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Two Sides of Inference

- Parametric
- Interval estimation, xbar
- Hypothesis testing, m0

- Nonparametric
- Interval estimates, EDF
- Hypothesis testing, P(X<Y) > P(X>Y)

Meaning of Nonparametric

- Not about parameters
- Methods for non-normal distributions
- Methods for ordinal data
- Data Scales
- Nominal, categorical, qualitative
- Ordinal
- Interval
- Ratio - natural zero

- Data Scales

Random Sample - Type 1

- Random 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 variables
- Inferences 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 Distributions

- Statistics support both sides of inference
- Estimators - random variables used to create interval estimates
- Test 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 m, the average sales tax paid for the population of invoices (with standard deviation s)
- Xbar is a test statistic for testing hypotheses
H0: m = m0

- Xbar is a random variable with sampling distribution asymptotically normal as n increases with mean m and standard deviation sn

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 (mX) of the RV X
- Xbar is a test statistic for testing hypotheses
H0: m = m0

- Xbar is a random variable with sampling distribution asymptotically normal as n increases with mean mX and standard deviationsX/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 largest
- Large body of literature on sampling distributions of order statistics

Estimation

- Definitions
- 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 a/2 quantile and q is the (1-a/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 s 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 - a/2 and 1-a/2 quantiles

Parametric vs nonparametric sampling distributions

- Exact distributions with approximate models
- Exact distributions with exact models (but usually small samples)
or

- Asymptotic distributions with exact models

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