Consistent estimator

{T₁, T₂, T₃, …} is a sequence of estimators for parameter θ₀, the true value of which is 4. This sequence is consistent: the estimators are getting more and more concentrated near the true value θ₀; at the same time, these estimators are biased. The limiting distribution of the sequence is a degenerate random variable which equals θ₀ with probability 1.

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.

Definition

Loosely speaking, an estimator T_n 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 {p_θ: θ ∈ Θ} is a family of distributions (the parametric model), and X^θ = {X₁, X₂, … : X_i ~ p_θ} is an infinite sample from the distribution p_θ. Let { T_n(X^θ) } be a sequence of estimators for some parameter g(θ). Usually T_n will be based on the first n observations of a sample. Then this sequence {T_n} 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: T_n = (X₁ + … + X_n)/n. This defines a sequence of estimators, indexed by the sample size n.

From the properties of the normal distribution, we know that T_n 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 T_n 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:

• In order to demonstrate consistency directly from the definition one can use the inequality ^[3]

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

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).

• Another useful result is the continuous mapping theorem: if T_n is consistent for θ and g(·) is a real-valued function continuous at point θ, then g(T_n) will be consistent for g(θ):^[4]

$T_n\ \xrightarrow{p}\ \theta\ \quad\Rightarrow\quad g(T_n)\ \xrightarrow{p}\ g(\theta)$

• Slutsky’s theorem can be used to combine several different estimators, or an estimator with a non-random covergent sequence. If T_n →^pα, and S_n →^pβ, then ^[5]

$\begin{align} & T_n + S_n \ \xrightarrow{p}\ \alpha+\beta, \\ & T_n S_n \ \xrightarrow{p}\ \alpha \beta, \\ & T_n / S_n \ \xrightarrow{p}\ \alpha/\beta, \text{ provided that }\beta\neq0 \end{align}$

• If estimator T_n is given by an explicit formula, then most likely the formula will employ sums of random variables, and then the law of large numbers can be used: for a sequence {X_n} of random variables and under suitable conditions,

$\frac{1}{n}\sum_{i=1}^n g(X_i) \ \xrightarrow{p}\ \operatorname{E}[\,g(X)\,]$

• If estimator T_n is defined implicitly, for example as a value that maximizes certain objective function (see extremum estimator), then a more complicated argument involving stochastic equicontinuity has to be used.^[6]

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 ${1 \over n} \sum x_i + {1 \over n}$ it is biased, but as $n \rightarrow \infty$, it approaches the correct value, and so it is consistent.

See also

• Fisher consistency — alternative, although rarely used concept of consistency for the estimators
• Consistent test — the notion of consistency in the context of hypothesis testing

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

• Amemiya, Takeshi (1985). Advanced econometrics. Harvard University Press. ISBN 0-674-00560-0. 
• Lehmann, E. L.; Casella, G. (1998). Theory of Point Estimation (2nd ed.). Springer. ISBN 0-387-98502-6. 
• Newey, W.; McFadden, D. (1994). Large sample estimation and hypothesis testing. In “Handbook of Econometrics”, Vol. 4, Ch. 36. Elsevier Science. ISBN 0-444-88766-0. 
• Nikulin, M.S. (2001), "Consistent estimator", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/C/c025240.htm