Lehmann–Scheffé theorem

In statistics, the Lehmann–Scheffé theorem is a prominent statement, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation.^[1] The theorem states that any estimator which is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers.^[2]^[3]

If T is a complete sufficient statistic for θ and E(g(T)) = τ(θ) then g(T) is the uniformly minimum-variance unbiased estimator (UMVUE) of τ(θ).

Statement

Let $\vec{X}= X_1, X_2, \dots, X_n$ be a random sample from a distribution that has p.d.f (or p.m.f in the discrete case) $f(x:\theta )$ where $\theta \in \Omega$ is a parameter in the parameter space. Suppose $Y=u({\vec {X}})$ is a sufficient statistic for θ, and let $\{f_{{Y}}(y:\theta ):\theta \in \Omega \}$ be a complete family. If $\phi :{\mathbb {E}}[\phi (Y)]=\theta$ then $\phi (Y)$ is the unique MVUE of θ.

Proof

By the Rao–Blackwell theorem, if $Z$ is an unbiased estimator of θ then $\phi (Y):={\mathbb {E}}[Z|Y]$ defines an unbiased estimator of θ with the property that its variance is not greater than that of $Z$ .

Now we show that this function is unique. Suppose $W$ is another candidate MVUE estimator of θ. Then again $\psi (Y):={\mathbb {E}}[W|Y]$ defines an unbiased estimator of θ with the property that its variance is not greater than that of $W$ . Then

{\mathbb {E}}[\phi (Y)-\psi (Y)]=0,\theta \in \Omega .

Since $\{f_{{Y}}(y:\theta ):\theta \in \Omega \}$ is a complete family

{\mathbb {E}}[\phi (Y)-\psi (Y)]=0\implies \phi (y)-\psi (y)=0,\theta \in \Omega

and therefore the function $\phi$ is the unique function of Y with variance not greater than that of any other unbiased estimator. We conclude that $\phi (Y)$ is the MVUE.

Example for when using a non-complete minimal sufficient statistic

An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that is not complete, was provided by Galili and Meilijson in 2016.^[4] Let $X_{1},\cdots X_{n}$ be a random sample from a scale-uniform distribution $X\sim U((1-k)\theta ,(1+k)\theta ),$ with unknown mean $E[X]=\theta$ and known design parameter $k\in (0,1)$ . In the search for "best" possible unbiased estimators for $\theta$ , it is natural to consider $X_{1}$ as an initial (crude) unbiased estimator for $\theta$ and then try to improve it. Since $X_{1}$ is not a function of $T=\left(X_{(1)},X_{(n)}\right)$ , the minimal sufficient statistic for $\theta$ (where $X_{(1)}=\min(X_{i})$ and $X_{(n)}=\max(X_{i})$ ), it may be improved using the Rao-Blackwell theorem as follows:

{\hat {\theta }}_{RB}=E_{\theta }[X_{1}|X_{(1)},X_{(n)}]={\frac {X_{1}+X_{n}}{2}}.

However, the following unbiased estimator can be shown to have lower variance:

{\hat {\theta }}_{LV}={\frac {1}{k^{2}{\frac {n-1}{n+1}}+1}}\cdot {\frac {(1-k)X_{(1)}+(1+k)X_{(n)}}{2}}.

And in fact, it could be even further improved when using the following estimator:

{\hat {\theta }}_{BAYES}={\frac {n+1}{n}}\left[1-{\frac {{\frac {X_{(1)}(1+k)}{X_{(n)}(1-k)}}-1}{\left({\frac {X_{(1)}(1+k)}{X_{(n)}(1-k)}}\right)^{n+1}-1}}\right]{\frac {X_{(n)}}{1+k}}

References

↑ Casella, George (2001). Statistical Inference. Duxbury Press. p. 369. ISBN 0-534-24312-6.
↑ Lehmann, E. L.; Scheffé, H. (1950). "Completeness, similar regions, and unbiased estimation. I.". Sankhyā. 10 (4): 305–340. JSTOR 25048038. MR 39201.
↑ Lehmann, E.L.; Scheffé, H. (1955). "Completeness, similar regions, and unbiased estimation. II.". Sankhyā. 15 (3): 219–236. JSTOR 25048243. MR 72410.
↑ Tal Galili & Isaac Meilijson (31 Mar 2016). "An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator". The American Statistician. 70 (1): 108–113. doi:10.1080/00031305.2015.1100683.

Statistics

Descriptive statistics

Continuous data

Center	Mean arithmetic geometric harmonic Median Mode

Dispersion	Variance Standard deviation Coefficient of variation Percentile Range Interquartile range

Shape	Moments Skewness Kurtosis L-moments

Count data

Index of dispersion

Summary tables

Dependence

Graphics

Data collection

Study design	Population Statistic Effect size Statistical power Sample size determination Missing data

Survey methodology	Sampling Standard error stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Point estimation	Estimating equations Maximum likelihood Method of moments M-estimator Minimum distance Unbiased estimators Mean-unbiased minimum-variance Rao–Blackwellization Lehmann–Scheffé theorem Median unbiased Plug-in

Interval estimation	Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife

Testing hypotheses	1- & 2-tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F

Goodness of fit	Chi-squared Kolmogorov–Smirnov Anderson–Darling Normality (Shapiro–Wilk) Likelihood-ratio test Model selection Cross validation AIC BIC

Rank statistics	Sign Sample median Signed rank (Wilcoxon) Hodges–Lehmann estimator Rank sum (Mann–Whitney) Nonparametric anova 1-way (Kruskal–Wallis) 2-way (Friedman) Ordered alternative (Jonckheere–Terpstra)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population statistics Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

Category
Portal
Commons
WikiProject

This article is issued from Wikipedia - version of the 8/22/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Lehmann–Scheffé theorem

Statement

Proof

Example for when using a non-complete minimal sufficient statistic

See also

References