Inference
Download
1 / 16

Inference - PowerPoint PPT Presentation


  • 117 Views
  • Uploaded on

Inference. Mary M. Whiteside, Ph.D. Nonparametric Statistics. Two Sides of Inference. Parametric Interval estimation, xbar Hypothesis testing, m 0 Nonparametric Interval estimates, EDF Hypothesis testing, P(X<Y) > P(X>Y). Meaning of Nonparametric. Not about parameters

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Inference' - traci


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Inference

Inference

Mary M. Whiteside, Ph.D.

Nonparametric Statistics


Two sides of inference
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
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


Random sample type 1
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
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
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
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
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 sn


Consider xbar a parametric statistic1
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
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
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
Estimation

  • Definitions

    • EDF

    • pth sample quantile

    • sample mean, variance, and standard deviation

    • unbiased estimators (S2 and s2)


Intervals for parameter estimation
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
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 estimator1
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
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