Question

Asked May 3, 2019

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Give and explain one way to find uniformly most powerful test (UMP).

Step 1

**The UMP test:**

While testing a null hypothesis, *H*_{0} against a composite alternative hypothesis, *H _{a}*, the test that is the most powerful against each of the simple alternatives contained in the composite alternative hypothesis, is called the UMP test or the uniformly most powerful test.

In other words, the UMP test has the highest power, (1 – *β*) [*β* = *P* (Type II error)] among all size *α* tests, that is, all tests at level of significance,* α*, for testing the particular null hypothesis *H*_{0} against the composite alternative hypothesis, *H _{a}*.

For a test of the parameter of interest, *θ*, when the composite alternative contains a set of values denoted by *ω*_{1}, there may be 2 possible tests, that is, 2 possible sets of hypotheses in this case:

*H*_{0}:*θ*=*θ*_{0}versus*H*:_{a}*θ*ϵ*ω*_{1}[Simple null versus composite alternative],*H*_{0}:*θ*ϵ*ω*_{0}versus*H*:_{a}*θ*ϵ*ω*_{1}[Composite null versus composite alternative].

Then, the UMP test of size *α* has the maximum power, say, *β *(*θ*_{1}), among all size *α* tests, for each simple alternative *θ*_{1} ϵ *ω*_{1}, defined in *H _{a}*.

The test function, which gives the rejection rule, when the rejection region is denoted by *R*, is given as:

Step 2

**Neyman-Pearson Lemma:**

According to Neyman-Pearson Lemma, the Likelihood Ratio Test (LRT) is the most powerful test for testing a simple null hypothesis against a simple alternative hypothesis.

**The Likelihood Ratio Test:**

The LRT statistic is the ratio of the supremum of likelihood function under the null model (i.e., under *H*0) to the supremum of likelihood function under the alternative model (i.e., under *Ha*).

Suppose *X*1, *X*2, …, *Xn* are the observations of a sample of size *n*. The likelihood function under the null model is denoted by *L* (*θ *| **x**; *θ* ϵ *ω*0) and that under the alternative model is denoted by *L* (*θ *| **x**; *θ* ϵ *ω*1). Then, the LRT statistic, *λ* is given as:

*λ*** = sup. { L (θ | x; θ ϵ ω0)}/ sup. { L (θ | x; θ ϵ ω1)}**.

**The decision rule:**

If *λ *(**x**) < *c*, reject *H*0. If *λ *(**x**) ≥ *c*, do not reject *H*0.

The test being a size *α* test, the *P* (Type I error) = *α*, that is, the probability of rejecting *H*0 when *H*0 is not false is *α*. The value of *c* can be obtained using this property as: *P* (*λ *(**x**) < *c *| *H*0) = *α*.

There may be one of the two relationships between *λ *(**x**) and *T *(**x**), where *T *(**x**) is a statistic or a function of *X*1, *X*2, …, *Xn*:

*λ*(**x**) is an increasing function of*T*(**x**), so that*λ*(**x**) <*c*implies and is implied by*T*(**x**) <*k*, such that*P*(*T*(**x**) <*k*|*H*0) =*α*.*λ*(**x**) is a decreasing function of*T*(**x**), so that*λ*(**x**) <*c*implies...

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