine our parameter estimation.
Here, the exponential distribution's properties simplify calculations using the log-likelihood function. For every uncensored observation, the data's contribution is calculated by looking at \(-\lambda Y_j + \log \lambda\), while censored observations contribute \(-\lambda c\).
The second derivative of this log-likelihood function with respect to \(\lambda\), evaluated at the estimated value, provides the observed information.
For Type I Censoring, it simplifies to:\[I_O(\lambda) = \frac{D}{\lambda^2}\]where \(D\) is the number of uncensored data points. This approach provides a measure of precision for our \(\lambda\) estimate, relying solely on the observed data that's not censored.

Gamma Distribution

The **Gamma Distribution** is versatile and extremely useful for modeling wait times and event counts, which perfectly aligns with studying censoring time. It's expressed with two parameters: the shape \(v\) and the rate \(\lambda \alpha\).
This distribution is employed when the censoring time isn't fixed but follows a probability distribution itself. If censoring occurs according to a gamma distribution, the statistical analysis incorporates the variability
across possible censoring times rather than a single cut-off point. The density function for gamma distribution is given by:\[f(c)=\frac{(\lambda \alpha)^{v} c^{v-1}}{\Gamma(v)} \exp (-c \lambda \alpha), \quad c>0, \alpha, v>0\]This function describes how likely different durations up to a particular time are, providing more nuance in evaluating expected information in the presence of natural variation.

Expected Information

**Expected Information** is a key metric in statistical inference, providing insight into how much uncertainty is reduced after making observations, even under a random censoring scenario. When censoring is not fixed, but a random variable with known distribution, expected information becomes the average information gained over all possible censoring scenarios.
For exponential distributions with gamma-distributed censoring, expected information is calculated by integrating observed information over the probability distribution of censoring times.
This results in the equation:\[i(\lambda) = n\left\{1-(1+1 / \alpha)^{-v}\right\} / \lambda^{2}\]This equation highlights how changes in gamma distribution parameters affect the information gained. For example, as \(\alpha\) approaches zero, most observation time occurs.
Therefore, information approaches its maximum value of \(n/\lambda^2\). Conversely, if \(\alpha\) increases indefinitely, censoring occurs immediately, reflecting in zero information. Expected information findings thus guide evaluating parameter precision in varying censor contexts.