Question

\(Y_{1}, Y_{2}\) are independent gamma variables with known shape parameters \(v_{1}, v_{2}\) and scale parameters \(\lambda_{1}, \lambda_{2}\), and it is desired to test the null hypothesis \(H_{0}\) that \(\lambda_{1}=\lambda_{2}=\lambda\), with \(\lambda\) unknown. Show that a minimal sufficient statistic for \(\lambda\) under \(H_{0}\) is \(Y_{1}+Y_{2}\), find its distribution, and show that it is complete. Hence show that the test is based on the conditional distribution of \(Y_{1}\) given \(Y_{1}+Y_{2}\) and that significance levels are computed from integrals of form $$ \frac{\Gamma\left(v_{1}+v_{2}\right)}{\Gamma\left(v_{1}\right) \Gamma\left(v_{2}\right)} \int_{0}^{y_{1} /\left(y_{1}+y_{2}\right)} u^{v_{1}-1}(1-u)^{\nu_{2}-1} d u $$ Explain how this argument is useful in comparison of the scale parameters of two independent exponential samples.

Short answer

A minimal sufficient statistic is \( Y_1 + Y_2 \) which is gamma distributed; testing uses the conditional beta distribution of \( Y_1 \) given this sum.

Step-by-step solution

Define the Problem

We need to show that a minimal sufficient statistic for \( \lambda \) under the null hypothesis \( H_0 \) is \( Y_1 + Y_2 \). These variables, \( Y_1 \) and \( Y_2 \), are independent gamma variables with known shape parameters \( v_1, v_2 \) and scale parameters \( \lambda_1, \lambda_2 \). Under \( H_0 \), these scale parameters are equal (i.e., \( \lambda_1 = \lambda_2 = \lambda \)).

Find the Joint Distribution

Since \( Y_1 \) and \( Y_2 \) are gamma distributed, their joint distribution is given by the product of their individual distributions when they are independent. Under the null hypothesis, the joint PDF of \( Y_1 \) and \( Y_2 \) is: \[ f(y_1, y_2 | \lambda) = \frac{\lambda^{v_1} y_1^{v_1 - 1} e^{-\lambda y_1}}{\Gamma(v_1)} \cdot \frac{\lambda^{v_2} y_2^{v_2 - 1} e^{-\lambda y_2}}{\Gamma(v_2)}. \] Simplifying, we get: \[ \frac{\lambda^{v_1 + v_2} (y_1 y_2)^{v_1 + v_2 - 2} e^{-\lambda (y_1 + y_2)}}{\Gamma(v_1)\Gamma(v_2)}. \] This suggests that \( Y_1 + Y_2 \) has a key role in the distribution.

Determine the Minimal Sufficient Statistic

Using the factorization theorem, the function \( Y_1 + Y_2 \) appears in the exponent of \( e^{-\lambda (y_1 + y_2)} \), indicating that it is sufficient. The factor that involves only \( y_1 + y_2 \) suggests it is also minimal. Therefore, \( T = Y_1 + Y_2 \) is a minimal sufficient statistic for \( \lambda \).

Find the Distribution of the Statistic

Under \( H_0 \), \( Y_1 + Y_2 \) is distributed as a gamma distribution with parameters \( v_1 + v_2 \) and scale \( \lambda \). The PDF of the sum is: \[ \frac{\lambda^{v_1 + v_2} (y_1 + y_2)^{v_1 + v_2 - 1} e^{-\lambda (y_1 + y_2)}}{\Gamma(v_1 + v_2)}. \] Hence, \( Y_1 + Y_2 \sim \text{Gamma}(v_1 + v_2, \lambda) \).

Show Completeness of the Statistic

A statistic \( T \) is complete if for any function \( g \), \( E[g(T)] = 0 \) implies \( g(T) = 0 \) for all \( T \). Given that the gamma distribution is a part of the exponential family, it is known that \( Y_1 + Y_2 \) being gamma-distributed is complete for its parameter \( \lambda \). Thus, \( T = Y_1 + Y_2 \) is complete.

Derive the Conditional Distribution

The conditional distribution of \( Y_1 \) given \( Y_1 + Y_2 \) is a beta distribution: \( \frac{Y_1}{Y_1 + Y_2} \sim \text{Beta}(v_1, v_2) \). Thus, under the null hypothesis, \( \frac{Y_1}{Y_1 + Y_2} = U \sim \text{Beta}(v_1, v_2) \).

Calculate Significance Levels

The significance level is determined by the integral: \[ \frac{\Gamma(v_1+v_2)}{\Gamma(v_1) \Gamma(v_2)} \int_{0}^{y_1 /(y_1+y_2)} u^{v_1-1}(1-u)^{v_2-1} \, du, \] which is the cumulative distribution function of the beta distribution.

Application to Exponential Samples

In the case of exponential distribution, the shape parameter is 1, and the problem simplifies to testing whether two independent exponential samples have the same scale parameter. Thus, the strategy described can be used to compare their scale parameters.

Key concepts

Minimal Sufficient Statistic

In statistical analysis, finding a minimal sufficient statistic is crucial as it helps condense the data without losing relevant information about the parameter of interest. Here, we look at the random variables \( Y_1 \) and \( Y_2 \), both of which are independent gamma distributed with known shape parameters and scale parameter \( \lambda \) under the null hypothesis \( H_0: \lambda_1 = \lambda_2 = \lambda \).

The combination \( Y_1 + Y_2 \) becomes particularly essential. By applying the factorization theorem, it shows up prominently in the factor of the joint probability distribution, indicating sufficiency. Since it cannot be reduced further and still suffice, it is minimal. Essentially, the statistic captures all the relevant data needed to estimate \( \lambda \) effectively.

Therefore, \( Y_1 + Y_2 \) serves as a minimal sufficient statistic for \( \lambda \) under the given hypothesis, ensuring that all the necessary information is compacted into this single measure.

Complete Statistic

Completeness of a statistic is a desirable property that ensures no unnecessary information is held beyond what's needed to understand the parameter. A statistic is complete if any function of it yielding an expected value of zero is genuinely zero for all values.

In our case, \( Y_1 + Y_2 \) aligns with the gamma distribution, a recognized member of the exponential family, which is known for offering complete statistics under certain conditions. This property allows us to say confidently that no information is lost, and the statistic \( Y_1 + Y_2 \) maintains all the complete information about \( \lambda \).

Completeness is beneficial in hypothesis testing because it provides robust parameter estimation and uniqueness in certain statistical procedures.

Beta Distribution

The construction of a conditional distribution reveals further insights into our variables, especially when examining dependent ratios. The ratio \( \frac{Y_1}{Y_1 + Y_2} \) follows a beta distribution, specifically \( \text{Beta}(v_1, v_2) \).

Beta distributions are particularly useful owing to their ability to handle proportions and weighted observations. In our hypothesis testing scenario, this distribution plays a pivotal role as it allows us to observe and evaluate the probabilities in a bounded interval \([0,1]\).

The significance levels in our tests are derived using the beta distribution's cumulative function. Understanding this transformation aids us in performing accurate analyses, where we assess how the proportion of two gamma variables distributes, thereby predicting outcomes under the null hypothesis.

Exponential Family

Many probability distributions, including the gamma distribution, belong to the exponential family. This family is a set of probability distributions defined by a particular functional form, which offers widespread applicability due to its mathematical convenience.

For our problem, it implies that \( Y_1 + Y_2 \), being gamma-distributed, falls within this category, providing complete sufficiency for \( \lambda \).

Using properties of the exponential family allows for simplified statistical inference. It is vital in deriving properties like sufficiency and completeness of statistics, critical in hypothesis testing as established in our scenario. The flexibility and robustness tied to the exponential family prove essential when dealing with diverse statistical problems.

Scale Parameter Comparison

In practical statistical studies, comparing scale parameters is crucial, particularly with distributions like gamma and exponential, which feature it prominently. Here, we aim to determine if the scale parameters from two independent gamma distributions are effectively equal.

Testing this null hypothesis involves determining if \( \lambda_1 = \lambda_2 \), and is made feasible by evaluating the minimal sufficient statistic \( Y_1 + Y_2 \).

Interestingly, when this context applies to exponential distributions (a special case of the gamma with shape parameter 1), the problem further simplifies, offering a streamlined method for scale comparisons. Understanding how to leverage these relationships is pivotal in statistical inference as it clarifies which distributions share attributes that can be tested collectively, optimizing our approach to hypothesis testing across different distributions.