Geodesic
A~remark on independence of a~tubular statistic and the sample mean
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 753-755

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Given a sample of size $n$ from a distribution with density $y(x)$, we show that if a definite $n-1$-dimensional tubular statistic (i.e. a continuous function on $R^n$ reduceable by an orthogonal transformation to a function on $R^{n-1}$ vanishing only at the origin) and the sample mean are independent then $y(x)$ is normal. 
@article{TVP_1971_16_4_a19,
     author = {L. B. Klebanov},
     title = {A~remark on independence of a~tubular statistic and the sample mean},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {753--755},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_4_a19/}
}
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L. B. Klebanov. A~remark on independence of a~tubular statistic and the sample mean. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 753-755. http://geodesic.mathdoc.fr/item/TVP_1971_16_4_a19/