Statistical Inference II J Lee Assignment 4 Problem 1 Exerci

Statistical Inference Ii J Lee Assignment 4problem 1 Exercise 12 I

Derive a likelihood ratio test of H0 : θ = θ0 versus HA : θ 6= θ0, and show that the rejection region is of the form {Xe−θ0X ≤ c} for a sample from an exponential distribution with the density function f(x|θ) = θe−θx. Determine which hypotheses are simple and which are composite among given examples, and derive likelihood ratio tests for other distribution models, analyzing their properties. Additionally, formulate hypothesis tests to assess if a sample from a normal distribution meets specified mean and variance criteria, or if a sample from a uniform distribution matches certain distributional forms, and evaluate the tests' power and error probabilities, especially under specified significance levels. Connect these methods to real-world scenarios such as vehicle fuel efficiency testing, and provide comprehensive explanations of their application, interpretation, and limitations in statistical inference contexts.

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Introduction

Statistical inference plays a vital role in analyzing data to make decisions about population parameters. Likelihood ratio tests (LRT) are a fundamental tool for hypothesis testing, allowing statisticians to determine whether observed data support a null hypothesis (H0) or favor an alternative hypothesis (HA). This paper explores the derivation of likelihood ratio tests for various distributions, their properties, and their applications in real-world contexts, as prompted by exercises from Rice's statistical textbook.

Likelihood Ratio Test for Exponential Distribution

Consider a sample X1, ..., Xn from an exponential distribution with density function f(x|θ) = θe^(−θx) for x > 0 and θ > 0. The likelihood function based on the sample is:

L(θ) = ∏_{i=1}^n θe^{−θx_i} = θ^n e^{−θ∑x_i}.

Under H0: θ = θ0, the likelihood function is L(θ0). Under the alternative HA: θ ≠ θ0, the maximum likelihood estimate (MLE) of θ is:

Έ = n / ∑x_i.

The likelihood ratio (LR) statistic is then:

\\[ \\Lambda = \\frac{L(θ_0)}{L(Έ)} = \\frac{θ_0^n e^{−θ_0 \\sum x_i}}{(n/\\sum x_i)^n e^{−n}}. \\]

Rearranging, the test rejects H0 for extremes of the sufficient statistic, which simplifies to the form:

\\[ \\text{Reject H0} \\text{ if } \\sum x_i \\notin \\text{intervals determined by } c. \\]

Expressed in terms of the order statistic X_{(1)}, the rejection region becomes {Xe^{−θ0}X ≤ c}, indicating that the test is based on the sum or maximum of observations.

Simple vs. Composite Hypotheses

In hypothesis testing, a simple hypothesis specifies the distribution completely, such as X ~ N(0,σ²), while a composite hypothesis involves a range of parameter values, e.g., X ~ N(0,σ²) for σ² > 10. Identifying whether hypotheses are simple or composite is crucial for choosing appropriate tests and understanding their properties.

Likelihood Ratio Tests in Other Distributions

For the double exponential (Laplace) distribution with density f(x) = (1/2λ ) e^{−λ|x|}, testing H0: λ=λ0 versus HA: λ=λ1 > λ0 involves deriving a likelihood ratio based on the joint density and the MLEs of λ under each hypothesis. The test's uniform most powerful (UMP) property depends on the distribution and alternative hypothesis; for λ > λ0, the likelihood ratio test is often UMP for the one-sided testing problem.

Testing Uniform Distributions and Their Power

In testing whether a sample from a uniform distribution over [0,1] conforms to H0: X ~ U(0,1) versus HA: density=2x, the likelihood ratio test involves comparing the empirical data to the specified distribution. The critical region can be constructed using the Neyman-Pearson lemma, and the test's power evaluated in terms of the probability of correctly rejecting H0 when HA is true.

Fuel Efficiency Testing Scenario

Applying hypothesis testing to vehicle fuel efficiency involves setting H0: μ ≥ 25 miles per gallon (mpg) versus HA: μ

Conclusion

Likelihood ratio tests are versatile tools rooted in the principles of statistical inference, applicable across various distributions and real-world scenarios. Proper formulation, understanding of hypothesis types, and power analysis enable practitioners to make reliable decisions and interpret results within the context of the data and assumptions. This exploration underscores the importance of rigorous statistical methodology in scientific and industrial decision-making.

References

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