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Chapter 6: Problem 4

Let \(Y_{1}, \ldots, Y_{n}\) represent the trajectory of a stationary two-state discrete-time Markov chain, in which $$ \operatorname{Pr}\left(Y_{j}=a \mid Y_{1}, \ldots, Y_{j-1}\right)=\operatorname{Pr}\left(Y_{j}=a \mid Y_{j-1}=b\right)=\theta_{b a}, \quad a, b=1,2 $$ note that \(\theta_{11}=1-\theta_{12}\) and \(\theta_{22}=1-\theta_{21}\), where \(\theta_{12}\) and \(\theta_{21}\) are the transition probabilities from state 1 to 2 and vice versa. Show that the likelihood can be written in form \(\theta_{12}^{n_{12}}\left(1-\theta_{12}\right)^{n_{11}} \theta_{21}^{n_{21}}\left(1-\theta_{21}\right)^{n_{22}}\), where \(n_{a b}\) is the number of \(a \rightarrow b\) transitions in \(y_{1}, \ldots, y_{n}\). Find a minimal sufficient statistic for \(\left(\theta_{12}, \theta_{21}\right)\), the maximum likelihood estimates \(\widehat{\theta}_{12}\) and \(\widehat{\theta}_{21}\), and their asymptotic variances.

Short Answer

Expert verified
The likelihood is expressed in the given form with transition counts. The MLEs are \(\widehat{\theta}_{12} = \frac{n_{12}}{n_{11} + n_{12}}\) and \(\widehat{\theta}_{21} = \frac{n_{21}}{n_{22} + n_{21}}\), with asymptotic variances depending on the sample size.

Step by step solution

01

Define the Transition Counts

First, identify the quantities \(n_{ab}\) which represent the number of transitions from state \(a\) to state \(b\). These counts are critical in determining the likelihood. Specifically, \(n_{12}\) is the count of transitions from state 1 to state 2, and \(n_{21}\) is the count from state 2 to state 1. Similarly, \(n_{11}\) and \(n_{22}\) are the counts of staying in state 1 and state 2, respectively.
02

Express the Likelihood Function

The likelihood is the product of the transition probabilities raised to their corresponding counts. For a sequence of observations, it is given by:\[L(\theta_{12}, \theta_{21}) = \theta_{12}^{n_{12}} (1-\theta_{12})^{n_{11}} \theta_{21}^{n_{21}} (1-\theta_{21})^{n_{22}}.\] This expression captures the likelihood of observing the sequence \(Y_1, \ldots, Y_n\) under the specified Markov chain model.
03

Identify the Minimal Sufficient Statistic

The minimal sufficient statistic is a function of the data that captures all necessary information for estimating the parameters. Here, the counts \(n_{12}, n_{21}, n_{11}, n_{22}\) serve as a sufficient statistic because they allow us to calculate the likelihood without loss of information about \(\theta_{12}\) and \(\theta_{21}\).
04

Calculate the Maximum Likelihood Estimates (MLE)

To find the MLEs for \(\theta_{12}\) and \(\theta_{21}\), we take derivatives of the likelihood, set them to zero, and solve the equations. The result is:\[\widehat{\theta}_{12} = \frac{n_{12}}{n_{11} + n_{12}}, \quad \widehat{\theta}_{21} = \frac{n_{21}}{n_{22} + n_{21}}.\] These estimators maximize the likelihood function for the observed data.
05

Determine the Asymptotic Variances

The asymptotic variance for the MLE of each parameter can be found using the observed information method. For a large sample size, these variances are:\[Var(\widehat{\theta}_{12}) = \frac{\theta_{12}(1-\theta_{12})}{n_{11} + n_{12}}, \quad \Var(\widehat{\theta}_{21}) = \frac{\theta_{21}(1-\theta_{21})}{n_{22} + n_{21}}.\] These expressions show that the variances decrease as the number of observed transitions increases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transition Probabilities
In a Markov chain, transition probabilities play a crucial role. They determine the likelihood of moving from one state to another in a sequence. Consider a two-state Markov chain, where the states are labeled as 1 and 2. The transition probability from state 1 to state 2 is denoted by \( \theta_{12} \), while the probability from state 2 back to state 1 is \( \theta_{21} \).
These probabilities must sum to one within each state because they represent all possible transitions. Hence, \( \theta_{11} = 1 - \theta_{12} \) for staying in state 1, and \( \theta_{22} = 1 - \theta_{21} \) for staying in state 2. By knowing these probabilities, we can predict the behavior of the Markov chain over time.
  • Understanding transition probabilities helps in modeling and predicting dynamic systems.
  • They are key to determining how systems evolve over time.
Models like these are widely used in various fields like genetics, weather forecasting, and economics.
Likelihood Function
The likelihood function in the context of a Markov chain is a way to quantify how probable a sequence of events is, given certain transition probabilities. When we observe a sequence of states \( Y_1, Y_2, \ldots, Y_n \), we want to determine the likelihood of observing this specific sequence under the model.
For our two-state Markov chain, the likelihood is represented by:\[L(\theta_{12}, \theta_{21}) = \theta_{12}^{n_{12}} (1-\theta_{12})^{n_{11}} \theta_{21}^{n_{21}} (1-\theta_{21})^{n_{22}}.\]This formula captures the likelihood as a product of the transition probabilities raised to their respective transition counts. Each term corresponds to how frequently those transitions appear in the observed sequence.
  • It is an essential tool for estimating parameters in probabilistic models.
  • The likelihood function aids in comparing how well different models explain the observed data.
A better understanding of this function allows for better model fitting and predictions.
Maximum Likelihood Estimation
Maximum Likelihood Estimation (MLE) is a method used to estimate the parameters of a statistical model. In the context of our Markov chain, it provides estimates for the transition probabilities \( \theta_{12} \) and \( \theta_{21} \).
To find the MLE, we calculate the derivatives of the likelihood function concerning the parameters and set them to zero. Solving these equations gives the MLEs:\[\widehat{\theta}_{12} = \frac{n_{12}}{n_{11} + n_{12}}, \quad \widehat{\theta}_{21} = \frac{n_{21}}{n_{22} + n_{21}}.\]These estimates maximize the probability of observing the given sequence according to the model. The larger the observed sample, the more reliable the estimates, as random fluctuations average out.
  • MLE is a powerful tool because it results in parameter estimates that create the highest likelihood of observing the given data.
  • Understanding MLE helps improve model accuracy and precision in statistical analysis.
This method is fundamental in various practical applications, such as genetics, finance, and machine learning.
Sufficient Statistic
A sufficient statistic provides a simplified representation of data that can capture all necessary information needed for parameter estimation. In our Markov chain scenario, the transition counts \( n_{12}, n_{21}, n_{11}, n_{22} \) serve as sufficient statistics for \( \theta_{12} \) and \( \theta_{21} \).
These counts allow us to calculate the likelihood function without losing any necessary information about the parameters. The concept of sufficient statistics ensures that we do not have to keep the entire data set, as these specific counts contain all the relevant information.
  • Utilizing sufficient statistics simplifies computations in statistical problems.
  • They reduce data complexity while retaining crucial information.
Understanding sufficient statistics is crucial in fields such as data science and engineering, allowing for more efficient data processing and analysis.

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Most popular questions from this chapter

Find the eigendecomposition of $$ P=\left(\begin{array}{ccc} 0 & 1 & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & 0 & \frac{1}{2} \end{array}\right) $$ and show that \(p_{11}(n)=a+2^{-n}\\{b \cos (n \pi / 2)+c \sin (n \pi / 2)\\}\) for some constants \(a, b\) and \(c\). Write down \(p_{11}(n)\) for \(n=0,1\) and 2 and hence find \(a, b\) and \(c\).

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