Numerical hazard function

This function calculates the hazard function based on a given probability density function (pdf). The hazard function represents the instantaneous rate at which an event occurs at a specific time, given that the event has not occurred before that time.

Usage

hazard_function(pdf)

Arguments

pdf

The probability density function used to model the survival time.

Value

A function that calculates the hazard function at a given time.

Details

The hazard function is calculated as the ratio of the probability density function to the survival function. The hazard function is useful for understanding the instantaneous rate of occurrence of an event at a specific time. The hazard function is also used to model the instantaneous rate of occurrence of an event in survival analysis. Just like the function survival_function(), the hazard_function() function is vectorized in \(t\), which allows the calculation of the hazard function for a vector of times. The hazard function is defined by:

\(h(t) = \frac{f_T(t)}{S_T(t)},\) where \(f_T(t)\) is the probability density function and \(S_T(t)\) is the survival function at time \(t\).

References

NADARAJAH, Saralees; KOTZ, Samuel. The beta exponential distribution. Reliability engineering & system safety, v. 91, n. 6, p. 689-697, 2006.

See also

survival_function().

Examples

# Saraless Nadarajah and Samnuel Kotz (2006)
beta_exponential <- function(x, a, b, lambda){
 lambda / beta(a, b) * exp(-b * lambda * x) * (1 - exp(-lambda * x))^(a - 1)
}

hazard_beta_exponential <- hazard_function(beta_exponential)

hazard_beta_exponential(
  t = seq(0.001, 1.5, length.out = 50L),
  a = 1.5,
  b = 1.8,
  lambda = 1.5
)
#>  [1] 0.1883102 0.9840887 1.3021254 1.5119839 1.6667616 1.7876074 1.8854069
#>  [8] 1.9665682 2.0351987 2.0940896 2.1452239 2.1900584 2.2296930 2.2649769
#> [15] 2.2965791 2.3250336 2.3507737 2.3741547 2.3954716 2.4149712 2.4328620
#> [22] 2.4493213 2.4645012 2.4785329 2.4915300 2.5035918 2.5148037 2.5252451
#> [29] 2.5349825 2.5440758 2.5525787 2.5605390 2.5679998 2.5749997 2.5815736
#> [36] 2.5877530 2.5935667 2.5990406 2.6041986 2.6090622 2.6136515 2.6179844
#> [43] 2.6220779 2.6259472 2.6296067 2.6330693 2.6363472 2.6394517 2.6423931
#> [50] 2.6451811
#> attr(,"time")
#>  [1] 0.00100000 0.03159184 0.06218367 0.09277551 0.12336735 0.15395918
#>  [7] 0.18455102 0.21514286 0.24573469 0.27632653 0.30691837 0.33751020
#> [13] 0.36810204 0.39869388 0.42928571 0.45987755 0.49046939 0.52106122
#> [19] 0.55165306 0.58224490 0.61283673 0.64342857 0.67402041 0.70461224
#> [25] 0.73520408 0.76579592 0.79638776 0.82697959 0.85757143 0.88816327
#> [31] 0.91875510 0.94934694 0.97993878 1.01053061 1.04112245 1.07171429
#> [37] 1.10230612 1.13289796 1.16348980 1.19408163 1.22467347 1.25526531
#> [43] 1.28585714 1.31644898 1.34704082 1.37763265 1.40822449 1.43881633
#> [49] 1.46940816 1.50000000
#> attr(,"class")
#> [1] "hazard_function"