By Grenander U.

ISBN-10: 0471082678

ISBN-13: 9780471082675

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**Additional info for Abstract inference**

**Example text**

N values of the random vector U ∼ P(n, π 0 ) are simulated and for each value of U the corresponding value of the statistics Xn2 can be computed. Suppose that M is the number of values greater then x2n . Then the P -value is approximated by M/N . The hypothesis H0′ is rejected with an approximated significance level α if M/N < α. The accuracy of this test depends on the number of simulations N . 3. If the hypothesis H0′ is rejected then the hypothesis H0 is also rejected because it is narrower.

For example, such a hypothesis is verified if we want to know whether realizations generated by a computer are obtained from the uniform U (0, 1), Poisson P(2), normal N (0, 1) or other completely specified distribution. The data are grouped in the following way: the abscissas axis is divided into a finite number of intervals using the points −∞ = a0 < a1 < ... < ak = ∞. , k i=1 So, instead of the fully informative data X, we use the grouped data U = (U1 , . . , Uk )T We can also say that the statistic U is obtained using a special data censoring mechanism, known as the mechanism of grouping data.

For example, the hypothesis may mean that the probability distribution of X belongs to the family of normal, exponential, Poisson, binomial or other distributions. , k. , θs ; s + 1 < k. 14] cannot be computed because the parameter θ is unknown. 14] by their estimators and to investigate the properties of the obtained statistic. If the maximum likelihood estimator of the parameter θ obtained from the initial non-grouped data is used then the limit distribution depends on the distribution F (x; θ) (so on the parameter θ).

### Abstract inference by Grenander U.

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