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The shifted exponential distribution is a modification of the regular exponential distribution. In mathematical terms, the shift parameter, \( \theta \), modifies the traditional exponential distribution to begin beyond zero. This functional adjustment allows for flexibility in modeling scenarios where the minimum of the distribution is not zero, but another value, \( \theta \).

For example, if you are measuring time between events in traffic flow as seen in Example 4.5, a shift might be needed since there is a minimal time gap (\( \theta \)) that is realistically possible.

Key characteristics of the shifted exponential distribution include:

• The probability density function (pdf) is only non-zero for values greater than or equal to \( \theta \). This property ensures that the function \/p within permissible values.
• The parameter \( \lambda \) remains the same as in the exponential distribution, controlling the rate of events occurring beyond \( \theta \).

Understanding the need for shifts helps tailor statistical models to real-world phenomena better.

The log-likelihood function is a handy tool for simplifying the computation of likelihoods, particularly when dealing with products of probabilities.

In our case, given a shifted exponential distribution, the likelihood function for a sample involves a product, as each data point contributes a probability, which can quickly become cumbersome to compute.

Formulaically, the likelihood function \( L(\lambda, \theta) \) is:

• \[ L(\lambda, \theta) = \lambda^n e^{-\lambda \sum_{i=1}^{n} (X_i - \theta)} \]

To simplify, we take its natural logarithm, turning products into sums and exponentials into linear terms. This leads us to the log-likelihood function \( \ell(\lambda, \theta) \):

• \[ \ell(\lambda, \theta) = n \ln \lambda - \lambda \sum_{i=1}^{n} (X_i - \theta) \]

Utilizing the log-likelihood is particularly useful when deriving parameter estimates, as differentiating this expression often results in straightforward mathematical manipulations.

Estimating parameters \( \lambda \) (rate) and \( \theta \) (shift) for a shifted exponential distribution revolves around determining values that maximize the likelihood of observing the given data.

### Estimating \( \lambda \)
The first step involves differentiating the log-likelihood function with respect to \( \lambda \) and setting the derivative to zero:

• \[ \frac{\partial \ell}{\partial \lambda} = \frac{n}{\lambda} - \sum_{i=1}^{n} (X_i - \theta) \]

Solving this provides the maximum likelihood estimator (MLE) for \( \lambda \):

• \[ \hat{\lambda} = \frac{n}{\sum_{i=1}^{n} (X_i - \theta)} \]

### Estimating \( \theta \)
The parameter \( \theta \) represents the shift and is estimated by considering the smallest observation in the data set, because the likelihood is zero for any observation smaller than \( \theta \):

• \[ \hat{\theta} = \min(X_1, X_2, \ldots, X_n) \]

### Application
For our example with the given observations, by following these steps, we estimated \( \hat{\theta} = 0.64 \) and \( \hat{\lambda} \approx 0.2065 \). These estimators help tailor the shifted exponential model to the specific data, offering insights into the rate and starting point of the observed phenomenon.