Question

Let \( \{0.13, 0.12, 0.78, 0.51\} \) be a realization of a random sample of size 4 from a population with cumulative distribution function \( F(.) \). Consider testing \( H_0: F = F_0 \) against \( H_1: F \neq F_0 \), where \[ F_0(x) = \begin{cases} 0 & \text{if } x < 0, \\ x & \text{if } 0 \leq x < 1, \\ 1 & \text{if } x \geq 1. \end{cases} \] Let \( D \) denote the Kolmogorov-Smirnov test statistic. If \( P(D > 0.669) = 0.01 \) under \( H_0 \) and \( \psi = \begin{cases} 1 & \text{if } H_0 \text{ is accepted at level 0.01}, \\ 0 & \text{otherwise} \end{cases} \)
then based on the given data, the observed value of } \( D + \psi \)
(rounded off to two decimal places) equals ...............

Hint:

- The Kolmogorov-Smirnov test compares the maximum deviation between the observed and expected cumulative distributions.
- The critical value depends on the sample size and the significance level.

Answer 1

1) Understanding the problem:
The problem provides a sample \( \{0.13, 0.12, 0.78, 0.51\} \) and asks us to compute the Kolmogorov-Smirnov (K-S) statistic \( D \) for testing the null hypothesis that the sample comes from a population with CDF \( F_0(x) \). The critical value for the test statistic \( D \) is given as 0.669, and the decision rule \( \psi \) depends on whether \( D \) exceeds this critical value at a significance level of 0.01.
2) Kolmogorov-Smirnov Statistic:
The Kolmogorov-Smirnov statistic is the maximum absolute difference between the empirical distribution function (EDF) and the cumulative distribution function (CDF) of the hypothesized distribution. The EDF for this sample is computed by sorting the data and calculating the proportion of observations less than or equal to each sample point: \[ \text{EDF}(x) = \frac{\text{Number of observations} \leq x}{\text{Total number of observations}}. \] For the sample \( \{0.13, 0.12, 0.78, 0.51\} \), we compute the EDF and compare it to \( F_0(x) \). The test statistic \( D \) is then the maximum of these differences.
3) Calculation of \( D \):
- The ordered sample is \( 0.12, 0.13, 0.51, 0.78 \).
- The EDF values at these points are \( \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4} \).
- The CDF values for these points based on \( F_0(x) \) are \( 0.12, 0.13, 0.51, 0.78 \), respectively.
- The Kolmogorov-Smirnov statistic \( D \) is the maximum absolute difference between the EDF and CDF, which is \( D = \max \left( | \text{EDF} - \text{CDF} | \right) \).
4) Decision Rule and \( \psi \):
- The critical value \( D_{critical} = 0.669 \) corresponds to a significance level of 0.01. Since the computed \( D \) is less than the critical value, we fail to reject the null hypothesis \( H_0 \).
- The value of \( \psi \) is 1 because \( H_0 \) is accepted at the 0.01 significance level.
5) Final Calculation:
The observed value of \( D + \psi \) is approximately \( 1.35 \) to \( 1.40 \).