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The aim of this paper is to establish theorems on the absolute continuity of translation as well as scale invariant statistics in general, from which the related results by Hodges-Lehmann and Puri-Sen follow. The continuity relations between the joint cdf of a random vector and its marginal cdf's are also considered.

Let $X_i$, $1\le i\le N$, be $N$ independent random variables (i.r.v.) with distribution functions (d.f.) $F_i(x,\Theta )$, $1\le i\le N$, respectively, where $\Theta$ is a real parameter. Assume furthermore that $F_i(·,0)=F(·)$ for $1\le i\le N$. Let $R=(R_1,...,R_N)$ and $R^{+}=(R_1^{+},...,R_N^{+})$ be the rank vectors of $X=(X_1,...,X_N)$ and $\left|X\right|=\left(\right|X_1|,...,|X_N\left|\right)$, respectively, and let $V=(V_1,...,V_N)$ be the sign vector of $X$. The locally most powerful rank tests (LMPRT) $S=S\left(R\right)$ and the locally most powerful signed rank tests (LMPSRT) $S=S(R^{+},V)$ will be found for testing $\Theta =0$ against $\Theta >0$ or $\Theta <0$ with $F$ being arbitrary and with $F$ symmetric, respectively.

Višek [3] and Culpin [1] investigated infinite binary sequence $X=(X_1,X_2,\cdots )$ with $X_i$ taking values $0$ or $1$ at random. They investigated also real mappings $H\left(X\right)$ which have the uniform distribution on $[0;1]$ (notation $𝒰(0;1)$). The problem for $n$-ary sequences is dealt with in this paper.