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Inference
Mary M. Whiteside, Ph.D.Nonparametric Statistics
Two Sides of InferenceParametric
Interval estimation, xbar Hypothesis testing, 0
Nonparametric Interval estimates, EDF Hypothesis testing, P(X<Y) > P(X>Y)
Meaning of Nonparametric
Not about parametersMethods for non-normal distributionsMethods for ordinal data
Data ScalesNominal, categorical, qualitativeOrdinalIntervalRatio - natural zero
Random Sample - Type 1Random sample from a finite population
Simple Stratified Cluster
Inferences are about the finite population Audit comprised of a sample from a
population of invoices Public opinion polls QC samples of delivered goods
Random Sample - Type 2
Observations of (iid) random variablesInferences are about the probability
distributions of the random variables Weekly average miles per gallon for your
new Lexus Chi square tests of independence in
medical treatment offered men and women Effect of female literacy on infant mortality
worldwide
Transition from data sets to distributions
All random variables, by definition, have probability functions (pmf or pdf) and cumulative probability distributions
Random variables defined on a random sample (Type 1 or 2) are called statistics with probability distributions that are called sampling distributions
Sampling DistributionsStatistics support both sides of
inferenceEstimators - random variables used
to create interval estimatesTest statistics - random variables
used to test hypotheses
Consider Xbar - a parametric statistic
Type I sample - subset of invoices where X = sales tax paid on an invoice randomly selected from a finite population Xbar is the average sales tax of n randomly selected
invoices Xbar is an estimator of , the average sales tax paid
for the population of invoices (with standard deviation )
Xbar is a test statistic for testing hypotheses H0: = 0
Xbar is a random variable with sampling distribution asymptotically normal as n increases with mean and standard deviation n
Consider Xbar - a parametric statistic
Type 2 sample - the complete set of miles per gallon observations made by you since buying your Lexus where X = mpg for your Lexus in a given week Xbar is the average mpg for n observations of X Xbar is an estimator of the expected value (X) of the
RV X Xbar is a test statistic for testing hypotheses
H0: = 0
Xbar is a random variable with sampling distribution asymptotically normal as n increases with mean X and standard deviation X/ n
X in the Type 1 sample
If X from a Type 1 sample is regarded as a random variable, then it has the discrete uniform distribution
Prob [X = x] = 1/N for all x in the population (where the N values of x are assumed to be unique)
Order statistics of rank k - a nonparametric statistic
the kth order statistic is the kth smallest observation
the first order statistic is the smallest observation in a sample
the nth order statistic is the largestLarge body of literature on sampling
distributions of order statistics
EstimationDefinitions
EDF pth sample quantile sample mean, variance, and standard
deviation unbiased estimators (S2 and s2)
Intervals for parameter estimation
(point estimate - r*standard error of the estimator, point estimate +q*standard error of the point estimate) where r is the /2 quantile and q is the (1-/2) quantile from the sampling distribution of the estimator r equals -q in symmetric distributions with
mean 0 (z = +/- 1.96 or t = +/-2.02581) r does not equal -q in skewed distributions
such as Chi squared and F
Sampling distribution of the estimator
Parametric procedures - Assumed normal or normal based from the Central Limit Theorem and sample size Xbar is approximately normal if n is large Xbar is t if X is normal and is unknown Xbar’s distribution is unknown if X’s
distribution is unknown and n is small
Sampling distribution of the estimator
Nonparametric distribution-free procedures I.e. the sampling distribution of the statistic (estimator or test statistic) is “free” from the distribution of X rank order statistics bootstrapped distributions - /2 and 1-
/2 quantiles
Parametric vs nonparametric sampling distributions
Exact distributions with approximate models
Exact distributions with exact models (but usually small samples)
orAsymptotic distributions with exact
models