Consistent estimator

In statistics, a sequence of estimators for parameter θ0 is said to be consistent (or asymptotically consistent) if this sequence converges in probability to θ0. It means that the distributions of the estimators become more and more concentrated near the true value of the parameter being estimated, so that the probability of the estimator being arbitrarily close to θ0 converges to one.

In practice one usually constructs a single estimator as a function of an available sample of size n, and then imagines being able to keep collecting data and expanding the sample ad infinitum. In this way one would obtain a sequence of estimators indexed by n and the notion of consistency will be understood as the sample size “grows to infinity”. If this sequence converges in probability to the true value θ0, we call it a consistent estimator; otherwise the estimator is said to be inconsistent.

The consistency as defined here is sometimes referred to as the weak consistency. When we replace the convergence in probability with the almost sure convergence, then the sequence of estimators is said to be strongly consistent.

Contents

Definition

Loosely speaking, an estimator Tn of parameter θ is said to be consistent, if it converges in probability to the true value of the parameter:[1]


    \underset{n\to\infty}{\operatorname{plim}}\;T_n = \theta.

A more rigorous definition takes into account the fact that θ is actually unknown, and thus the convergence in probability must take place for every possible value of this parameter. Suppose {pθ: θ ∈ Θ} is a family of distributions (the parametric model), and Xθ = {X1, X2, … : Xi ~ pθ} is an infinite sample from the distribution pθ. Let { Tn(Xθ) } be a sequence of estimators for some parameter g(θ). Usually Tn will be based on the first n observations of a sample. Then this sequence {Tn} is said to be (weakly) consistent if [2]


    \underset{n\to\infty}{\operatorname{plim}}\;T_n(X^{\theta}) = g(\theta),\ \ \text{for all}\ \theta\in\Theta.

This definition uses g(θ) instead of simply θ, because often one is interested in estimating a certain function or a sub-vector of the underlying parameter. In the next example we estimate the location parameter of the model, but not the scale:

Example: sample mean for normal random variables

Suppose one has a sequence of observations {X1, X2, …} from a normal N(μ, σ2) distribution. To estimate μ based on the first n observations, we use the sample mean: Tn = (X1 + … + Xn)/n. This defines a sequence of estimators, indexed by the sample size n.

From the properties of the normal distribution, we know that Tn is itself normally distributed, with mean μ and variance σ2/n. Equivalently, \scriptstyle (T_n-\mu)/(\sigma/\sqrt{n}) has a standard normal distribution. Then


    \Pr\!\left[\,|T_n-\mu|\geq\varepsilon\,\right] = 
    \Pr\!\left[ \frac{\sqrt{n}\,\big|T_n-\mu\big|}{\sigma} \geq \sqrt{n}\varepsilon/\sigma \right] = 
    2\left(1-\Phi\left(\frac{\sqrt{n}\,\varepsilon}{\sigma}\right)\right) \to 0

as n tends to infinity, for any fixed ε > 0. Therefore, the sequence Tn of sample means is consistent for the population mean μ.

Establishing consistency

The notion of asymptotic consistency is very close, almost synonymous to the notion of convergence in probability. As such, any theorem, lemma, or property which establishes convergence in probability may be used to prove the consistency. Many such tools exist:


    \Pr\!\big[h(T_n-\theta)\geq\varepsilon\big] \leq \frac{\operatorname{E}\big[h(T_n-\theta)\big]}{\varepsilon},

the most common choice for function h being either the absolute value (in which case it is known as Markov inequality), or the quadratic function (respectively Chebychev's inequality).


    T_n\ \xrightarrow{p}\ \theta\ \quad\Rightarrow\quad g(T_n)\ \xrightarrow{p}\ g(\theta)
\begin{align}
  & T_n %2B S_n \ \xrightarrow{p}\ \alpha%2B\beta, \\
  & T_n   S_n \ \xrightarrow{p}\ \alpha \beta, \\
  & T_n / S_n \ \xrightarrow{p}\ \alpha/\beta, \text{ provided that }\beta\neq0
  \end{align}
\frac{1}{n}\sum_{i=1}^n g(X_i) \ \xrightarrow{p}\ \operatorname{E}[\,g(X)\,]

Bias versus consistency

Unbiased but not consistent

An estimator can be unbiased but not consistent. For example, for an iid sample {x
1
,..., x
n
} one can use T(X) = x
1
as the estimator of the mean E[x]. This estimator is obviously unbiased, and obviously inconsistent.

Biased but consistent

Alternatively, an estimator can be biased but consistent. For example if the mean is estimated by {1 \over n} \sum x_i %2B {1 \over n} it is biased, but as n \rightarrow \infty, it approaches the correct value, and so it is consistent.

See also

Notes

  1. ^ Amemiya 1985, Definition 3.4.2
  2. ^ Lehman & Casella 1998, p. 332
  3. ^ Amemiya 1985, equation (3.2.5)
  4. ^ Amemiya 1985, Theorem 3.2.6
  5. ^ Amemiya 1985, Theorem 3.2.7
  6. ^ Newey & McFadden (1994, Chapter 2)

References