Question

Show that the product of the sample observations is a sufficient statistic for \(\theta>0\) if the random sample is taken from a gamma distribution with parameters \(\alpha=\theta\) and \(\beta=6\).

Short answer

To show that the product of the sample observations is a sufficient statistic, apply the Neyman-Fisher Factorization Theorem. It states that a statistic is a sufficient statistic for a parameter if the joint PDF can be factorised in a certain way. After identifying the PDF of the Gamma distribution, derive the joint PDF of the sample observations and try to factorise it according to the theorem. If the factorisation is possible, then by the theorem, the product of the sample observations is indeed a sufficient statistic for \(\theta\).

Step-by-step solution

Identify the Probability Density Function (PDF)

The PDF of a single observation X from a Gamma distribution with parameters \(\alpha = \theta\) and \(\beta = 6\) can be written as: \(f(x|\theta) = \frac{1}{6^\theta \Gamma(\theta)}x^{\theta-1}e^{-x/6}\)

Derive the Joint Probability Density Function (PDF) of the Sample Observations

If X1, X2, ..., Xn is a random sample from this distribution, the joint PDF of these sample observations is the product of the individual PDFs. Hence, the joint PDF can be written as: \(f(x_1, x_2, ..., x_n|\theta) = \prod_{i=1}^{n}f(x_i|\theta)\)

Apply the Neyman-Fisher Factorization Theorem

The Neyman-Fisher Factorization Theorem states that a statistic T(X) is a sufficient statistic for \(\theta\) if the joint PDF can be factorized as a product of two factors, one of which depends on the sample x only through T(x), and the other depends on x only through \(\theta\). So try to factorize \(f(x_1, x_2, ..., x_n|\theta)\) accordingly to confirm if the product of sample observations is indeed a sufficient statistic.

Factorize the Joint PDF

Factorize \(f(x_1, x_2, ..., x_n|\theta)\) as \(f(x_1, x_2, ..., x_n|\theta) = g(T(x)|\theta)h(x)\), where T(x) is the product of sample observations and g and h are some appropriate functions. This step may involve some algebraic manipulations.

Confirm Sufficiency

If it's possible to factorize the joint PDF as described in Step 4, then T(x) is a sufficient statistic for \(\theta\) by the Neyman-Fisher Factorization Theorem.

Key concepts

Understanding Probability Density Function (PDF)

Within the field of statistics, the Probability Density Function (PDF) is a fundamental concept used to describe the likelihood of a continuous random variable taking on a particular value. Essentially, a PDF provides a curve where the probability of the variable falling within a certain range can be found by calculating the area under the curve for that range.

For example, the PDF of a gamma distribution with parameters \(\alpha = \theta\) and \(\beta = 6\) is given by: \[f(x|\theta) = \frac{1}{6^\theta \Gamma(\theta)}x^{\theta-1}e^{-x/6}\] In this expression, \(\Gamma(\theta)\) is the gamma function, which generalizes the factorial function to continuous values, and \(\frac{1}{6^\theta \Gamma(\theta)}\) serves as a scaling factor to ensure that the total area under the PDF curve equals 1, which is a requirement for all probability distributions.

Gamma Distribution Insights

The gamma distribution is a versatile probability distribution that includes exponential and chi-squared distributions as special cases. It is particularly useful for modeling the time until an event occurs, such as the lifespan of a product or the time between occurrences of an event.

The gamma distribution is characterized by two parameters: \(\alpha\) and \(\beta\), which influence the shape and scale of the distribution, respectively. When a random variable X is said to follow a gamma distribution with parameters \(\alpha = \theta\) and \(\beta = 6\), it implies a certain skewness and decay behavior of the distribution, signified by the chosen values of \(\theta\) and 6.

Neyman-Fisher Factorization Theorem Explained

The Neyman-Fisher Factorization Theorem is a key result in the theory of statistical estimation, which establishes a criterion for a statistic to be considered sufficient for a parameter \(\theta\). A sufficient statistic is capable of summarizing all the information in a sample that is relevant for estimating a population parameter.

To utilize this theorem, one needs to express the joint PDF of sample observations as a product of two separate functions: one that depends solely on the data through the sufficient statistic and another that depends only on the parameter. This factorization makes the task of statistical inference more efficient, as it allows us to focus solely on the sufficient statistic rather than the entire dataset.

Joint PDF of Sample Observations

In practice, statisticians often work with samples rather than individual observations. In such cases, the joint PDF of sample observations is an essential tool. For independent and identically distributed (i.i.d.) samples from a gamma distribution, the joint PDF is simply the product of each individual's PDF.

Mathematically, for a random sample \(X_1, X_2, ..., X_n\), the joint PDF would be: \[f(x_1, x_2, ..., x_n|\theta) = \prod_{i=1}^{n}f(x_i|\theta)\] This product allows us to treat the entire sample as a single entity in our calculations, which is particularly useful when applying techniques like the Neyman-Fisher factorization.

Statistical Inference Fundamentals

Statistical inference is a cornerstone of data analysis, enabling us to draw conclusions about a population from a sample. Types of inference include estimating population parameters, testing hypotheses, and predicting future observations. The inference process is guided by the choice of a sufficient statistic, as it summarizes all necessary information from the sample related to the parameter of interest.

In the context of the gamma distribution example, if the product of the sample observations is a sufficient statistic for \(\theta\), it means that this product captures all the information needed to infer about \(\theta\). In other words, no other information from the sample can provide additional clues about the value of the parameter, which greatly simplifies the inferential process.