Conducting Actuarial Studies — Part 5: Statistical Inference: Estimation and Hypothesis Testing
Statistical inference involves hypothesis testing (evaluating some idea about a population using a sample) and estimation (estimating the value or potential range of values of some characteristic of the population based on that of a sample).
1. Introduction
Statistical inference involves two areas: estimation and hypothesis testing. This topic provides insight into how risk managers make portfolio decisions on the basis of statistical analysis of samples of investment returns or other random economic and financial variables.
We will first focus on the construction of confidence intervals for population parameters based on sample statistics. We then discuss hypothesis testing procedures used to conduct tests concerned with population means and population variances.
Specific tests reviewed include the z-test, t-test, chi-square test, and F-test. We belive that it important to know how to construct and interpret a confidence interval and when and how to apply each of the test statistics discussed when conducting a hypothesis test.
2. Point Estimates
Point estimates are single value estimates of population parameters, and confidence intervals are ranges of estimated values within which the actual value of the parameter will lie with a given probability. Desirable statistical properties of an estimator include unbiasedness, efficiency, and consistency.
3. T-Values
The t-distribution is similar, but not identical, to the normal distribution in shape — it is defined by the degrees of freedom, has a lower peak, and has fatter tails.
4. Confidence Interval
Thet-distribution is used to construct confidence intervals for the population mean when the population variance is not known. The (1-α) confidence interval for the population mean, μ, is:
Use the t-distribution if:
- Population distribution is normal with an unknown variance (large or small sample).
- Population distribution is nonnormal with unknown variance, but the sample is large (n >30).
5. The z-Test
The standard normal distribution (z-distribution) is used to construct confidence intervals for for the population mean when the population variance is known. The (1-α) confidence interval for the population mean, μ, is:
6. Hypothesis Testing
The hypothesis testing process requires a statement of a null and an alternative hypothesis, the selection of the appropriate test statistic, specification of the significance level, a decision rule, the calculation of a sample statistic, a decision regarding the hypotheses based on the test, and a decision based on the test results.
The null hypothesis is what the researcher wants to reject. The alternative hypothesis is what the researcher wants to prove, and it is accepted when the null hypothesis is rejected.
A two-tailed test results from a two-sided alternative hypothesis.
A one-tailed test results from a one-sided alternative hypothesis:
The decision rule depends on the alternative hypothesis and the distribution of the test statistic.
7. Type I and Type II Errors
A Type I error is the rejection of the null hypothesis when it is actually true, while a Type II error is the failure to reject the null hypothesis when it is actually false.
My famous example of type I error and type II error (the example with which I have been teaching statistics for over a decade and a half) has to do with a Lakers game in the era of Magic Johnson and Kareem Abdul-Jaber.
The underlying assumption here is that I am a Lakers fan and I bought a ticket to a Lakers game (In the picture below, it’s me with the Lakers hat as part of a 6th grade trip to Emek Hula, in northern Israel).
Type I Error — I went to the Lakers game but the Lakers lost. So my loss is that I went to see a bad game.
Type II error — I didn’t go to the Lakers game and the Lakers won. So my loss is that I lost a great game
8. Significance and Confidence Levels
The significance level can be interpreted as the probability that a test statistic will reject the null hypothesis by chance when the null hypothesis is actually true (i.e., the probability of a Type I error.). I refer to the significance level as an incorrect decision (rejecting the null hypothesis) conditional on the null hypothesis being true.
The confidence level can be interpreted as the probability that a test statistic will fail to reject the null hypothesis when the null hypothesis is true. I refer to the confidence level as a correct decision (not rejecting the null hypothesis) conditional on the null hypothesis being true.
9. The Power of Test
The power of a test is the probability of rejecting the null when it is false. The power of a test = 1 — P(Type II error).
10. The Relation Between the Confidence Interval and Hypothesis Tests
Hypothesis testing compares a computed test statistic to a critical value at a stated level of significance, which is the decision rule for the test.
A hypothesis about a population parameter is rejected when the sample statistic lies outside a confidence interval around the hypothesized value for the chosen level of significance.
11. Unknown and Known Population Variances
With unknown population variance, the t-statistic is used for tests of the mean of a normally distributed population:
If the population variance is known, the appropriate test statistic the z-statistic for tests of the mean of a population:
12. Chi-Square Test
The test of a hypothesis about the population variance for a normally distributed population uses a chi-square test statistic:
13. F-Test
The F-test comparing two variances based on independent samples from two normally distributed populations uses an F-distributed test statistic:
About the Author
Roi Polanitzer, CFV, QFV, FEM, F.IL.A.V.F.A., FRM, CRM, PDS, is a well-known authority in Israel the field of business valuation and has written hundreds of papers that articulate many of the concepts used in modern business valuation around the world. Mr. Polanitzer is the Owner and Chief Appraiser of Intrinsic Value — Independent Business Appraisers, a business valuation firm headquartered in Rishon LeZion, Israel. He is also the Owner and Chief Data Scientist of Prediction Consultants, a consulting firm that specializes in advanced analysis and model development.
Over more than 17 years, he has conduct actuarial studies which include: constructing discrete and continuous probability distributions, applying population and sample statistics, making statistical inference and hypothesis testing, estimating the parameters of distributions, creating graphical representation of statistical relationships, training linear regression with single and multiple regressors (the Ordinary Least Squares (OLS) method, interpreting and using regression coefficients, the t-statistic, and other output, hypothesis testing and confidence intervals, heteroskedasticity and multicollinearity), applying Monte Carlo methods, estimating correlation and volatility using EWMA and GARCH models and building volatility term structures.
Mr. Polanitzer holds an undergraduate degree in economics and a graduate degree in business administration, majoring in finance, both from the Ben-Gurion University of the Negev. He is a Full Actuary (Fellow), a Corporate Finance Valuator (CFV), a Quantitative Finance Valuator (QFV) and a Financial and Economic Modeler (FEM) from the Israel Association of Valuators and Financial Actuaries (IAVFA). Mr. Polanitzer is the Founder of the IAVFA and currently serves as its chairman.
Mr. Polanitzer’s professional recognitions include being designated a Financial Risk Manager (FRM) by the Global Association of Risk Professionals (GARP), a Certified Risk Manager (CRM) by the Israel Association of Risk Managers (IARM), as well as being designated a Python Data Analyst (PDA), a Machine Learning Specialist (MLS), an Accredited in Deep Learning (ADL) and a Professional Data Scientist (PDS) by the Professional Data Scientists’ Israel Association (PDSIA). Mr. Polanitzer is the Founder of the PDSIA and currently serves as its CEO.
He is the editor of IAVFA’s weekly newsletter since its inception (primarily for the professional appraisal community in Israel).
Mr. Polanitzer develops and teaches business valuation professional trainings and courses for the Israel Association of Valuators and Financial Actuaries, and frequently speaks on business valuation at professional meetings and conferences in Israel. He also developed IAVFA’s certification programs in the field of valuation and he is responsible for writing the IAVFA’s statement of financial valuation standards.