Biased Estimators in Statistics

Questions and Answers

What does the Mean Square Error (MSE) of a biased estimator consist of?

Only the variance of the estimator

Only the bias error

The sum of the variance and the square of the bias error (correct)

The determinant of the estimator matrix

What is required for one estimator to be uniformly preferable to another?

It has a larger bias error

It has a higher variance for all values of θ

It has a smaller MSE for all admissible values of the parameter θ (correct)

It must be a biased estimator

In the expression for MSE, what does 'mT(y)' represent?

The maximum likelihood estimator of θ

The expectation of the estimator T(y) (correct)

The variance of the estimator

The trace of the estimator matrix

What does the term 'bias error' refer to in the context of an estimator?

The difference between the estimator and the true parameter value Signup and view all the answers

How is the MSE of a vector parameter θ defined?

MSE is defined using a norm squared of the difference of the estimator from θ Signup and view all the answers

What happens to the MSE when an estimator improves its variance but increases bias?

MSE can either increase or decrease Signup and view all the answers

Which notation represents the trace of a matrix?

tr(M) Signup and view all the answers

What is the significance of the term 'deterministic' in the MSE equation of a biased estimator?

It shows that the bias error is a constant value Signup and view all the answers

Under what condition does the Maximum Likelihood estimator coincide with the Gauss-Markov estimator?

When observations are corrupted by additive Gaussian noise. Signup and view all the answers

What is the form of the Maximum Likelihood estimator when observations depend linearly on θ?

$θ̂M L = arg min(y - Uθ)T Σ^{-1}(y - Uθ)$ Signup and view all the answers

What happens to the Gauss-Markov estimator when the measurements are independent and identically distributed Gaussian noise?

It becomes identical to the Least Squares estimator. Signup and view all the answers

What property does the Gauss-Markov estimator possess under the conditions specified?

It is efficient and UMVUE. Signup and view all the answers

What does Eθ[∂ ln fyθ(y) / ∂θ] equal to in the setting of Gaussian measurement noise?

U T Σ^{-1} U Signup and view all the answers

What must the sum of the coefficients ai equal for the estimator to be unbiased?

1 Signup and view all the answers

What form does the BLUE estimator of m take?

A linear combination of observed values Signup and view all the answers

What condition must be met for the variance of T(y) to be minimized?

The observed variables must be independent Signup and view all the answers

What is the variance of the estimator T(y) when y i are independent?

The quadratic sum of the weights ai Signup and view all the answers

In the context of the BLUE estimator, what dictates that T(y) be an unbiased estimator of m?

The expected value Eθ[T(y)] must equal the true mean m Signup and view all the answers

What does maximizing the probability density function correspond to in terms of parameter θ?

Choosing θ that maximizes the likelihood of measurement y. Signup and view all the answers

What is the effect of choosing coefficients ai that do not satisfy the constraint ai = 1?

The estimator will be biased Signup and view all the answers

Which statement about the log-likelihood function is true?

It achieves its maximum in the same values as L(θ|y). Signup and view all the answers

Which statement about the BLUE estimator is correct?

It has a simple form that is easy to compute for known variances Signup and view all the answers

What condition must be satisfied if θ̂ is a maximum for L(θ|y)?

The equations must equal zero for all dimensions of θ. Signup and view all the answers

What may occur when computing the maximum likelihood estimator for certain parameters?

The likelihood function may not be differentiable everywhere. Signup and view all the answers

What does T(y) represent in the context of estimating m?

The weighted sum of the observations Signup and view all the answers

In the context of independent Gaussian random variables, what is the goal when computing the ML estimator of the mean?

To maximize the likelihood function with respect to the mean. Signup and view all the answers

What is the role of the parameter vector p in the estimation process?

It determines the level of complexity for the estimation. Signup and view all the answers

Which of the following best describes the natural logarithm's effect on the likelihood function?

It simplifies calculations without changing the positions of maxima. Signup and view all the answers

What could happen if the domain Θ is not an open set?

The maximum could be achieved at the boundary of Θ. Signup and view all the answers

What condition must be satisfied for equality to be obtained in the given formulation?

Set $k = i - τ$ Signup and view all the answers

Which function represents the contribution of the time series when $τ ≥ 0$?

$a^2i−τ σe^2$ Signup and view all the answers

What does the term $|p| < 1$ signify in the context of time series analysis?

The stability of the process Signup and view all the answers

Which equation describes the time series contribution when $τ < 0$?

$a^2i σe$ Signup and view all the answers

What does the summation symbol $ ext{X}$ indicate in the equations provided?

An infinite series Signup and view all the answers

What is the probable role of the variable $a$ in the equations?

It acts as a coefficient in the series Signup and view all the answers

What condition is implied when $τ$ is less than 0?

Data is analyzed in reverse order Signup and view all the answers

Which aspect does $σ$ represent in the time series equations?

The variance or standard deviation Signup and view all the answers

Study Notes

Estimators and Mean Square Error (MSE)

A biased estimator's Mean Square Error (MSE) combines both variance and bias error, represented as:
MSE(T(·)) = Eθ[(T(y) - mT(y))²] + (mT(y) - θ)²
The first term represents the variance of the estimator while the second term is the square of the bias error.
Trade-off between variance and bias is crucial in estimation problems.
MSE is useful for comparing different estimators; an estimator T1(·) is preferred over T2(·) if:
Eθ[(T1(y) - θ)²] ≤ Eθ[(T2(y) - θ)²], for all θ in Θ.

Mean Square Error in Higher Dimensions

For a parameter vector θ in Rp, MSE definition extends to:
MSE(T(·)) = Eθ[||T(y) - θ||²] = tr{Eθ[(T(y) - θ)(T(y) - θ)ᵀ]}
"tr" indicates the trace of a matrix, summing diagonal elements.

Best Linear Unbiased Estimator (BLUE)

BLUE estimators are linear and computationally straightforward by determining optimal coefficients.
An unbiased estimator T(·) requires the constraint:
∑(ai) = 1 for coefficients ai in the form T(y) = ∑(ai y_i).
Minimum variance coincides with maximizing the likelihood under certain conditions involving independent variables with known variances.

Maximum Likelihood Estimator (MLE)

MLE of an unknown parameter θ is derived as:
TM L(y) = arg max L(θ|y).
Log-likelihood function ln L(θ|y) is often maximized instead, as it simplifies calculations.
Optimal parameters θ̂ for MLE solve the equations derived from the derivative conditions:
∂L(θ|y)/∂θi = 0, i = 1,..., p.

Gaussian Observations and Estimation

For independent Gaussian variables, solving optimizations reveals the MLE coincides with the Gauss-Markov estimator.
MLE, under additive Gaussian noise conditions with linear observations, aligns with the Least Squares estimator.
In Gaussian noise scenarios, the Gauss-Markov estimator gains efficiency and stands as the Uniformly Minimum Variance Unbiased Estimator (UMVUE).

Time Series Analysis

Auto-covariance Ry(t + τ, t) describes the correlation between time series observations.
Formulas include summation expressions defining correlations based on different lag parameters τ, capturing structural relationships in data.

These notes encapsulate critical concepts and frameworks for understanding estimators, the MSE, and the application of maximum likelihood methods in statistical analysis.

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Description

This quiz explores the concept of biased estimators in statistics, focusing on the Mean Squared Error (MSE) formula. It delves into the mathematical expectations and their implications on statistical estimations. Test your understanding of these critical statistical concepts.