Simulation Study

1. The single simple random sample of length n = 50 is:

(4, 2, 4, 5, 6, 5, 4, 0, 5, 4, 5, 3, 3, 3, 5, 4, 3, 3, 6, 4, 4, 6, 2, 3, 3, 3, 4, 5, 5, 5, 4, 3, 3, 4, 7, 5, 6, 6, 7, 4, 6, 4, 6, 5, 4, 3, 4, 6, 6, 4).

2. From this single random sample, the estimate of k is \(\hat k\) = 7.67 and the estimate of p is \(\hat p\) = 0.56.
3. From the N = 1000 samples of size n = 50, we get 1000 estimates of k and p. The following is a printing for each of the first 10 estimates \(\hat k\) and \(\hat p\):
##        khat phat
##  [1,] 11.90 0.36
##  [2,] 10.89 0.39
##  [3,] 19.06 0.23
##  [4,] 11.37 0.38
##  [5,] 10.64 0.40
##  [6,]  9.46 0.45
##  [7,]  8.25 0.52
##  [8,] 11.21 0.38
##  [9,]  7.06 0.61
## [10,]  6.64 0.65
4. From the N = 1000 samples of size n = 100, we get 1000 estimates of k and p. The following is a printing for each of the first 10 estimates \(\hat k\) and \(\hat p\):
##        khat phat
##  [1,]  7.78 0.55
##  [2,]  9.48 0.45
##  [3,] 11.18 0.38
##  [4,] 10.15 0.42
##  [5,]  8.68 0.50
##  [6,]  9.72 0.44
##  [7,]  8.86 0.49
##  [8,]  9.16 0.47
##  [9,] 11.15 0.39
## [10,] 12.51 0.34
From the N = 1000 samples of size n = 250, we get 1000 estimates of k and p. The following is a printing for each of the first 10 estimates \(\hat k\) and \(\hat p\):
##        khat phat
##  [1,] 10.22 0.42
##  [2,]  9.21 0.47
##  [3,] 10.91 0.39
##  [4,] 10.96 0.39
##  [5,] 10.54 0.41
##  [6,]  9.36 0.46
##  [7,] 11.59 0.37
##  [8,]  9.06 0.47
##  [9,]  9.88 0.44
## [10,]  9.75 0.44
5. For each sample size, we get the following biases and mean squared errors (MSE) for each estimator for both k and p:
\(\hat k\)

  • n = 50: bias = 0.89 and MSE = 23.31

  • n = 100: bias = 0.56 and MSE = 4.95

  • n = 250: bias = 0.33 and MSE = 1.49

\(\hat p\)

  • n = 50: bias = 0.0288 and MSE = 0.0118

  • n = 100: bias = 0.022 and MSE = 0.0061

  • n = 250: bias = 0.0215 and MSE = 0.0025

The \(\hat k\) estimators seem to overestimate the parameter k for each sample size n. Whereas, the \(\hat p\) estimators seem to consistently have very small overestimates or underestimates of the parameter and are therefore closer to being considered unbiased because the calculated biases and mean squared errors (MSE) are approximately 0 for each sample size. As the sample size increases, the bias and MSE usually decrease and become closer to 0 for both \(\hat k\) and \(\hat p\).

6. View the following plots for each part below:

  1. We can see from both of these plots that as the sample size increases the estimates of \(\hat k\) and \(\hat p\) both turn out to be nearer to the given values of the parameters k and p. In other words, the plots both show that as the sample size n gets larger, then the estimators \(\hat k\) and \(\hat p\) become closer to the parameters k = 10 and p = 0.4.