Question

The pdf depicted in Figure \(7.9 .1\) is given by $$f_{m_{2}}(x)=e^{x}\left(1+m_{2}^{-1} e^{x}\right)^{-\left(m_{2}+1\right)}, \quad-\infty0\), (the pdf graphed is for \(m_{2}=0.1\) ). This is a member of a large family of pdfs, \(\log F\) -family, which are useful in survival (lifetime) analysis; see Chapter 3 of Hettmansperger and McKean (1998). (a) Let \(W\) be a random variable with pdf \((7.9 .2) .\) Show that \(W=\log Y\), where \(Y\) has an \(F\) -distribution with 2 and \(2 m_{2}\) degrees of freedom. (b) Show that the pdf becomes the logistic (6.1.8) if \(m_{2}=1\). (c) Consider the location model where $$X_{i}=\theta+W_{i} \quad i=1, \ldots, n$$ where \(W_{1}, \ldots, W_{n}\) are iid with pdf \((7.9 .2)\). Similar to the logistic location model, the order statistics are minimal sufficient for this model. Show, similar to Example \(6.1 .4\), that the mle of \(\theta\) exists.

Short answer

a) The pdf of F-distribution and the transformed Y are identical, confirmimg that Y follows an F-distribution. b) Substituting \(m_2=1\) converts the given pdf into the logistic distribution. c) The MLE of \(\theta\) exists as it's proven to be the sample median of the \(X_i - W_i\)'s.

Step-by-step solution

Identification of the random variable transformation

We need to show that a random variable \(W\) having the given pdf (7.9.2) can be written as \(W = log Y\), where \(Y\) follows an F-distribution with 2 and \(2m_2\) degrees of freedom. For this, we take \(Y = e^W\). Hence, the pdf of \(Y\) will be given by \(g_Y(y) = e^{-y}(1 + m_2^{-1}y)^{-\left(m_2+1\right)}\) for \(y > 0\) and \(0\) elsewhere.

Comparison of pdfs

To show that \(Y\) has an F-distribution, its pdf needs to be compared with the pdf of an F-distribution, \(f(y) = 2m_2^{2m_2} / B(m_2, 1) * y^{2m_2-1} *(1+m_2 y)^{-2m_2}\) for \(y > 0\). After simplifying the pdf of \(Y\), we realize that the pdfs of \(Y\) and \(f(y)\) match, implying that \(Y\) follows an F-distribution with 2 and \(2m_2\) degrees of freedom.

Changing the parameters

To show that the given pdf becomes a logistic distribution when \(m_2=1\), we need to substitute \(m_2\) with \(1\) in the pdf function, giving us \(f(x) = 1/(1+e^{-x})\) which is the pdf of the logistic distribution.

Establishing the location model

We need to consider the model \(X_i = \theta + W_i\), where \(W_i\) is a random variable having the given pdf and the \(W_i\)'s are IID. The task is to show that the MLE of \(\theta\) exists under conditions similar to Example 6.1.4.

Identifying MLE

According to Example 6.1.4, for order statistics which are minimal sufficient, the MLE of the intercept \(\theta\) is the sample median. Here, as the \(W_i\)'s are IID and the order statistics are minimal sufficient, the MLE of \(\theta\) will be the sample median of the \(X_i - W_i\)'s, thus showing that the MLE of \(\theta\) exists.

Key concepts

Log F-distribution

The Log F-distribution is a continuous probability distribution commonly used in the field of survival analysis. It is derived from the logarithm of a variable that follows an F-distribution. In our exercise, the probability density function (pdf) of the log F-distribution is defined as: \[ f_{m_{2}}(x)=e^{x}\left(1+m_{2}^{-1} e^{x}\right)^{-\left(m_{2}+1\right)}, \quad-\infty 0\) is a parameter that influences the shape of the distribution. A particular case is when \(m_{2} = 0.1\), which is the scenario depicted in the given exercise. To interpret these properties better, consider that this distribution can describe situations where we want to model the natural logarithm of variables that follow an F-distribution with given degrees of freedom. Thus, understanding the log F-distribution helps us gain insights into the underlying behavior of these random processes, especially in survival analysis contexts.

Survival analysis

Survival analysis is a branch of statistics focused on analyzing the expected duration of time until one or more events happen, such as death in biological organisms or failure in mechanical systems. It is especially useful in medical research and reliability engineering. This analysis uses probability distributions, like the Log F-distribution in our exercise, to model the time until events happen. The Log F-distribution is part of a larger family of distributions that are particularly suited to survival data. These probability density functions (pdf) help to better understand and model the time-dependent nature of such data. There are several applications of survival analysis:

• Estimating the survival time for patients in medical research
• Investigating the reliability and failure time of mechanical components
• Understanding the profitability period in business settings
• Assessing time-to-failure data for risk management

The objective is often to estimate the survival function or hazard function, which provides helpful insights into the probability that a subject survives past a certain time. Deploying distributions like the Log F-distribution can enhance the understanding and predictability of these phenomena.

Maximum Likelihood Estimation

Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a statistical model. The main aim is to find the parameter values that maximize the likelihood of making the observations given the model. MLE operates on the principle that the observed data is most probable under the true set of parameters. In our exercise, MLE is employed to estimate \(\theta\) in a location model where \(X_i = \theta + W_i\) with \(W_i\) following a specified distribution. Here's how it works:

• The likelihood function is composed using observed data points.
• This function is subsequently maximized concerning the model parameters.
• Tools such as calculus often assist in finding where this maximum occurs.

For symmetric distributions like the logistic, it often turns out that the maximum likelihood estimator for location parameters (like \(\theta\)) is the sample median of the observed data. This is because, in symmetric settings, the median balances the data to minimize the sum of absolute deviations from the central value, thus providing the most likely estimate. Understanding MLE's robust and consistent nature helps ensure the parameter estimates are reliable and valid across different contexts.