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Statistical Inference and Regression Analysis: Stat-GB.3302.30, Stat-UB.0015.01

Statistical Inference and Regression Analysis: Stat-GB.3302.30, Stat-UB.0015.01. Professor William Greene Stern School of Business IOMS Department Department of Economics. Part 4 – Statistical Inference. 4 .1 – The Normal Family of Distributions. Normal. Standard Normal.

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Statistical Inference and Regression Analysis: Stat-GB.3302.30, Stat-UB.0015.01

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  1. Statistical Inference and Regression Analysis: Stat-GB.3302.30, Stat-UB.0015.01 Professor William Greene Stern School of Business IOMS Department Department of Economics

  2. Part 4 – Statistical Inference

  3. 4.1 – The Normal Family of Distributions

  4. Normal

  5. Standard Normal

  6. Chi Squared 1 = Square of N(0,1)

  7. Limit Result for Square of N(0,1)

  8. Sum of Two Independent Chi Squared(1) Variables

  9. Sum of N Independent Chi Squareds

  10. Limit Result for Square of Normal

  11. Noncentral Chi Squared

  12. t distribution If v=1, t=N[0,1]/N[0,1] = Cauchy. No finite moments.

  13. Limiting Form of t

  14. F Distribution

  15. Limiting Form of F

  16. Special Case of F

  17. Independence of Sample Mean and Variance in Normal Sampling

  18. Useful Result

  19. Distribution of the t statistic

  20. 4.2 – Interval Estimation

  21. Estimation • Point Estimator: Provides a single estimate of the feature in question based on prior and sample information. • Interval Estimator: Provides a range of values that incorporates both the point estimator and the uncertainty about the ability of the point estimator to find the population feature exactly.

  22. Obtaining a Confidence Interval • Pivotal quantity f(estimator, parameters) that has a known distribution free of parameters and data • Probability statement can be made about the pivotal quantity • Manipulate the interval to describe the parameter.

  23. Example – Normal Mean

  24. t distribution – values of t*

  25. Normal Variance

  26. GSOEP Income Data Descriptive Statistics for 1 variables --------+--------------------------------------------------------------------- Variable| Mean Std.Dev. Minimum Maximum Cases Missing --------+--------------------------------------------------------------------- HHNINC| .353343 .157058 .035000 1.500000 24 0 --------+--------------------------------------------------------------------- For the mean, t* for 24-1 = 23 degrees of freedom = 2.069 Confidence interval for mean is .353343 +/- 2.069 * (.15708/sqr(24)) = .353343 +/- .032064 Confidence interval for variance: Critical values from chi squared 23 are 11.69 and 38.08. Confidence interval for 2is (24-1).157082/38.08 to (24-1).157082/11.69 = .014903 to .048546 Confidence interval for  is .122078 to .220332 Notice, not symmetric around s2 or s.

  27. Large Sample Results • There are almost no other cases in which there exists an exact pivotal quantity • Most estimators rely on large sample results based on central limit theorems (estimator – parameter) ----------------------------------------  N(0,1) standard error of estimator

  28. Confidence Intervals

  29. Interpretation of The Interval • Not a statement about probabilities that  will lie in specific intervals. • (1-) percent of the time, the interval will contain the true parameter

  30. Application: Credit Modeling • 1992 American Express analysis of • Application process: Acceptance or rejection; X = 0 (reject) or 1 (accept). • Cardholder behavior • Loan default (D = 0 or 1). • Average monthly expenditure (E = $/month) • General credit usage/behavior (Y = number of charges) • 13,444 applications in November, 1992

  31. 0.7809 is the true proportion in the population of 13,444 we are sampling from.

  32. Estimates plus and minus 1 and 2 standard errors

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