# Calculate the density - Unbiased estimator for θ

• MHB
• mathmari
In summary: Keep up the good work!In summary, the conversation discusses a statistical product model with three estimators proposed for a parameter theta. The conversation also explores the density of the maximum value of a set of variables and the process of calculating probability from a density function. Finally, the conversation mentions the concept of expected value and how it relates to determining the unbiasedness of an estimator.
mathmari
Gold Member
MHB
Hey! :giggle:

For $n \in \mathbb{N}$ we consider the statistical product model $(X ,(P_{\theta})_{\theta\in\Theta})$ with $X = (0,\infty)^n$, $\Theta = (0,\infty)$ and densities $f_{\theta}(x_i) = \frac{1}{\theta} \textbf{1}_{(0,\theta)}(x_i)$ for all $x_i \in (0,\infty)$, $\theta \in \Theta$. It will be the three estimators $T$, $\tilde{T}$ , $\hat{T}$ : $X \rightarrow \mathbb{R}$, $$T(x)=\frac{2}{n}\sum_{i=1}^nx_i, \ \ \ \tilde{T}(x)=c\cdot \max (x_1,\ldots ,x_n), \ \ \ \hat{T}(x)=2x_1$$ proposed for $\theta$, where $c \in (0,\infty)$ is a constant.

(a) Show that $f : \mathbb{R} \rightarrow [0,\infty)$, $f(y) = \frac{n}{\theta^n} y^{n-1}\textbf{1}_{(0,\theta)}(y)$, is the density of $\max(X_1, \ldots , X_n)$ by first calculating the distribution function $F(y) = P_{\theta}[\max(X_1,\ldots , X_n) \leq y]$ and then deriving it.

(b) For which $c \in (0,\infty)$ is $\tilde{T}$ unbiased for $\theta$ ?

(c) Calculate for $T$, $\tilde{T}$ and $\hat{T}$ each the mean square deviation at $\theta$.
At (a) we have :

We have that $\max(X_1,\ldots , X_n) \leq y$ if $X_i\leq y$ for all $1\leq i\leq n$, right?
Therefore we get $F(y) = P_{\theta}[\max(X_1,\ldots , X_n) \leq y]=\left (P_{\theta}[X_1\leq y]\right )^n$.
Do we know that probability? How could we continue? :unsure: At (b) do we check if $E(T)=\theta$ , $E(\tilde{T})=\theta$ and $E(\hat{T})=\theta$ ? Or what are we supposed to check ? :unsure:

Hi mathmari,

Good work so far. Here are a few ideas to keep things moving.

(a) You're correct that $\max(X_1,\ldots , X_n) \leq y$ iff $X_i\leq y$ for all $1\leq i\leq n$. It is also true in this case that $F(y) = \left (P_{\theta}[X_1\leq y]\right )^n$, but this does not follow just because $\max(X_1,\ldots , X_n) \leq y$ iff $X_i\leq y$ for all $1\leq i\leq n$. The reason this is true is because each of the variables $X_{i}$ is distributed identically/the same way (in this case uniformly on the interval $(0,\theta)$) according to the probability distributions/densities $f_{\theta}(x_{i})$. If this were not the case, you could not conclude $F(y)=\left (P_{\theta}[X_1\leq y]\right )^n$.

In general (i.e., this applies broadly and not just to this problem), to obtain a probability from a density, we integrate the density over an appropriate interval. For example, if we wanted to know the probability that $0.25\leq X_{i}\leq 0.75$, we would calculate $$\int_{0.25}^{0.75}f_{\theta}(x_{i})dx_{i}.$$
Try using this information to determine what $P_{\theta}[X_1\leq y]$ is. The cumulative probability is then $F(y) = \left (P_{\theta}[X_1\leq y]\right )^n.$ By definition of probability densities (see Wikipedia - Probability Density), the probability density of $\max(X_1,\ldots , X_n)$ is the derivative of $F(y)$ with respect to $y$.

(b) You need to determine the value of $c$ so that $\theta = E(\tilde{T})$. In general, if a random variable $X$ has a probability density function $f(x)$, then the expected value of $X$ is $$E[X] = \int_{-\infty}^{\infty}x\cdot f(x)dx$$
You will want to use your answer from part (a) to calculate $E(\tilde{T})$.

Feel free to let me know if you have any other questions.

Last edited:

## 1. What is the purpose of calculating the density?

The purpose of calculating the density is to determine the amount of mass per unit volume of a substance. This is an important measurement in many scientific fields, as it can help identify and classify materials, as well as provide insight into their physical properties.

## 2. What is an unbiased estimator for θ?

An unbiased estimator for θ is a statistical method used to estimate the true value of a parameter (θ) without any systematic errors or biases. In other words, it is a method that, on average, produces estimates that are close to the true value of θ.

## 3. How is the density calculated?

The density is calculated by dividing the mass of a substance by its volume. The formula for density is: density = mass / volume. The units for density are typically grams per cubic centimeter (g/cm3) or kilograms per cubic meter (kg/m3).

## 4. Why is it important to use an unbiased estimator for θ?

Using an unbiased estimator for θ is important because it ensures that the estimated value is as close to the true value as possible. This is crucial in scientific research, as biased estimates can lead to incorrect conclusions and potentially impact the validity of a study.

## 5. Are there any limitations to using an unbiased estimator for θ?

Yes, there are limitations to using an unbiased estimator for θ. For example, the accuracy of the estimate may be affected by the sample size and the variability of the data. Additionally, certain assumptions must be met for the estimator to be unbiased, so it may not be suitable for all types of data.

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