Monotone Likelihood Ratios-Most Powerful Test

In summary, the use of monotonic likelihood ratios in hypothesis testing involves comparing the likelihood functions L(thetha) under Ho and L(thetha) under Ha, where a positive derivative indicates a monotonically increasing function. The Uniform Most Powerful Test's critical region takes the form of "above ratio <= C" due to the fact that as the ratio increases, the likelihood of the null hypothesis being true also increases. Therefore, to reject the null hypothesis, the ratio must be less than a certain value. This approach is based on the concept that a larger likelihood ratio supports the acceptance of the null hypothesis.
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mckammer
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Monotone Likelihood Ratios--Most Powerful Test

This isn't really a problem, its more of a theory question. I'm having trouble understanding the reading in my textbook (Introduction to Mathematical Statistics, by Hogg Craig, and McKean).

I'm looking at the section of using concepts dealing with Monotonic Likelihood Ratios, to conduct hypothesis tests of the form

Ho: thetha <= thetha-naught
Ha: thetha > thetha-naught

I know that if the likelihood functions L(thetha) under Ho/L(thetha) under Ha has a positive derivative (so its monotonically increasing), the Uniform Most Powerful Test's critical region takes the form of

above ratio <= C, where C is a constant.

What is the reasoning behind this?

I tried to justify it to myself, but the furthest I got was:
1) The ratio is the ratio of likelihood under Ho to likelihood under Ha.
2) Normally, for simple hypothesis, if this is large, then we accept the null hypothesis.
3) Here, the ratio is increasing, for increases in thetha, so the chance of the null being true is increasing as thetha is increasing?
 
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4) So, the ratio must be less than a certain value to reject the null hypothesis? Any help would be appreciated.
 

Related to Monotone Likelihood Ratios-Most Powerful Test

1. What is a Monotone Likelihood Ratio-Most Powerful Test?

A Monotone Likelihood Ratio-Most Powerful Test is a type of statistical test used to determine whether a parameter in a probability distribution has a certain value or falls within a specific range. It is based on the likelihood ratio principle, which compares the likelihood of the observed data under the null hypothesis to the likelihood under an alternative hypothesis.

2. How does a Monotone Likelihood Ratio-Most Powerful Test work?

A Monotone Likelihood Ratio-Most Powerful Test works by calculating the likelihood ratio for the observed data and comparing it to a critical value. If the likelihood ratio is greater than the critical value, the null hypothesis is rejected in favor of the alternative hypothesis. This test is considered the most powerful because it has the highest probability of correctly rejecting the null hypothesis when it is false.

3. When is a Monotone Likelihood Ratio-Most Powerful Test used?

A Monotone Likelihood Ratio-Most Powerful Test is typically used in situations where the parameter of interest has a natural ordering, such as in testing for trends or differences between two groups. It is also commonly used in medical and scientific research to determine the effectiveness of a treatment or intervention.

4. What are the assumptions of a Monotone Likelihood Ratio-Most Powerful Test?

The main assumptions of a Monotone Likelihood Ratio-Most Powerful Test include a random and independent sample, a continuous probability distribution, and the null and alternative hypotheses being simple and nested. Additionally, the likelihood function must be monotonic, meaning that it either increases or decreases as the parameter of interest increases.

5. How is the power of a Monotone Likelihood Ratio-Most Powerful Test calculated?

The power of a Monotone Likelihood Ratio-Most Powerful Test is calculated by using the non-central chi-square distribution and the degrees of freedom associated with the test. It can also be calculated using statistical software or online calculators. The power of the test is affected by factors such as the sample size, significance level, and the effect size of the parameter being tested.

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