Learn PyINLA

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between independent events that occur at a constant average rate. It is characterized by its memoryless property, meaning the probability of an event occurring in the future is independent of the past. This distribution is widely utilized in reliability analysis and queuing theory.

← Back to Likelihoods

The exponential distribution is a continuous probability distribution often used to model the time between independent events that occur at a constant average rate. It is characterized by its memoryless property, meaning the probability of an event occurring in the future is independent of the past. This distribution is widely utilized in reliability analysis and queuing theory.

Parametrization

The probability density function (PDF) for the exponential distribution, considering \(\pmb{y}\) as a vector of responses, is given by:

\[f(y_i \mid \lambda_i) = \lambda_i \exp(-\lambda_i y_i), \quad y_i > 0, \; \lambda_i > 0, \; i = 1, 2, \dots, n,\]

where:

  • \(\pmb{y} = (y_1, y_2, \ldots, y_n)\) represents the observed response times.

  • \(\pmb{\lambda} = (\lambda_1, \lambda_2, \ldots, \lambda_n)\) represents the rate parameters for each observation.

In survival analysis, models are often specified through the hazard function. For the exponential model, the baseline hazard is constant over time, and the hazard function is:

\[h(y_i) = \lambda_i, \quad i = 1, 2, \ldots, n. \\ \]

Assume \(n\) = 1. To demonstrate the shape of the Exponential distribution for various rate parameters \(\lambda\), Figure 1 displays several Exponential PDFs.

Exponential PDFs for different rate parameters \(\lambda\). As \(\lambda\) increases, the distribution places more mass near \(y=0\). For smaller \(\lambda\), the distribution decays more slowly.

The rate parameter \(\lambda_i\) is linked to the linear predictor \(\eta_i\) using the default log link:

\[\lambda_i = \exp(\eta_i), \quad i = 1, 2, \ldots, n.\]

The available link functions are: default, log.

Hyperparameters

The Exponential likelihood has no hyperparameters. The rate parameter \(\lambda\) is fully determined by the linear predictor \(\eta\) through the link function.

Validation Rules

pyINLA enforces several validation rules for Exponential models to ensure correct specification:

No Hyper Configuration

The Exponential likelihood has no hyperparameters. Do NOT provide control['family']['hyper'] configuration.

Not Allowed Arguments

The following arguments are not allowed for exponential likelihoods and will raise PyINLAError:

  • E (exposure) - Only allowed for poisson/nbinomial

  • scale - Only allowed for gaussian/nbinomial/gamma/beta/logistic/t

  • Ntrials - Only allowed for binomial/betabinomial/nbinomial2

  • control['family']['variant'] - Not supported for exponential

Allowed Link Functions

Key: control['family']['link']

  • default

  • log

Response Values

Response variable \(\pmb{y}\) must be strictly positive (\(y_i > 0\)). pyINLA will raise PyINLAError if any response value is zero or negative.

Specification

  • family="exponential"
  • Required arguments:
    • \(\pmb{y}\): response vector of strictly positive values