A cumulative ${}^{1}3$C breath test can be used to detect a pancreatic disorder, for example. The test is burdened by uncertain input data. Among them, the estimation of CO${}_2$ production rate is a key issue. An established estimate based on the body surface area could be replaced by an estimate using the basal metabolic rate. Although the latter estimate might be considered more appropriate than the former, the differences between them are large in some sex and agegroups. Such disagreements pose...

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.

CONTENTS1. Introduction.............................................................................................................................................52. Assumptions and notations................................................................................................................63. An important asymptotic distribution..................................................................................................84. The asymptotic distribution of $K{*}_{\mu ,\nu }$ under the...

We consider the likelihood ratio test (LRT) process related to the test of the absence of QTL (a QTL denotes a quantitative trait locus, i.e. a gene with quantitative effect on a trait) on the interval $[0,T]$ representing a chromosome. The originality is in the fact that some genotypes are missing. We give the asymptotic distribution of this LRT process under the null hypothesis that there is no QTL on $[0,T]$ and under local alternatives with a QTL at $t^{☆}$ on $[0,T]$. We show that the LRT process is asymptotically...

We describe a primality test for $M={\left(2p\right)}^{2^n}+1$ with an odd prime p and a positive integer n, which are a particular type of generalized Fermat numbers. We also present special primality criteria for all odd prime numbers p not exceeding 19. All these primality tests run in deterministic polynomial time in the input size log₂M. A special 2pth power reciprocity law is used to deduce our result.

In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), $$u_{tt}+f\left(x\right)u_t+{(-\Delta )}^{\alpha /2}\left(u^m\right)=h(t,x){\left|u\right|}^p,$$ posed in $(0,T)\times {ℝ}^N,$ where ${(-\Delta )}^{\alpha /2},0<\alpha \le 2$ is $\alpha /2$-fractional power of $-\Delta .$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times 2$ system of the same type.

Let U₀ be a random vector taking its values in a measurable space and having an unknown distribution P and let U₁,...,Uₙ and $V₁,...,V_m$ be independent, simple random samples from P of size n and m, respectively. Further, let $z₁,...,z_k$ be real-valued functions defined on the same space. Assuming that only the first sample is observed, we find a minimax predictor d⁰(n,U₁,...,Uₙ) of the vector $Y^m={\sum }_{j=1}^m{(z₁\left(V_j\right),...,z_k\left(V_j\right))}^T$ with respect to a quadratic errors loss function.