What is the difference between an estimate and an estimator? Explain all the properties of an estimator with reference to BLUE.

An “estimate” and an “estimator” are terms commonly used in statistics to describe different aspects of the process of determining an unknown quantity or parameter based on available data. Understanding the nuances between these two terms and their properties is crucial in statistical analysis.

### 1. **Estimate vs. Estimator:**

An estimate is a specific value calculated from sample data that serves as the best guess for an unknown population parameter. For instance, if you want to estimate the average height of adults in a city, you might collect data from a sample of individuals and calculate the average height from that sample. This calculated value would be your estimate of the population parameter (average height of all adults in the city).

An estimator is a rule, formula, or method used to calculate an estimate. It’s a statistical function or procedure that generates an estimate based on sample data. For instance, the sample mean is an estimator used to estimate the population mean.

### 2. **Properties of an Estimator:**

Estimators are evaluated based on several properties that determine their reliability, accuracy, and efficiency. The most common properties include:

#### Unbiasedness:
An estimator is unbiased if, on average, it gives the correct estimate of the parameter. Mathematically, an estimator \( \hat{\theta} \) for a parameter θ is unbiased if \( \mathbb{E}(\hat{\theta}) = \theta \), where \( \mathbb{E} \) represents the expected value.

#### Consistency:
Consistency refers to the property where an estimator becomes more accurate as the sample size increases. In other words, as more data is collected, the estimator’s value converges to the true parameter value. Formally, an estimator \( \hat{\theta} \) is consistent if \( \hat{\theta} \xrightarrow{P} \theta \), where \( \xrightarrow{P} \) denotes convergence in probability.

#### Efficiency:
Efficiency refers to the ability of an estimator to attain a small variance. Among unbiased estimators, the one with the smallest variance is considered the most efficient.

#### Sufficiency:
An estimator is sufficient if it contains all the information in the sample relevant to estimating the parameter. A sufficient statistic summarizes all the information about the parameter contained in the data.

### 3. **BLUE (Best Linear Unbiased Estimator):**

The concept of BLUE is particularly relevant in the context of linear regression. In linear regression models, the aim is to estimate parameters that define the relationship between variables.

– **Best:** The estimator should have the smallest variance among all unbiased estimators.
– **Linear:** The estimator is a linear combination of the observed values.
– **Unbiased:** The estimator’s expected value should equal the true parameter value.
– **Efficient:** The estimator should have the smallest variance among all other unbiased estimators.

In ordinary least squares (OLS) regression, the coefficients obtained are BLUE estimators when certain assumptions (such as normality, homoscedasticity, and no multicollinearity) hold true. These estimators possess the properties of being unbiased, efficient (have the smallest variance), and are obtained through a linear combination of the observed values.

In summary, estimators are mathematical methods or rules used to calculate estimates, while estimates are specific values obtained from these methods using sample data. Evaluating estimators involves examining properties like unbiasedness, consistency, efficiency, and sufficiency. BLUE, specifically in linear regression, refers to estimators that possess optimal properties among unbiased estimators.