Advanced Statistics (17 problems)

Consider a random sample of size n from a Bernoulli distribution, Xi ~ BIN (1,p).

(a) Find the CRLB (Cramero-Rao Lower Bound) for the variances of unbiased estimators of p.

(b) Find the CRLB for the variances of unbiased estimators of p(1-p).

(c) Find a UMVUE of p.

Consider a random sample of size n from a Poisson distribution, Xi ~ POI ().

(a) Find the CRLB for the variances of unbiased estimators of

(b) Find the CRLB for the variances of unbiased estimators of.

(c) Find a UMVUE of .

(d) Find the MLE of.

(e) Isan unbiased estimator of?

(f) Isasymptotically unbiased?

(g) Show thatis an unbiased estimator of theta.

(h) Find Var () and compare to the CRLB of (b).

A high-end market research firm has contacted your boss and is trying to sell some business to your organization. Upper management does not want to appear incompetent, so they have asked you to research and explain three major ways multivariate statistics are used in a business.

Research the Cybrary and provide at least 1 example of how a real company has used each of the following multivariate techniques: factor analysis, multi-dimensional scaling, and cluster analysis. Companies that provide statistics software websites and market research firm websites usually include case studies and customer testimonials.

TWO PAGES FOCUS HERE IS EXPLAINING AND GIVE EXAMPLES OF THE FOLLOWING FOUR MULTIVARIANT TECHNIQUES (A) FACTOR ANALYSIS, (B) MULTI-DIMENTIONAL SCALING(C) CLUSTER ANALYSIS IN A REAL COMPANY.

Let X ~ POI (mu), and let theta = P[X = 0] =.

(a) Is =an unbiased estimator of theta?

(b) Show that is an unbiased estimator of theta, where u(0) = 1 and u(x) = 0 if x = 1,2,…..

(c) Compare the MSEs of theta hat and theta tilde for estimating theta = e^-mu when mu = 1 and mu = 2.

Let X ~ BIN (n,p) and = X/n.

(a) Find a constant c so that E[c()(1-)] = p(1-p).

(b) Find an unbiased estimator of Var (X).

(c) Consider a random sample of size N from BIN (n,p). Find unbiased estimators of p and Var (X) based on the random sample.

Let X₁, X₂,…, Xn be a random sample of size n from a normal distribution, Xi ~ , and define and

(a) Find a statistic that is a function of U and W and unbiased for the parameter

(b) Find a statistic that is unbiased for

Let X₁,…..Sn be a random sample from EXP (theta) and define = and =.

(a) Find the variances of and.

(b) Find the MSEs of and.

(c) Compare the variances of and. for n = 2.

(d) Compare the MSEs of and.for n = 2.

Let be a random sample from a uniform distribution, Xi ~ UNIF ( – 1, + 1).

(a) Show that the sample mean, X bar, is an unbiased estimator of theta.

(b) Show that the “midrange,” , is an unbiased estimator of.

Let X be a random variable with CDF.

(a) Show that is minimized by the value c = E(X).

(b) Assuming that X is continuous, show that E[absolute value of (X-c)] is minimized if c is the median, that is, the value such that F(c) = ½.

Assume that X₁ and X₂ are independent normal random variables, Xi ~ , and let Y₁ = X₁ + X₂ and Y2 = X₁ – X₂. Show that Y₁ and Y₂ are independent and normally distributed.

Newfood Corporation

Test marketing, marketing mix

Product development at Newfood is considering adding a new “nutritional food” product to their snack/diet food product line. The product is tentatively labeled K-Pack and is packaged to look like a typical candy bar.

Newfood is interested in determining the demand and a good marketing mix (e.g. a price, advertising and promotion strategy) for K-Pack. Management believes the product will have very little direct competition in the diet-snack food category. Further, a pre-test market study was very encouraging. Management predicts sales of 2 million cases (24 packages in a case) in 1999 of K-Pack using a mix of 48 cents per package and a $3 million in advertising per year.

The projected P&L for the year (national sales) is:

Sales 2 million cases

Revenue $16.12 million (assume about 70% of the retail price is revenue to the manufacturer)

Manufacturing Costs $3 million ($1 million fixed manufacturing cost plus $1 per case variable)

Advertising $3 million

Net Margin $10.12 million.

Newfood plans to subcontract the production and distribution of K-Pack. The total annual cost of contracting (production and distribution) is given in the above projected Profit & Loss (P&L) as the fixed manufacturing cost (=$1 million in 1999). This cost is expected to change very little over the next 3 years.

Prior to making the “go-no-go” decision on K-Pack, Newfood plans to conduct a 6-month test market study in 2 different market regions (New York and Los Angeles). The objective of the tests are:

· Obtain better estimates of the first year

· Study the effects of marketing mix variables on sales

· Estimate the long-run potential of the product

Newfood believes the above objectives can be achieved using a “controlled” introduction of the K-pack into four cities in each marketing region. Marketing mix variables are to be varied across grocery stores in each city and in each region. Sales of K-Pack will be measured using store audits from a panel of stores in each city.

Design of the Test Market

Marketing mix variables are: price, advertising, and location of K-Pack within each store. Considered are: three prices (48 cents, 58 cents, 68 cents), two levels of advertising (a simulation of a $3 million introduction plan and a $6 million plan), 2 store locations (L!: K-Pack in the bakery section versus L2: K-Pack in the breakfast section). Prices and location are to be varied across stores within cities while advertising will be varied across cities. Advertising will be in the form of spot TV ads. Ad levels will be selected that simulate on a local basis the impact of national ads at the level of $3 million and $6 million. Due to differences in costs across markets, and costs between spot and network ads, Newfood plans to measure and equate the effect of advertising, normalized for market size, on consumer impression of K-Pack.

Controls

In the selection of cities (and stores within each city) within each region for the test, Newfood plans to “match” cities on the variables: store size, number of checkout counters, and characteristics f the market region. Newfood hopes to obtain useful information, about the “relationships between” the mix variables and sales that can be used in introducing and managing K-Pack nationally.

Test Design

The test design to be used in each region is given below. Each cell contains three stores and each store will be audited monthly.

P=price; L=location with 3 stores per cell; a, b, c, d are unique city labels determined by random selection without replacement from the list: 1, 2, 3 and 4.

The preceding test design will generate the following sample sizes for each region:

The advertising plan and actual GRPs achieved by the city within the region are given in the following tables.

Results from the Test Marketing

Table 1A: Los Angeles Region*

Table 1B: New York Region*

*GRP = reach x Frequency where Reach is a measure of the % of the target audience reached by the advertising and Frequency is the number of times those reached view the ad. E.g. a business with a 60% reach and a frequency of four produces an impact of 240 GRPS. This is a far better measure of advertising effectiveness than the dollar spent on advertising because GRPs better measure the level of impact delivered by target market Reach and Frequency.

Let X₁,…..Xn be a random sample from a normal distribution, N(0,).

(a) Is the MLE, theta hat, an unbiased estimator of theta?

(b) Is theta hat a UMVUE of theta?

Consider a random sample of size n from a Pareto distribution, Xi ~ PAR (theta, 2).

(a) Find the ML Equation d/d (ln L()) = 0

(b) From the data of Example 4.6.2, compute the ML estimate, theta hat, to three decimal places. Note: the ML equation cannot be solved explicitly for theta hat, but it can be solved numerically, by an iterative method, or by trial and error.

Find the asymptotic distribution of the MLE of p with Xi was ~ BIN (1, p).

Find method of moment estimators of based on a random samplefrom

The Pelnor Corporation is the nation’s largest manufacturer of industrial-size washing machines. A main ingredient in the production process is 8- by 10-foot sheets of stainless steel. The steel is used for both interior washer drums and outer casings.

Steel is purchased weekly on a contractual basis from the Smith-Layton Foundry, which, because of limited availability and lot sizing, can ship either 8,000 or 11,000 square feet of stainless steel each week. When Pelnor’s weekly order is placed, there is a 45% chance that 8,000 square fet will arrive and a 55% chance of received the larger size order.

Pelnor uses the stainless steel on a stochastic (nonconstant) basis. The probabilities of demand each week follow:

Pelnor has a capacity to store no more than 25,000 square feet of steel at any time. Because of the contract, orders must be placed each week regardless of the on-hand supply.

a) Simulate stainless steel order arrivals and use for 20 weeks. (Begin the first week with a starting inventory of 0 stainless steel.) If an end-of-week inventory is ever negative, assume that back order are permitted and fill the demand from the next arriving order.

b) Should Pelnor add more storage area? If so, how much? If not, comment on the system.

Suppose that the random variable

(a) Find the moment-generating function of X.

(b) Using this moment generating function verify that the mean of the random variable isand the variance is.

(c) Suppose that. Verify that Y has a distribution with 1 degree of freedom.