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Teoriya Veroyatnostei i ee Primeneniya, 1980, Volume 25, Issue 2, Pages 359–364
(Mi tvp1164)
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This article is cited in 2 scientific papers (total in 2 papers)
Short Communications
Lower bounds for average sample size in the tests of invariability
I. N. Volodin Kazan
Abstract:
This article is a continuation of a lower bounds research for average sample size in specific procedures of statistical inference (see [1] and [2]). Let $\mathscr P$ be a family of absolutely continuous distributions $P$ on a measurable space $\langle\mathscr X,\mathscr A\rangle$ and $\mathfrak S$ is a group of transformations of $\mathscr X$. The problem of testing hypothesis on an invariance (with respect
to $\mathfrak S$) of distribution $P$ of the observed random variable is considered. The hypothesis is
a statement
$$
P\in\{P:P(A)=P(sA)\ \forall\,A\in\mathscr A,\ \forall\,s\in\mathfrak\sigma\},
$$
the alternative is $P\in\{P:\exists\,t\in\mathfrak S,\ \sup_A|P(A)-P(tA)|\ge\Delta\}$. The problem of verifying the homogeneity of distributions $P_1$ and $P_2$ of two random variables is also considered. In this case the hypothesis is a statement
$$
(P_1,P_2)\in\{(P_1,P_2):\exists\,t\in\mathfrak S,\ P_1(A)=P_2(tA)\ \forall\,A\in\mathscr A\},
$$
the alternative is $(P_1,P_2)\in\{(P_1,P_2):\sup_A|P_1(A)-P_2(sA)|\ge\Delta\ \forall\,s\in\mathfrak S$. Lower
bounds for average sample size which is sufficient for the acceptance of decition on the propound hypothesis trustiness with guaranteed restrictions $(\alpha,\beta)$ on the probabilities of errors is established.
Received: 11.04.1978
Citation:
I. N. Volodin, “Lower bounds for average sample size in the tests of invariability”, Teor. Veroyatnost. i Primenen., 25:2 (1980), 359–364; Theory Probab. Appl., 25:2 (1981), 356–360
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https://www.mathnet.ru/eng/tvp1164 https://www.mathnet.ru/eng/tvp/v25/i2/p359
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