• Kolmogorov-Smirnov Test vs. Chi-Square Goodness-of-Fit:
• The Kolmogorov-Smirnov (K-S) test is a non-parametric test used to test whether a sample comes from a specific continuous distribution, functioning as the non-parametric analogue to the Chi-square goodness-of-fit test.
• Key Advantages of K-S Test:
• Does not require binning data into arbitrary intervals or categories (unlike Chi-square).
• Works effectively with small sample sizes.
• Compares the empirical cumulative distribution function $F_n(x)$ with the theoretical CDF $F_0(x)$ using the supremum distance statistic: $D = \sup_x |F_n(x) - F_0(x)|$.
• Other Options:
• Median test / Mann-Whitney U test / Wilcoxon test: Used for testing differences in location/medians between two independent or paired groups, not goodness of fit.
• Conclusion: The analogue is the Kolmogorov-Smirnov test (Option 2).

Correct option: (2) Kolmogorov-Smirnov test

• Degrees of Freedom in Contingency Tables:
• For an $r \times c$ two-way contingency table with $r$ rows and $c$ columns testing for independence of attributes using the Chi-square statistic:

• Given $r = 5$ and $c = 5$:

• Conclusion: The number of degrees of freedom is 16 (Option 3).

Correct option: (3) 16

• Type I and Type II Errors in Hypothesis Testing:
• Type I Error ($\alpha$): Probability of rejecting the null hypothesis ($H_0$) when $H_0$ is actually true (producer's risk or false positive).
• Type II Error ($\beta$): Probability of accepting (failing to reject) the null hypothesis ($H_0$) when $H_0$ is false (consumer's risk or false negative).
• Power of a Test: The probability of correctly rejecting a false null hypothesis, defined as $1 - \beta$.
• Critical Region: The set of sample outcomes for which the null hypothesis is rejected.
• Conclusion: Probability of accepting $H_0$ when it is false is Type II error (Option 4).

Correct option: (4) Type Il error

• Properties of Likelihood Ratio Tests (LRT):
• Statement (I): Likelihood Ratio test is generally an UMP if an UMP test at all exists. $\implies$ Correct.
• By the Karlin-Rubin theorem and Neyman-Pearson lemma, for families with Monotone Likelihood Ratio (MLR), the Likelihood Ratio Test reduces to the Uniformly Most Powerful (UMP) test for one-sided composite hypotheses.
• Statement (II): In a Likelihood Ratio test, probability of type I error is controlled by effectively choosing a suitable cutoff point. $\implies$ Correct.
• The critical region is defined by $\lambda(x) \le k$, where the threshold $k$ is chosen such that $\sup_{\theta \in \Theta_0} P_\theta(\lambda(X) \le k) = \alpha$, strictly controlling the size (Type I error probability) of the test.
• Conclusion: Both Statement (I) and Statement (II) are correct (Option 1).

Correct option: (1) Both Statement (I) and Statement (I) are correct