Let the random vector $X = (X_1 \cdots X_p)'$ have a multivariate normal distribution with unknown mean $\xi = (\xi_1 \cdots \xi_p)'$ and unknown nonsingular covariance matrix $\Sigma$. Write $\Sigma^{-1}\xi = \Gamma = (\Gamma_1 \cdots \Gamma_p)'$. The problem considered here is that of testing the hypothesis $H_0 : \Gamma_{q + 1} = \cdots = \Gamma_p = 0$ against the alternative $H_1 : \Gamma_{p' + 1} = \cdots = \Gamma_p = 0$ when $p \geqq p' > q$ and $\xi, \Sigma$ are both unknown. This problem arises in discriminating between two multivariate normal populations with the same unknown covariance matrix when one is interested to test whether the variables $X_{q + 1} \cdots X_{p'}$ contribute significantly to the discrimination. For a comprehensive treatment of this subject, the reader is referred to Rao (1952), Chapter 7. In this paper we will find the likelihood ratio test of $H_0$ against $H_1$ and show that this test is uniformly most powerful similar invariant. The problem of testing $H_0$ against $H_1$ remains invariant under the groups $G_1$ and $G_2$ where $G_1$ is the group of $p' \times p'$ non-singular matrices $g = \begin{pmatrix}g_{11} & 0\\g_{22} & g_{22}\end{pmatrix}$ (with $g_{11}$ a $q \times q$ matrix) which transform the coordinates $X_1 \cdots X_{p'}$ of $X$ and $G_2$ is the group of translations of the coordinates $X_{p' + 1} \cdots X_p$ of $X$. We may restrict our attention to the space of the sufficient statistic $(\bar X, S)$ of $(\xi, \Sigma)$. A maximal invariant under $G_1$ and $G_2$ in the space of $(\bar X, S)$ is $R = (R_1, R_2)'$, and a corresponding maximal invariant in the parametric space of $(\xi, \Sigma)$ is $\delta = (\delta_1, \delta_2)'$, where $R_i \geqq 0, \delta_i \geqq 0$ are defined in Section 2. In Section 1, we will find the likelihood ratio test of $H_0$ against $H_1$ in the usual way. The likelihood ratio test is invariant under all transformations which keep the problem invariant, and hence is a function only of $R$. In Section 2, we will find the joint density of $R_1$ and $R_2$ under the hypothesis and under the alternatives and then follow Neyman's approach of invariant similar regions to show that the likelihood ratio test in this case is uniformly most powerful similar invariant. In terms of maximal invariants, the above problem reduces to that of testing $H_0 : \delta_2 = 0, \delta_1 > 0$ against the alternative $H_1 : \delta_2 > 0, \delta_1 > 0$. According to a Fisherian philosophy of statistical inference applied to invariant procedures, it is reasonable to think of $R_1$ as giving information about the discriminating ability of the set of variables $(X_1 \cdots X_q)$, but no information about parameters governing additional discriminating ability from variables $X_{q + 1} \cdots X_{p'}$. Thus Fisher might call $R_1$ ancillary for the problem at hand and condition on it. We are not concerned here with the philosophical issues of statistical inference; instead, we will find (in Section 3) the distribution of the likelihood ratio conditional on $R_1$ which forms the basis of inference in a Fisherian approach. It will be shown that in this conditional situation, the likelihood ratio test is uniformly most powerful invariant. A more general statement of this same problem is to find the likelihood ratio test of the hypothesis $H'_0 : \Gamma \varepsilon \mathscr{Z}'$ that $\Gamma$ belongs to $\mathscr{Z}'$ against the alternative $H'_1 : \Gamma \varepsilon \mathscr{Y}'$ that $\Gamma$ belongs to $\mathscr{Y}'$, when $\xi, \Sigma$ are both unknown and $\mathscr{Z}' \subset \mathscr{Y}'$ are linear sub-spaces of the adjoint space $\mathscr{X}'$ of the space of $X$'s, and are of dimensions $q$ and $p'$ respectively. This problem can be easily reduced to that above by a proper choice of coordinate system, depending on the particular forms of $\mathscr{Z}'$ and $\mathscr{Y}'$. One could have worked with this general formulation instead of that above but the author did not find it convenient for computational purposes. As a corollary, if $q = 0$ then $H_0$ falls back to the usual null hypothesis of multivariate analysis of variance. It is easy to see that the likelihood ratio test for $q = 0$ reduces to the usual Hotelling's $T^2$ test which is uniformly most powerful invariant (Lehmann (1959)). Fisher (1938) has dealt with a particular case of the general formulation where $\mathscr{Z}'$ is the one-dimensional linear sub-space of $\mathscr{X}'$, and a test based on a discriminant function was suggested by him. The problem of testing $H_0$ against $H_1$ has been dealt with by Rao (1949) and a test depending on the ratio of Mahalanobis' $D^2$ statistics based on the first $q$ and $p'$ components of $X$ (which is related to Fisher's discriminant function in a simple manner) was suggested by him. It will be seen that both the tests are the likelihood ratio test.
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In statistical analysis, the law of large numbers is important because it gives validity to your sample size. When working with a small amount of data, the assumptions you make may not appropriately translate to the actual population. Therefore, it is important to make sure enough data points are being captured to adequately represent the entire data set.In business, the law of large numbers is important when setting targets or goals. A company may double its revenue in a single year. Should the company obtain only 50% growth in revenue the next year, it will have earned the same amount of money each of the last two years. Therefore, it is important to be mindful that percentages can be misleading as large dollar values escalate."}},{"@type": "Question","name": "How Can Companies Overcome the Challenge of the Law of Large Numbers?","acceptedAnswer": {"@type": "Answer","text": "Companies often strive to overcome the challenge of the law of large numbers by acquiring smaller growth companies that can infuse scalable growth. They also attempt to become more efficient and utilize their size for manufacturing, ordering, or distribution benefits. Last, companies can be more attentive to dollar goals as opposed to percent goals."}},{"@type": "Question","name": "What Is the Law of Small Numbers?","acceptedAnswer": {"@type": "Answer","text": "The law of small numbers is the theory that people underestimate the variability in small sample sizes. This means that when people study a sample size that is too small, they usually overestimate the population's value based on the incorrect sample size."}},{"@type": "Question","name": "What Is the Law of Large Numbers in Psychology?","acceptedAnswer": {"@type": "Answer","text": "Similar to other examples above, the law of large numbers in psychology translates to how a larger number of trials often leads to a more accurate expected value. As more trials are performed, the closer the projection is to being a correct medical assessment."}}]}]}] Investing Stocks Bonds ETFs Options and Derivatives Commodities Trading FinTech and Automated Investing Brokers Fundamental Analysis Technical Analysis Markets View All Simulator Login / Portfolio Trade Research My Games Leaderboard Banking Savings Accounts Certificates of Deposit (CDs) Money Market Accounts Checking Accounts View All Personal Finance Budgeting and Saving Personal Loans Insurance Mortgages Credit and Debt Student Loans Taxes Credit Cards Financial Literacy Retirement View All News Markets Companies Earnings CD Rates Mortgage Rates Economy Government Crypto ETFs Personal Finance View All Reviews Best Online Brokers Best Savings Rates Best CD Rates Best Life Insurance Best Personal Loans Best Mortgage Rates Best Money Market Accounts Best Auto Loan Rates Best Credit Repair Companies Best Credit Cards View All Academy Investing for Beginners Trading for Beginners Become a Day Trader Technical Analysis All Investing Courses All Trading Courses View All LiveSearchSearchPlease fill out this field.SearchSearchPlease fill out this field.InvestingInvesting Stocks Bonds ETFs Options and Derivatives Commodities Trading FinTech and Automated Investing Brokers Fundamental Analysis Technical Analysis Markets View All SimulatorSimulator Login / Portfolio Trade Research My Games Leaderboard BankingBanking Savings Accounts Certificates of Deposit (CDs) Money Market Accounts Checking Accounts View All Personal FinancePersonal Finance Budgeting and Saving Personal Loans Insurance Mortgages Credit and Debt Student Loans Taxes Credit Cards Financial Literacy Retirement View All NewsNews Markets Companies Earnings CD Rates Mortgage Rates Economy Government Crypto ETFs Personal Finance View All ReviewsReviews Best Online Brokers Best Savings Rates Best CD Rates Best Life Insurance Best Personal Loans Best Mortgage Rates Best Money Market Accounts Best Auto Loan Rates Best Credit Repair Companies Best Credit Cards View All AcademyAcademy Investing for Beginners Trading for Beginners Become a Day Trader Technical Analysis All Investing Courses All Trading Courses View All EconomyEconomy Government and Policy Monetary Policy Fiscal Policy Economics View All Financial Terms Newsletter About Us Follow Us Table of ContentsExpandTable of ContentsThe Law of Large NumbersPrinciplesStatistical AnalysisCentral Limit TheoremBusiness GrowthBusiness Example (Tesla)InsuranceLaw of Large Numbers FAQsThe Bottom LineCorporate FinanceFinancial AnalysisLaw of Large Numbers: What It Is, How It's Used, ExamplesBy be457b7860
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