Abstract:
 Let $p_{1}$,...,$p_{m}$ be a discrete probability distribution; all $p_{i}$>0 and $\sum_{i=1}^{m}$ $p_{i}$=1. Denote $v_{1n}$,...,$v_{mn}$ the corresponding frequencies in a sample of size n and consider the vector Yn of components of chisquare statistic $Y_{in}=\frac{v_{in}np_{i}}{\sqrt{np_{i}}}$, i= 1,...,m. As n$\rightarrow \infty$ 1 the vector $Y_{n}$ has the limit distribution of 0mean Gaussian vector Y= $(Y_{1},...,Y_{m})^T$ such that Y=X$(X,\sqrt{p})\sqrt{p}, where X= (X_{1},...,X_{m})^T$ is a vector of independent $\mathcal{N}$(0;1) random variables, $\sqrt{p}=(\sqrt{p}_1,...,\sqrt{p}_m)^T$, and (a,b) denotes inner product of vectors a and b. The distribution of Y depends on $\sqrt{p}$it is only its sum of squares (Y,Y) = $\sum_{i=1}^{m} Y_i^2$; which is chisquare distributed and hence has distribution free from $\sqrt{p}$. It is for this reason that we do not have any other asymptotically distribution free goodness of fit test for the discrete distributions but the chisquare statistic. In this presentation we introduce a vector $Z_{n}$= $\{Z_{in}}_{i=1}^{m}$ as follows: let $r$ be the unit length "diagonal" vector with all coordinates 1/$\sqrt{m}$, and put $Z_{n}$=$Y_{n}$$(Y_{n},r)\frac{1}{1+(\sqrt{p},r)}(r+\sqrt{p})$. Then (i) $any$ statistic based on $Z_{n}$ is asymptotically distribution free; (ii)asymptotically, the partial sums $\sum_{i=1}^{k} Z_{in}$, k$\leq$ m; will behave as a discrete time analog of the standard Brownian bridge; (iii) the transformation from $Y_{n}$ to $Z_{n}$ is onetoone. We will explain the nature of transformation of $Y_{n}$ to $Z_{n}$, give different form of such transformations and then show that it produces new results also for continuous distributions in $\mathbb{R}^d$ and for both simple and parametric hypothesis. Therefore, a unified approach to distribution free goodness of fit testing for continuous and discrete distirbutions in $\mathbb{R}^d$ is now possible. This talk is based on Khmaladze, E.V., Note on distribution free testing for discrete distributions, Ann. Stat. 2013, 29792993.
