Finding UMVUE Of A Poisson Parameter

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Introduction

In this article, we will discuss how to find the uniformly minimum variance unbiased estimator (UMVUE) of a Poisson parameter. We will start by defining the problem and the necessary concepts.

Problem Definition

Suppose we have a random variable XX that follows a Poisson distribution with parameter λ\lambda, where λ>0\lambda > 0. We are interested in finding the UMVUE of the parameter θ=λ3e−2λ\theta = \lambda^3e^{-2\lambda}.

Part a: Showing that S is an unbiased estimator of θ

To show that S=(−1)X−1X(X−1)(X−2)S = (-1)^{X-1}X(X-1)(X-2) is an unbiased estimator of θ\theta, we need to prove that the expected value of SS is equal to θ\theta.

Expected Value of S

The expected value of SS can be calculated as follows:

E[S]=E[(−1)X−1X(X−1)(X−2)]E[S] = E[(-1)^{X-1}X(X-1)(X-2)]

Using the properties of the Poisson distribution, we can rewrite the expected value as:

E[S]=∑x=3∞(−1)x−1x(x−1)(x−2)e−λλxx!E[S] = \sum_{x=3}^{\infty} (-1)^{x-1} x(x-1)(x-2) \frac{e^{-\lambda} \lambda^x}{x!}

Simplifying the expression, we get:

E[S]=∑x=3∞(−1)x−1(x−1)(x−2)e−λλx(x−3)!E[S] = \sum_{x=3}^{\infty} (-1)^{x-1} (x-1)(x-2) \frac{e^{-\lambda} \lambda^x}{(x-3)!}

Using the fact that ∑x=3∞λx(x−3)!=e2λ\sum_{x=3}^{\infty} \frac{\lambda^x}{(x-3)!} = e^{2\lambda}, we can rewrite the expected value as:

E[S]=∑x=3∞(−1)x−1(x−1)(x−2)e−λe2λ(x−3)!E[S] = \sum_{x=3}^{\infty} (-1)^{x-1} (x-1)(x-2) \frac{e^{-\lambda} e^{2\lambda}}{(x-3)!}

Simplifying further, we get:

E[S]=∑x=3∞(−1)x−1(x−1)(x−2)e−λ+2λ(x−3)!E[S] = \sum_{x=3}^{\infty} (-1)^{x-1} (x-1)(x-2) \frac{e^{-\lambda + 2\lambda}}{(x-3)!}

E[S]=∑x=3∞(−1)x−1(x−1)(x−2)eλ(x−3)!E[S] = \sum_{x=3}^{\infty} (-1)^{x-1} (x-1)(x-2) \frac{e^{\lambda}}{(x-3)!}

Using the fact that ∑x=3∞eλ(x−3)!=e2λ\sum_{x=3}^{\infty} \frac{e^{\lambda}}{(x-3)!} = e^{2\lambda}, we can rewrite the expected value as:

E[S]=∑x=3∞(−1)x−1(x−1)(x−2)eλ(x−3)!E[S] = \sum_{x=3}^{\infty} (-1)^{x-1} (x-1)(x-2) \frac{e^{\lambda}}{(x-3)!}

Simplifying further, we get:

E[S]=eλ∑x=3∞(−1)x−1(x−1)(x−2)1(x−3)!E[S] = e^{\lambda} \sum_{x=3}^{\infty} (-1)^{x-1} (x-1)(x-2) \frac{1}{(x-3)!}

Using the fact that ∑x=3∈(−1)x−1(x−1)(x−2)1(x−3)!=−3e−2λ\sum_{x=3}^{\in} (-1)^{x-1} (x-1)(x-2) \frac{1}{(x-3)!} = -3e^{-2\lambda}, we can rewrite the expected value as:

E[S]=eλ(−3e−2λ)E[S] = e^{\lambda} (-3e^{-2\lambda})

E[S]=−3eλ−2λE[S] = -3e^{\lambda - 2\lambda}

E[S]=−3e−λE[S] = -3e^{-\lambda}

Using the fact that θ=λ3e−2λ\theta = \lambda^3e^{-2\lambda}, we can rewrite the expected value as:

E[S]=−3e−λE[S] = -3e^{-\lambda}

E[S]=−31λ3λ3e−2λE[S] = -3 \frac{1}{\lambda^3} \lambda^3e^{-2\lambda}

E[S]=−3θE[S] = -3 \theta

Since E[S]=−3θE[S] = -3\theta, we can conclude that SS is not an unbiased estimator of θ\theta. However, we can modify SS to make it an unbiased estimator.

Part b: Modifying S to make it an unbiased estimator of θ

To modify SS to make it an unbiased estimator of θ\theta, we can multiply it by a constant. Let's call the modified estimator TT.

T=kST = kS

where kk is a constant.

We want to find the value of kk that makes TT an unbiased estimator of θ\theta.

Expected Value of T

The expected value of TT can be calculated as follows:

E[T]=E[kS]E[T] = E[kS]

E[T]=kE[S]E[T] = kE[S]

Since E[S]=−3θE[S] = -3\theta, we can rewrite the expected value as:

E[T]=k(−3θ)E[T] = k(-3\theta)

We want to find the value of kk that makes E[T]=θE[T] = \theta.

k(−3θ)=θk(-3\theta) = \theta

−3kθ=θ-3k\theta = \theta

−3k=1-3k = 1

k=−13k = -\frac{1}{3}

So, the modified estimator TT is:

T=−13ST = -\frac{1}{3}S

T=13(−1)X−1X(X−1)(X−2)T = \frac{1}{3}(-1)^{X-1}X(X-1)(X-2)

Conclusion

In this article, we discussed how to find the uniformly minimum variance unbiased estimator (UMVUE) of a Poisson parameter. We started by defining the problem and the necessary concepts. We then showed that S=(−1)X−1X(X−1)(X−2)S = (-1)^{X-1}X(X-1)(X-2) is not an unbiased estimator of θ=λ3e−2λ\theta = \lambda^3e^{-2\lambda}. However, we modified SS to make it an unbiased estimator T=13(−1)X−1X(X−1)(X−2)T = \frac{1}{3}(-1)^{X-1}X(X-1)(X-2).

UMVUE of θ

To find the UMVUE of θ\theta, we need to find the estimator that has the minimum variance among all unbiased estimators.

Rao-Blackwell Theorem

The Rao-Blackwell theorem states that if we have an unbiased estimator TT and a sufficient statistic SS for the parameter θ\theta, then the estimator E[T∣S]E[T|S] is also unbiased and has a smaller variance than TT.

In this case, we have sufficient statistic S=(−1)X−1X(X−1)(X−2)S = (-1)^{X-1}X(X-1)(X-2) and the unbiased estimator T=13ST = \frac{1}{3}S. We can use the Rao-Blackwell theorem to find the UMVUE of θ\theta.

UMVUE of θ

Using the Rao-Blackwell theorem, we can find the UMVUE of θ\theta as follows:

E[T∣S]=E[13S∣S]E[T|S] = E\left[\frac{1}{3}S|S\right]

E[T∣S]=13E[S∣S]E[T|S] = \frac{1}{3}E[S|S]

Since E[S∣S]=SE[S|S] = S, we can rewrite the UMVUE as:

E[T∣S]=13SE[T|S] = \frac{1}{3}S

E[T∣S]=13(−1)X−1X(X−1)(X−2)E[T|S] = \frac{1}{3}(-1)^{X-1}X(X-1)(X-2)

So, the UMVUE of θ\theta is:

θ^=13(−1)X−1X(X−1)(X−2)\hat{\theta} = \frac{1}{3}(-1)^{X-1}X(X-1)(X-2)

Final Answer

In conclusion, we have found the UMVUE of the Poisson parameter θ=λ3e−2λ\theta = \lambda^3e^{-2\lambda} as:

θ^=13(−1)X−1X(X−1)(X−2)\hat{\theta} = \frac{1}{3}(-1)^{X-1}X(X-1)(X-2)

This estimator has the minimum variance among all unbiased estimators and is the best estimator for the parameter θ\theta.

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Q: What is the uniformly minimum variance unbiased estimator (UMVUE) of a Poisson parameter?

A: The UMVUE of a Poisson parameter is an estimator that has the minimum variance among all unbiased estimators. In this case, the UMVUE of the Poisson parameter θ=λ3e−2λ\theta = \lambda^3e^{-2\lambda} is:

θ^=13(−1)X−1X(X−1)(X−2)\hat{\theta} = \frac{1}{3}(-1)^{X-1}X(X-1)(X-2)

Q: Why is the UMVUE important?

A: The UMVUE is important because it provides the best possible estimate of the parameter with the minimum variance. This means that the UMVUE is the most reliable and accurate estimator for the parameter.

Q: What is the Rao-Blackwell theorem?

A: The Rao-Blackwell theorem is a statistical theorem that states that if we have an unbiased estimator TT and a sufficient statistic SS for the parameter θ\theta, then the estimator E[T∣S]E[T|S] is also unbiased and has a smaller variance than TT.

Q: How is the UMVUE related to the Rao-Blackwell theorem?

A: The UMVUE is related to the Rao-Blackwell theorem in that it is the result of applying the theorem to the estimator T=13ST = \frac{1}{3}S. The Rao-Blackwell theorem states that E[T∣S]E[T|S] is the UMVUE, and in this case, E[T∣S]=13SE[T|S] = \frac{1}{3}S.

Q: What is the relationship between the UMVUE and the sufficient statistic?

A: The UMVUE is the result of conditioning the estimator TT on the sufficient statistic SS. This means that the UMVUE is a function of the sufficient statistic SS and is the best possible estimate of the parameter with the minimum variance.

Q: How is the UMVUE used in practice?

A: The UMVUE is used in practice to estimate the parameter of a Poisson distribution. It is the most reliable and accurate estimator for the parameter and is used in a variety of applications, including statistics, engineering, and economics.

Q: What are some common applications of the UMVUE?

A: Some common applications of the UMVUE include:

  • Estimating the parameter of a Poisson distribution
  • Modeling the number of events in a fixed interval
  • Analyzing the distribution of counts in a sample
  • Estimating the probability of a rare event

Q: What are some common challenges in finding the UMVUE?

A: Some common challenges in finding the UMVUE include:

  • Identifying the sufficient statistic
  • Finding the UMVUE using the Rao-Blackwell theorem
  • Verifying the unbiasedness and minimum variance of the UMVUE
  • Applying the UMVUE in practice

Q: How can the UMVUE be used to improve statistical analysis?

A: The UMVUE can be used to improve statistical analysis by providing the most reliable and accurate estimate of the parameter. This can lead to better decision-making and more accurate predictions.

Q: What are some common misconceptions about the UMVUE?

A: Some common misconceptions about the UMVUE include:

  • Believing that the UMVUE is always the best estimator
  • Thinking that the UMVUE is only used in theoretical statistics
  • Assuming that the UMVUE is always easy to find

Q: How can the UMVUE be used to improve statistical modeling?

A: The UMVUE can be used to improve statistical modeling by providing the most reliable and accurate estimate of the parameter. This can lead to better model selection and more accurate predictions.

Q: What are some common applications of the UMVUE in real-world scenarios?

A: Some common applications of the UMVUE in real-world scenarios include:

  • Estimating the number of defects in a manufacturing process
  • Modeling the number of customers in a store
  • Analyzing the distribution of counts in a sample
  • Estimating the probability of a rare event

Q: How can the UMVUE be used to improve data analysis?

A: The UMVUE can be used to improve data analysis by providing the most reliable and accurate estimate of the parameter. This can lead to better decision-making and more accurate predictions.

Q: What are some common challenges in applying the UMVUE in practice?

A: Some common challenges in applying the UMVUE in practice include:

  • Identifying the sufficient statistic
  • Finding the UMVUE using the Rao-Blackwell theorem
  • Verifying the unbiasedness and minimum variance of the UMVUE
  • Applying the UMVUE in complex data sets

Q: How can the UMVUE be used to improve statistical inference?

A: The UMVUE can be used to improve statistical inference by providing the most reliable and accurate estimate of the parameter. This can lead to better decision-making and more accurate predictions.

Q: What are some common applications of the UMVUE in statistical inference?

A: Some common applications of the UMVUE in statistical inference include:

  • Estimating the parameter of a Poisson distribution
  • Modeling the number of events in a fixed interval
  • Analyzing the distribution of counts in a sample
  • Estimating the probability of a rare event

Q: How can the UMVUE be used to improve statistical modeling?

A: The UMVUE can be used to improve statistical modeling by providing the most reliable and accurate estimate of the parameter. This can lead to better model selection and more accurate predictions.

Q: What are some common challenges in applying the UMVUE in statistical modeling?

A: Some common challenges in applying the UMVUE in statistical modeling include:

  • Identifying the sufficient statistic
  • Finding the UMVUE using the Rao-Blackwell theorem
  • Verifying the unbiasedness and minimum variance of the UMVUE
  • Applying the UMVUE in complex data sets

Q: How can the UMVUE be used to improve statistical analysis?

A: The UMVUE can be used to improve statistical analysis by providing the most reliable and accurate estimate of the parameter. This can lead to better decision and more accurate predictions.

Q: What are some common applications of the UMVUE in statistical analysis?

A: Some common applications of the UMVUE in statistical analysis include:

  • Estimating the parameter of a Poisson distribution
  • Modeling the number of events in a fixed interval
  • Analyzing the distribution of counts in a sample
  • Estimating the probability of a rare event