Given the function f(x)=(θ)/(x^2), x ≥θ, θ>0, (a) Verify that it is a density function. (b)

step 1 For exercise 7a, we have F of x equals 1 half, 1 plus Theta x, where we have negative 1 is less

Given the function f(x)=(θ)/(x^2), x ≥θ, θ>0, (a) Verify...

Given the function $$ f(x)=\frac{\theta}{x^2}, \quad x \geq \theta, \quad \theta>0, $$ (a) Verify that it is a density function. (b) Find the maximum likelihood estimators of $\theta$ and $1 / \theta$ for random samples of size $n$. (c) Is the maximum likelihood estimator of $\theta$ unbiased? (d) Find the method of moments estimators of $\theta$ and $1 / \theta$.

Given the function
$$
f(x)=\frac{\theta}{x^2}, \quad x \geq \theta, \quad \theta>0,
$$
(a) Verify that it is a density function.
(b) Find the maximum likelihood estimators of $\theta$ and $1 / \theta$ for random samples of size $n$.
(c) Is the maximum likelihood estimator of $\theta$ unbiased?
(d) Find the method of moments estimators of $\theta$ and $1 / \theta$.

Key Concepts

Method of Moments Estimation

The method of moments is an alternative estimation technique that involves equating sample moments (such as the sample mean) with their corresponding theoretical moments and solving these equations to obtain parameter estimates. This approach simplifies estimation by leveraging relationships between moments and parameters.

Unbiasedness of Estimators

An estimator is unbiased if its expected value is equal to the parameter it is intended to estimate. Assessing unbiasedness involves calculating or analyzing the expectation of the estimator and determining whether there is systematic deviation from the true parameter value.

Order Statistics

Order statistics, particularly the sample minimum or maximum, often play a crucial role in parameter estimation when the support of the distribution depends on the parameter. In some cases, the MLE can correspond to one of these order statistics, such as the minimum observed value, due to the structure of the likelihood function.

Probability Density Function

A probability density function (pdf) describes the relative likelihood for a continuous random variable to take on a given value. One must ensure that the pdf is non-negative over its domain and that it integrates to one over the entire support. Verifying these properties confirms that the function properly defines a distribution.

Integration and Normalization

Normalization involves integrating the proposed density function over its entire support and verifying that the result is equal to one. This step is essential in establishing that the function is a legitimate density function and lays the groundwork for further statistical analysis.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method of estimating parameters by maximizing the likelihood function, which measures how likely the observed data is given a set of parameter values. This methodology involves constructing the likelihood from the product of individual densities and solving for the parameter that maximizes this function.

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