Goodness-of-Fit Tests

Prove that under the assumption of the hypothesis $H_{0}: F=F_{0}$, for any $x \in \mathbb {R}$ we have

Using theorem $12.1$, prove the consistency of the Kolmogorov test.

There is a sample $X_{1}, X_{2}, X_{3}$ of size 3. To test the hypothesis that the sample is drawn from the uniform distribution on $[0,1]$, the following version of the Kolmogorov test is used: the hypothesis of uniformity is rejected if

What is the size of this test?

Prove that under the condition $0 \leq X_{(1)} \leq X_{(n)} \leq 1$ the following equality holds

(using this representation, the value of the statistic $\omega^{2}$ used in the Cramér-von Mises-Smirnov test is often computed).

Using theorem $12.2$, prove the consistency of the chi-squared test.

The digits $0,1,2, \ldots , 9$ appeared $74,92,83,79,80,73,77,75,76,91$ times, respectively, among the first 800 decimal digits of $\pi$. Using the chi-squared test, test the hypothesis that these data agree with the uniform distribution on the set $\left\{ 0,1, \ldots , 9\right\}$ at significance level (a) $0.05$, (b) $0.5$, (c) $0.8$.

A professional dentist has learned to knock out wisdom teeth with his fist. It is known that he knocked out 52 wisdom teeth on the first attempt, 31 on the second, 3 on the third, and the remaining 5 teeth required more than 4 attempts. Test the hypothesis that the dentist knocks out any given wisdom tooth with probability $\frac{2}{3}$, at significance level $0.1$.

Among 5000 families with three children, there are exactly 1010 families with three boys, 2200 families with two boys and one girl, and 950 families with one boy and two girls (in all remaining families all the children are girls). Can we, at significance level $\alpha =0.02$, consider that the number of boys $\xi$ in a family with three children has the following distribution

where $\theta \in (0, \frac{1}{4})$