How to find ks test statistic using the given maximum likelihood estimator values and a sorted data in R?

Question

The random variable Y is said to have a two-parameter APE distribution denoted by APE(α, λ), with the shape and scale parameters as α > 0 and λ > 0, respectively, if the PDF of Y for y > 0 is

f(y; α, λ) =  (log α /α−1 )λe^(−λy)α^(1−e^(−λy)) ,if α = 1

           =  λe^(−λy) ,if α = 1

           =  0,otherwise

The CDF of Y for y > 0 becomes

F(y; α, λ) = [α^(1−e^(−λy))−1] /(α−1), if α = 1

           =  1 − e^(−λy),if α = 1

y=1 4 4 7 11 13 15 15 17 18 19 19 20 20 22 23 28 29 31 32 36 37 47 48 49 50 54 54 55 59 59 61 61
66 72 72 75 78 78 81 93 96 99 108 113 114 120 120 120 123 124 129 131 137 145 151 156 171
176 182 188 189 195 203 208 215 217 217 217 224 228 233 255 271 275 275 275 286 291 312
312 312 315 326 326 329 330 336 338 345 348 354 361 364 369 378 390 457 467 498 517 566
644 745 871 1312 1357 1613 1630.

n=109

alpha estimated =0.00366583  
lambda estimated =0.0009550325

When I tried coding this using R, I first found the cdf; #CDF OF APE

cdf <- function(alpha,lambda){

  if(alpha!=1){

    apecdf<-((alpha^(1-exp(-lambda*y)))-1)/ (alpha-1)}else if(alpha==1){

     apecdf<- 1-(exp(-lambda*y))}

  return(apecdf)

}
#k-s test statistic and p-Value for APE

t <- ks.test(y,cdf(0.00366583,0.0009550325),shape=0.00366583,scale=0.0009550325)

t

RESULT:

    Two-sample Kolmogorov-Smirnov test

data:  y and cdf(0.00366583, 0.0009550325)

D = 1, p-value < 2.2e-16  
alternative hypothesis: two-sided

What am I doing wrong here?

I need to get the ks-statistic= 0.0742, p-value=0.5852 (somewhat near these)

Answer 1

In my opinion you have to perform a one-sample Kolmogorov-Smirnov Test. You have a sample of the random variable Y and you want to check if the random variable is two-parameter APE distributed. So I've updated your cdf such that it has a further argument for the values of the sample, called argument y.

However, you have some problems performing the KS-Test. First problem is that this random sample y has ties (meaning repeated values). Since the KS-Test is designed for continuous distributions your y should not contain repeated values. Second problem is that you are using estimates that are computed on the basis of y.

In the attached reprex I've perfomed a one-sample KS-Test. But I can not reproduce the desired test statistic D and p-value. I've inserted some comments.

cdf <- function(y, alpha,lambda){ 
  
  if(alpha!=1){
    
    apecdf<-((alpha^(1-exp(-lambda*y)))-1)/ (alpha-1)}else if(alpha==1){
      
      apecdf<- 1-(exp(-lambda*y))}
  
  return(apecdf)
  
}

# given: 
y <- c(1, 4, 4, 7, 11, 13, 15, 15, 17, 18, 19, 19, 20, 20, 22, 23, 28, 29, 31, 32, 
       36, 37, 47, 48, 49, 50, 54, 54, 55, 59, 59, 61, 61, 66, 72, 72, 75, 78, 78, 
       81, 93, 96, 99, 108, 113, 114, 120, 120, 120, 123, 124, 129, 131, 137, 145, 
       151, 156, 171, 176, 182, 188, 189, 195, 203, 208, 215, 217, 217, 217, 224, 228, 
       233, 255, 271, 275, 275, 275, 286, 291, 312, 312, 312, 315, 326, 326, 329, 330, 
       336, 338, 345, 348, 354, 361, 364, 369, 378, 390, 457, 467, 498, 517, 566, 644, 
       745, 871, 1312, 1357, 1613, 1630)
alpha <- 0.00366583
lambda <- 0.0009550325

# Case 1: Perform a one-sample KS-Test: 
t1 <- ks.test(x = y,y = "cdf", alpha, lambda)
#> Warning in ks.test(x = y, y = "cdf", alpha, lambda): ties should not be present
#> for the Kolmogorov-Smirnov test
t1
#> 
#>  One-sample Kolmogorov-Smirnov test
#> 
#> data:  y
#> D = 0.061673, p-value = 0.8014
#> alternative hypothesis: two-sided

## Problems: 
### 1) KS-Test is for continuous distributions and hence your y should not contain the repeated values (ties)!
### 2) Parameters should not be estimated from data 
###    (specified in ?ks.test... "If a single-sample test is used, the parameters 
###    specified in ... must be pre-specified and not estimated from the data. 
###    There is some more refined distribution theory for the KS test with estimated 
###    parameters (see Durbin, 1973), but that is not implemented in ks.test.")

# Case 2: Perform a one-sample KS-Test adding some variation in y: 
y_var <- y + rnorm(length(y), sd = 0.005) #  because of the ties problem! 
colMeans(cbind(y, y_var)) 
#>        y    y_var 
#> 233.3211 233.3211
apply(cbind(y, y_var), 2, sd)   
#>        y    y_var 
#> 296.4344 296.4344
# pretty similar!

t2 <- ks.test(x = y_var, y = "cdf", alpha, lambda)
t2
#> 
#>  One-sample Kolmogorov-Smirnov test
#> 
#> data:  y_var
#> D = 0.061669, p-value = 0.8015
#> alternative hypothesis: two-sided

## Problem: 
### Second Problem of Case 1 from line 30 - 34!

^{Created on 2020-07-15 by the reprex package (v0.3.0)}

Hope it helps!