## Maximum Likelihood Estimation

Let X1, . . . Xn be IID with PDF f(x; θ). The likelihood function is defined byThe log-likelihood function

Continue ReadingLet X1, . . . Xn be IID with PDF f(x; θ). The likelihood function is defined byThe log-likelihood function

Continue ReadingTill now, we have covered the estimation of statistical functionals i.e. functions of CDF – Fx. They are Non-parametric Inference.

Continue ReadingNormal Interval The normal confidence interval is defined asTn ± zα/2sêbootwhere sêboot is the bootstrap estimate of standard error. Pivotal

Continue ReadingBootstrap is a non-parametric method for estimating accuracy defined in terms of standard error, bias, variance, confidence interval, etc. Suppose

Continue ReadingWhen starting with the inference problem, the most basic is the non-parametric estimation of CDF and functions of CDF. Let

Continue ReadingFor a parameter θ, a 1-α confidence interval is Cn = (a, b)where a = A(X1,. . , Xn) and

Continue ReadingPoint estimation refers to the use of sample data to provide a single best guess (known as point estimate) of

Continue ReadingA statistical model is a set of distribution or a set of densities. A parametric model is a statistical model

Continue ReadingThe Weak Law of Large Numbers (WLLN) If X1, X2, . . . , Xn are IID, then This theorem

Continue ReadingPointwise or sure convergence A sequence of random variables {Xn}n∈N is said to converge point-wise or surely to X if Xn(ω) → X(ω), ∀

Continue ReadingMarkov’s Inequalities Let X be a non-negative random variable and suppose that E(X) exists. For any t > 0, Chebyshev’s

Continue ReadingVariance Variance means spread of a distribution Let X be a random variable with mean μ. The variance of X

Continue ReadingThe expected value, or mean, or first moment, of X is defined to beassuming that the sum (or integral) is

Continue ReadingSuppose X is a random variable with PDF fX and CDF FX. Let Y = r(X) be a function of

Continue ReadingLet X = (X1, . . . , Xn) where X1, . . . , Xn are random variables. We

Continue ReadingIndependent Random Variables Two random variables are X and Y are independent if, for every A and B,P(X ∈ A,

Continue ReadingJoint Mass Function Remember the probability mass function definition. That is the study of one random variable. Given two discrete

Continue ReadingUniform Probability Distribution X has Uniform(a, b) distribution, written X~Uniform(a, b), ifwhere a < b. The distribution function is The

Continue ReadingIn this section, we are going to cover some important Discrete Random Variables. Note that we will be writing X

Continue ReadingRandom Variable A random variable is a mappingX : Ω→Rthat assigns a real number X(ω) to each outcome ω Getting

Continue Reading