Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample on \(X\) that has a \(\Gamma(\alpha=4, \beta=\theta)\) distribution, \(0<\theta<\infty\) (a) Determine the mle of \(\theta\). (b) Suppose the following data is a realization (rounded) of a random sample on \(X\). Obtain a histogram with the argument \(\mathrm{pr}=\mathrm{T}\) (data are in ex6111.rda). \(\begin{array}{lllllllllll}9 & 39 & 38 & 23 & 8 & 47 & 21 & 22 & 18 & 10 & 17 & 22 & 14\end{array}\) \(\begin{array}{llllllllllll}9 & 5 & 26 & 11 & 31 & 15 & 25 & 9 & 29 & 28 & 19 & 8\end{array}\) (c) For this sample, obtain \(\hat{\theta}\) the realized value of the mle and locate \(4 \hat{\theta}\) on the histogram. Overlay the \(\Gamma(\alpha=4, \beta=\hat{\theta})\) pdf on the histogram. Does the data agree with this pdf? Code for overlay: \(\mathrm{xs}=\) sort \((\mathrm{x}) ; \mathrm{y}=\mathrm{dgamma}(\mathrm{xs}, 4,1 / \mathrm{betahat}) ;\) hist \((\mathrm{x}, \mathrm{pr}=\mathrm{T}) ;\) lines \(\left(\mathrm{y}^{\sim} \mathrm{xs}\right)\).

Short answer

The MLE of \(\theta\) is calculated using the formula \(\frac{1}{n}\sum_{i=1}^{n} X_i\). A histogram of the given data is obtained, and the \(\Gamma(\alpha=4, \beta=\hat{\theta})\) PDF is overlayed. The agreement of the data with this PDF is analysed visually by looking at the similarity between the histogram and the overlayed PDF.

Step-by-step solution

Calculation of MLE of theta

In a Gamma distribution \(\Gamma(\alpha=4, \beta=\theta)\), the MLE of \(\theta\) can be obtained by using the following expression: \(\hat{\theta} = \frac{1}{n}\sum_{i=1}^{n} X_i\). Here, \(n\) is the total number of samples.

Converting the data into a usable sample

The provided data must be converted into a usable sample in R. The list of all the given numbers is combined to form the data sample.

Creating a histogram

The histogram is created using the hist() function in R with argument pr=T. This will provide a probability density rather than a frequency, which is essential in comparing the histogram to the overlayed gamma distribution.

Calculating the realized value of MLE

The realized value of the MLE, \(\hat{\theta}\), can be calculated for this sample by substituting the data into the MLE formula from step 1.

Overlay the PDF on the histogram

The Gamma distribution PDF with parameters \(\alpha=4,\beta=\hat{\theta}\) is overlayed on the histogram. The PDF is generated using the function dgamma() in R, and then the line plot of \(y\sim xs\) is superimposed on top of the histogram.

Compare the PDF with the Data

By visually inspecting the overlayed histogram and PDF, it can be observed if the given data sample agrees with the PDF of the Gamma distribution with specified parameters.

Key concepts

Gamma Distribution

The Gamma distribution is a two-parameter family of continuous probability distributions used widely in statistical models for a variety of disciplines. It's especially relevant when dealing with variables that are skewed and must be positive since the Gamma distribution is defined only for positive real numbers.

Represented as \( \Gamma(\alpha, \beta) \) with shape parameter \( \alpha \) and scale parameter \( \beta \) which are both positive real numbers, this distribution captures the time until an event occurs given a certain average frequency of occurrence.

Understanding this distribution is critical for maximum likelihood estimation (MLE) when you have a dataset that follows it, as in the exercise where sample data is assumed to follow a Gamma distribution with a known \( \alpha=4 \) and unknown scale parameter \( \theta \).

Probability Density Function

The probability density function (PDF) is a function that describes the relative likelihood for a random variable to take on a given value. For continuous variables, the PDF provides the heights of the curve such that the area under the curve in any interval gives the probability of the variable falling within that interval.

In our specific context of the Gamma distribution, the PDF is crucial for visualization. It describes the distribution of our random variable, and by overlaying the PDF of a Gamma distribution on a histogram of sampled data, we can observe how well the model fits the actual data.

Getting hands-on with real data and applying the concept of PDF as in the provided exercise (step 5), where \(\mathrm{dgamma}()\) function in R is used to overlay the model's PDF over the data histogram, directly illustrates the practical application of probability density functions in data analysis and model fitting.

Statistical Inference

Statistical inference involves using data from a sample to make generalizations about a larger population. It's a fundamental aspect of statistics that encompasses various methodologies like point estimation, confidence intervals, and hypothesis tests.

Maximum likelihood estimation is a method of statistical inference used to find the parameter values that make the observed data most probable. In our exercise example, the MLE of \(\theta\) gives us the value that would make the observed sample most likely if the population indeed follows a Gamma distribution. Essentially, statistical inference helps provide the justification and methodology to arrive at conclusions about the broader context from which the sample is drawn.

When we overlay the PDF and inspect the fit to the data (see step 6), we are using the inferred model to validate our statistical assumptions against empirical evidence, a fundamental process in statistical inference.