The Likelihood Function - Introduction
• Recall: a statistical model for some data is a set { f θ : θ ∈ Ω} of distributions, one of which corresponds to the true unknown distribution that produced the data.
• The distribution fθ can be either a probability density function or a probability mass function.
• The joint probability density function or probability mass function
of iid random variables X1, …, Xn is n θ ()1 ,..., n = ∏ θ xfxxf i . () i=1
week 3 1 The Likelihood Function
•Let x1, …, xn be sample observations taken on corresponding random variables X1, …, Xn whose distribution depends on a parameter θ. The likelihood function defined on the parameter space Ω is given by
(θ 1,...,| n )= θ ( 1 xxfxxL n ).,...,
• Note that for the likelihood function we are fixing the data, x1,…, xn, and varying the value of the parameter.
•The value L(θ | x1, …, xn) is called the likelihood of θ. It is the probability of observing the data values we observed given that θ is the true value of the parameter. It is not the probability of θ given that we
observed x1, …, xn.
week 3 2 Examples
• Suppose we toss a coin n = 10 times and observed 4 heads. With no knowledge whatsoever about the probability of getting a head on a single toss, the appropriate statistical model for the data is the Binomial(10, θ) model. The likelihood function is given by
• Suppose X1, …, Xn is a random sample from an Exponential(θ) distribution. The likelihood function is
week 3 3 Sufficiency - Introduction
• A statistic that summarizes all the information in the sample about the target parameter is called sufficient statistic.
• An estimator θ ˆ is sufficient if we get as much information about θ ˆ from θ as we would from the entire sample X1, …, Xn.
• A sufficient statistic T(x1, …, xn) for a model is any function of the data x1, …, xn such that once we know the value of T(x1, …, xn), then we can determine the likelihood function.
week 3 4 Sufficient Statistic
•A sufficient statistic is a function T(x1, …, xn) defined on the sample space, such that whenever T(x1, …, xn) = T(y1, …, yn), then
(θ ,...,| 21 )= ⋅ (θ 1,...| yyLcxxL n ) for some constant c.
• Typically, T(x1, …, xn) will be of lower dimension than x1, …, xn, so
we can consider replacing x1, …, xn by T(x1, …, xn) as a data reduction and this simplifies the analysis.
•Example…
week 3 5 Minimal Sufficient Statistics
•A minimal sufficient statistic T for s model is any sufficient
statistic such that once we know a likelihood function L(θ|x1, …, xn) for the model and data then we can determine T(x1, …, xn). • A relevant likelihood function can always be obtained from the value of any sufficient statistic T, but if T is minimal sufficient as well, then we can also obtain the value of T from any likelihood function. • It can be shown that a minimal sufficient statistics gives the maximal reduction of the data. •Example…
week 3 6 Alternative Definition of Sufficient Statistic
•Let X1, …, Xn be a random sample from a distribution with unknown
parameter θ. The statistic T(x1, …, xn) is said to be sufficient for θ if
the conditional distribution of X1, …, Xn given T does not depend on θ.
• This definition is much harder to work with as the conditional
distribution of the sample X1, …, Xn given the sufficient statistics T is often hard to derive.
week 3 7 Factorization Theorem
•Let T be a statistic based on a random sample X1, …, Xn. Then T is a sufficient statistic for θ if
(θ 1,...,| n )= ( θ )(1,..,; xxhTgxxL n )
i.e. if the likelihood function can be factored into two nonnegative
functions one that depend on T(x1, …, xn) and θ and one that depend only on the data x1, …, xn. • Proof:
week 3 8 Examples
week 3 9 Minimum Variance Unbiased Estimator
• MVUE for θ is the unbiased estimator with the smallest possible variance. We look amongst all unbiased estimators for the one with the smallest variance.
week 3 10 The Rao-Blackwell Theorem
ˆ • Let θ ˆ be an unbiased estimator for θ such that Var (θ ) ∞< . If T is a sufficient statistic for θ, define * = (θθ ˆˆ | TE ) . Then, for all θ,
E(ˆ* )= θθ and Var( * )≤ Var(θθˆˆ ).
• Proof:
week 3 11