# A test of uniformity for 1D data based on domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - Binomial Approximation An object of class `"htest"` (i.e., hypothesis test) function which performs a hypothesis test of uniformity of `Xp` points in the support interval $(a,b)$). The support interval $(a,b)$ is partitioned as `(b-a)*(0:nint)/nint` where `nint=round(sqrt(nx),0)` and `nx` is number of `Xp` points, and the test is for testing the uniformity of `Xp` points in the interval $(a,b)$. The null hypothesis is uniformity of `Xp` points on $(a,b)$. The alternative is deviation of distribution of `Xp` points from uniformity. The test is based on the (asymptotic) binomial distribution of the domination number of PE-PCD for uniform 1D data in the partition intervals based on partition of $(a,b)$. The function yields the test statistic, $p$-value for the corresponding alternative, the confidence interval, estimate and null value for the parameter of interest (which is $Pr($domination number$\le 1)$), and method and name of the data set used. Under the null hypothesis of uniformity of `Xp` points in the support interval, probability of success (i.e., $Pr($domination number$\le 1)$) equals to its expected value) and `alternative` could be two-sided, or left-sided (i.e., data is accumulated around the end points of the partition intervals of the support) or right-sided (i.e., data is accumulated around the centers of the partition intervals). PE proximity region is constructed with the expansion parameter $r \ge 1$ and centrality parameter `c` which yields $M$-vertex regions. More precisely $M_c=a+c(b-a)$ for the centrality parameter `c` and for a given $c \in (0,1)$, the expansion parameter $r$ is taken to be $1/\max(c,1-c)$ which yields non-degenerate asymptotic distribution of the domination number. The test statistic is based on the binomial distribution, when success is defined as domination number being less than or equal to 1 in the one interval case (i.e., number of failures is equal to number of times restricted domination number = 1 in the intervals). That is, the test statistic is based on the domination number for `Xp` points inside the partition intervals for the PE-PCD. For this approach to work, `Xp` must be large for each partition interval, but 5 or more per partition interval seems to work in practice. Probability of success is chosen in the following way for various parameter choices. `asy.bin` is a logical argument for the use of asymptotic probability of success for the binomial distribution, default is `asy.bin=FALSE`. When `asy.bin=TRUE`, asymptotic probability of success for the binomial distribution is used. When `asy.bin=FALSE`, the finite sample probability of success for the binomial distribution is used with number of trials equals to expected number of `Xp` points per partition interval. ```r PEdom.num.binom.test1Dint( Xp, support.int, c = 0.5, asy.bin = FALSE, alternative = c("two.sided", "less", "greater"), conf.level = 0.95 ) ``` ## Arguments - `Xp`: A set of 1D points which constitute the vertices of the PE-PCD. - `support.int`: Support interval $(a,b)$ with $a

PEdom.num.binom.test1Dint function