# Reference for statistical inference concerning these topics

I am looking for book recommendations /lecture notes for self studying Inference. Following are the main topics of concern:

Parametric models, parameters, random sample and its likelihood, statistic and its sampling distributions, problems of inference. Examples from standard discrete and continuous models such as Bernoulli, Binomial, Poisson, Negative Binomial, Normal, Exponential, Gamma, Weibull, Pareto etc. Concept of sufficiency, minimal sufficiency, Neyman factorization criterion, Fisher information, exponential families. Maximum likelihood estimators, method of moment estimators, percentile estimators, least squares estimators, minimum mean squares estimators, uniformly minimum variance unbiased estimators, Rao Blackwell theorem, Cramer Rao lower bond, different examples. Statistical Hypotheses simple and composite, statistical tests, critical regions, Type I and Type II errors, size and power of a test, Neyman Pearson lemma and its different applications. Most powerful test, uniformly most powerful test, unbiased test and uniformly most unbiased test. Likelihood ratio test. Interval estimation, confidence intervals, construction of confidence intervals, shortest expected length confidence interval, most accurate one sided confidence interval and its relation to UMP test.

Kindly suggest some good texts or lecture notes to help me break ice into this topic.

• "Statistical Inference" by Casella and Berger is a standard reference and a good book. Dec 22, 2020 at 1:18
• math.stackexchange.com/q/51785/321264 Feb 1, 2021 at 14:06

I would recommend Probability and Statistical Inference by Hogg, Tenis, and Zimmerman (10th edition specifically!!). I am not sponsored, but this book helped me a lot when I was trying to self study. Steps were easy to follow.

Chapter 1 introduces the the properties of the probability, counting rues, independent/dependent/mutually exclusive events, Bayes etc.

Then the Chapter 2 introduces all the Discrete distributions and the properties of them like you mentioned - Bernoulli, Binomial, Hypergeometric, Negative Binomial, and Poison distributions.

Chapter 3 introduces all the continuous distributions like Beta, Gamma, Exponential, Chi-square, Normal and Uniform distributions.

Chapter 5 introduces transformations and functions of random variables, moment generating function, central limit theorem, sampling distribution, Normal distribution, Z-distribution, and derivation of student t-distribution chebyshe's inequality, and more.

Chapter 6 introduces descriptive statistics(sample mean, sample variance, relative frequency, empirical rules, etc), sample percentile, quartiles, five-number summaries, Min, Max, Range, IQR, Box plot, order statistics, Maximum likelihood and estimators, method of moments estimation, biased/unbiased estimators,assymptotic distributions of maximum likelihood estimators, sufficient statistics, simple regression, and Bayesian estimation and more.

Chapter 7 basically talks about confidence interval for various distributions that we talked about chapter 2&3, sample size, and resampling methods.

Chapter 8 introduces definition of hypothesis and different types of statistical tests - test of equality of two means, test of equal variance, test of equal proportions and many more. Also talks about distribution-free hypothesis tests(nonparametric methods) - Wilcoxon's signed rank test and Wilcoxon's rank sum test.It also talks about critical region, type 1 error, type 2 error, significance level, p-value, power of a statistical test. When and how to reject the null hypothesis.

I haven yet studied chapter 9, but I hope you may find this book covers some of keywords that you mentions above and more.

Thank.