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 L|(θ x1 ,..., xn ) = θ f( 1,..., xn ) x . • 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 L x|(θ 1 ,..., x 2) = c⋅ (|θ L1 ,... yn ) y 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 L x|(θ 1 x ,..., n ) = g;( Tθ )(1 h ,.., n x ) x 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 • ed estimatorθˆ for Let be an unbiasθ such that Var (θ < ) ∞ . If T is a * sufficient statistic for θ, defineθ ˆ = ET ( θ ˆ| ) . Then, for all θ, * * Eθ( ˆ )= θ and Varθ(ˆ)≤ Var( θ ˆ). • Proof: week 3 11.