The procedure of many hypotheses logarithmically asymptotically optimal (LAO)
testing for a model consisting of three or more independent objects is analyzed. It is supposed that

In [

Investigation of testing of the distributions of many uniform objects is an interesting not yet fulfilled task. It is natural to begin this study with the simplest case of statistically independent objects.

Let

Let us recall main definitions from [

The exponential decrease of the error probabilities as

The matrix

The sequence of tests

Let us now consider the model with three objects. Let

Let

Let

We study corresponding reliabilities

Definitions (

We call the test sequence

Our aim is to analyze the reliability matrix

We consider the problem for three objects for brevity; the generalization of the problem for

We define the divergence (Kullback-Leibler distance)

For given positive diagonal elements

We recall the theorem concerning one object.

If the distributions

(a)when the given numbers

(b) even if one of the conditions (

The diagonal elements of the reliability matrix of the LAO test in each row are equal only to the element in the same row and in the last column:

That is, the elements of the last column are equal to the diagonal elements of the same row and due to (

For

Applying Kuhn-Tucker theorem for (

Now let us consider the model of three independent objects and

If the elements

Equalities (

It follows from the independence of the objects that

Remark that here we consider also the probabilities of right (not erroneous) decisions. Because

Now we will show how we can erect the LAO test from the set of compound tests when

The following subset of tests:

Really, because

The following theorem is the main result of the present paper. It is a generalization and improvement of the corresponding theorem proved in [

If all distributions

(a) when given strictly positive elements

(b) when even one of the inequalities (

The test

Let the test

Taking into account (

According to Corollary

(b) When even one of the inequalities (

Theorem

When we consider the model with

If elements

For given

Some illustrations of exposed results are in examples concerning two objects. The set

We exposed a solution of multiple hypothesis LAO testing problem for many objects. The first idea may be to study matrix

Proposed approach to the problem gives also the possibility to define the LAO tests for each of the separate objects. It must be noted that the approach with renumbering of the triples of hypotheses does not have this opportunity.

In applications one of two approaches may be used in conformity with preferences of the investigator.