Consistency refers to the property of an estimator that as the sample size increases, the estimator converges in probability to the true value of the parameter being estimated. In other words, a consistent estimator will yield results that become more accurate and stable as more data points are collected.
Characteristics of a Consistent Estimator
A consistent has some important characteristics:
- Convergence: The estimator will produce values that get closer to the true parameter value with larger samples.
- Reliability: Provides reassurance that the estimates will be valid as more data is accounted for.
Examples of Consistent Estimators
- Sample Mean ($\overline{x}$): The sample mean is a consistent estimator of the population mean ($\mu$). A larger sample from a population converges to the actual population mean, compared to a smaller smaller.
- Sample Proportion ($\hat{p}$): The sample proportion is also a consistent estimator of the true population proportion. As the number of observations increases, the sample proportion gets closer to the true population proportion.
Question: $\hat{\theta}$ is a consistent estimator of the parameter $\theta$ of a given population if
- $\hat{\theta}$ is unbiased, and
- $var(\hat{\theta}) \rightarrow 0$ when $n\rightarrow \infty$
Answer: Suppose $X$ is random with mean $\mu$ and variance $\sigma^2$. If $X_1,X_2,\cdots,X_n$ is a random sample from $X$ then
\begin{align*}
E(\overline{X}) &= \mu\\
Var(\overline{X}) & = \frac{\sigma^2}{n}
\end{align*}
That is $\overline{X}$ is unbiased and $\lim\limits_{n\rightarrow\infty} Var(\overline{X}) = \lim\limits_{n\rightarrow\infty} \frac{\sigma^2}{n} =0$
Question: Show that the sample mean $\overline{X}$ of a random sample of size $n$ from the density function $f(x; \theta) = \frac{1}{\theta} e^{-\frac{x}{\theta}}, \qquad 0<x<\infty$ is a consistent estimator of the parameter $\theta$.
Answer: First, we need to check that $E(\overline{x})=\theta$, that is, the sample mean $\overline{X}$ is unbiased.
\begin{align*}
E(X) &= \mu = \int x\cdot f(x; \theta) dx = \int\limits_{0}^{\infty}x\cdot \frac{1}{\theta} e^{-\frac{x}{\theta}}dx\\
&= \frac{1}{\theta} \int\limits_{0}^{\infty} xe^{-\frac{x}{\theta}}dx\\
&= \frac{1}{\theta} \left[ \Big| -\theta x e^{-\frac{x}{\theta}}dx\Big|_{0}^{\infty} + \theta \int\limits_{0}^{\infty} e^{-\frac{x}{\theta}}dx \right]\\
&= \frac{1}{\theta} \left[0+\theta(-\theta) e^{-\frac{x}{\theta}}\big|_0^{\infty} \right] = \theta\\
E(X^2) &= \int x^2 f(x; \theta)dx = \int\limits_{0}^{\infty}x^2 \frac{1}{\theta} e^{-\frac{x}{\theta}}dx\\
&= \frac{1}{\theta}\left[ \Big| – x^2 \theta e^{-\frac{x}{\theta} }\Big|_{0}^{\infty} + \int\limits_0^\infty 2x\theta e^{-\frac{x}{\theta}}dx \right]\\
&= \frac{1}{\theta} \left[ 0 + 2\theta^2 \int\limits_0^\infty \frac{x}{\theta} e^{-\frac{x}{\theta}}dx\right]
\end{align*}
The expression is to be integrated into $E(X)$ which equals 0. Thus
\begin{align*}
E(X^2) &=\frac{1}{\theta} 2\theta^2\theta = 2\theta^2\\
Var(X) &=E(X^2) – [E(X)]^2 = 2\theta^2 – \theta^2 = \theta^2
and \quad Var(\overline{X}) &= \frac{\sigma^2}{n}\\
\lim\limits_{n\rightarrow \infty} \,\, Var(\overline{X}) &= \lim\limits_{n\rightarrow \infty} \frac{\sigma^2}{n} = 0
\end{align*}
Since $\overline{X}$ is unbiased and $Var(\overline{X})$ approaches 0 and $n\rightarrow \infty$, the $\overline{X}$ is a consistent estimator of $\theta$.
Importance of Consistency in Statistics
The following are a few key points about the importance of consistency in statistics:
Reliable Inferences: Consistent estimators ensure that as sample size increases, the estimates become closer and closer to the true population value/parameters. This helps researchers and statisticians to make sound inferences about a population based on sample data.
Foundation for Hypothesis Testing: Most of the statistical methods rely on consistent estimators. Consistency helps in validating the conclusions drawn from statistical tests, leading to confidence in decision-making.
Improved Accuracy: Since more data points are available due to the increase in sample size, the more consistently the estimates will converge to the true value. All this leads to more accurate statistical models, which can improve analysis and predictions.
Mitigating Sampling Error: Consistent estimators help to reduce the impact of random sampling error. As sample sizes increase, the variability in estimates tends to decrease, leading to more dependable conclusions.
Building Statistical Theory: Consistency is a fundamental concept in the development of statistical theory. It provides a rigorous foundation for designing and validating statistical methods and procedures.
Trust in Results: Consistency builds trust in the findings of statistical analyses. It is because the results are stable and reliable across different samples (due to large samples), therefore it is more likely to accept and act upon those results.
Framework for Model Development: In statistics and data science, developing models based on consistent estimators results in models with more accuracy.
Long-Term Decision Making: Consistency in data interpretation supports long-term planning, risk assessment, and resource allocation. It is required that businesses and organizations often make strategic decisions based on statistical analyses.
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